UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS MATEMÁTICAS TESIS DOCTORAL Growth in groups of non-positive curvature Crecimiento en grupos de curvatura negativa o nula Croissance dans les groupes à courbure négative ou nulle MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Xabier Legaspi Juanatey DIRECTORES Yago Antolín Pichel Rémi Coulon © Xabier Legaspi Juanatey, 2023 Universidad Complutense de Madrid Facultad de Ciencias Matemáticas Programa de Doctorado en Investigación Matemática Instituto de Ciencias Matemáticas TESIS DOCTORAL Growth in Groups of Non-positive Curvature Crecimiento en grupos de curvatura negativa o nula Croissance dans les groupes à courbure négative ou nulle Memoria para optar al grado de doctor presentada por Xabier Legaspi Juanatey Directores : Prof. Yago Antolín Pichel y Prof. Rémi Coulon · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · ·· · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · ·· · · · · ·· · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 232 · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ······· · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · ·· · · · ·· · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · ·· · · · · · · ·· · · · · ·· · · · · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · · · ·· THÈSE DE DOCTORAT DE L’UNIVERSITÉ DE RENNES DÉLIVRÉE CONJOINTEMENT AVEC UNIVERSIDAD COMPLUTENSE DE MADRID ÉCOLE DOCTORALE NO 601 Mathématiques, Télécommunications, Informatique, Signal, Systèmes, Électronique Spécialité : Mathématiques et leurs interactions Par Xabier LEGASPI JUANATEY Growth in Groups of Non-positive Curvature Croissance dans les groupes à courbure négative ou nulle Crecimiento en grupos de curvatura negativa o nula Thèse présentée et soutenue à Madrid, le 12 juillet 2023 Unité de recherche : Institut de Recherche Mathématique de Rennes (IRMAR - UMR CNRS 6625) Facultad de Ciencias Matemáticas de la Universidad Complutense de Madrid Rapporteurs avant soutenance : François DAHMANI Professeur, Université de Grenoble-Alpes Tatiana SMIRNOVA-NAGNIBEDA Professeure, Université de Genève Composition du Jury : Président : José Javier ETAYO GORDEJUELA Catedrático de universidad, Universidad Complutense de Madrid Secrétaire : Alejandra GARRIDO ÁNGULO Profesor contratado doctor, Universidad Complutense de Madrid Examinateurs : Andrei JAIKIN-ZAPIRAIN Catedrático de universidad, Universidad Autónoma de Madrid Tatiana SMIRNOVA-NAGNIBEDA Professeure, Université de Genève Juan SOUTO Directeur de recherche CNRS, Université de Rennes 1 Dir. de thèse : Rémi COULON Chargé de recherche CNRS, Université de Rennes 1 Co-dir. de thèse : Yago ANTOLÍN PICHEL Profesor titular, Universidad Complutense de Madrid Á miña nai, quen me ensinou a resolver crebacabezas, e espertou á miña curiosidade polas palabras. E ao meu pai, quen me ensinou a desfrutar da vida, sen perder nunca de vista os soños. Friends, Romans, countrymen, lend me your ears; I come to bury Caesar, not to praise him. The evil that men do lives after them; The good is oft interred with their bones; So let it be with Caesar. from Julius Caesar of William Shakespeare, spoken by Marc Antony 7 ACKNOWLEDGEMENT To ... ... my PhD supervisors Yago Antolín and Rémi Coulon, for being, respectively, the place to stand, and the lever long enough that allowed me to move the world. Thanks for the many suggestions, directions, indirections, comments, disagreements, ideas, jokes, moments of hope and desperation, insistence in working less, etc. Thanks Rémi for drawing the non-positive curvature on the blackboard and printing it. Thanks Yago for showing me the ways of the Force. The best directors. ... François Dahmani and Tatiana Nagnibeda, for accepting to be the referees of the thesis, as well as to the members of the jury Javier Etayo, Alejandra Garrido, Andrei Jaikin, Juan Souto. I am specially grateful to François Dahmani for co-organising the thematic program “Groups Acting on Fractals, Hyperbolicity and Self-similarity” in Paris in Spring 2022; and to Juan Souto, for being honest with me a warm day of August 2018, throwing me the book “Sur les Groupes Hyperboliques d’après Mikhael Gromov” and then telling me that I was a bit lost. Now I understand why you threw me that book. Thanks to Alex Sisto, for an inspiring beer. I was shocked to meet you in person. I am grateful to Chris Cashen for all his support and encouragement during this year. Thanks to Daniel Groves for spending some time in front of my poster in YGGT XI nodding with the head and saying “wow”- that was really encouraging and fun at that point. Thanks to Fernando Alcalde for sending me to Rennes after my undergraduate. Thanks to Rob Kropholler for doing what he did. Special thanks to Mark Hagen for some moral support in dark times. ... Titouan Serandour, for proving your existence more than once with a swing, for your patience, for listening, for your time, for the carrot juice, for the views from the window of your office, etc ... Markus Steenbock, for being THE collaborator and my old sibling in math. You are a wonderful living being and a great friend. Thanks for introducing me to the secrets of small cancellation theory. At the beginning of my thesis, I didn’t know that part of my job would consist on 9 talking to many mathematicians from different parts of the world. During these years, I met some wonderful people that influenced my research and mathematical tastes in one way or another. Thanks to Aaron Messerla for his “it is OK” life philosophy lesson; Abdul Zalloum for his curiosity and for being as enthusiastic and joyful as hard-worker; Alex Edletzberger, for our ability to complain deeply and fix the world. Thank you for the support in desperate times. Thanks to Alice Kerr and Pénélope Azuelos for some great math discussions. I remember hiking with Alice during YGGT IX in Saint-Jacut-de-la-Mer and getting really wet. Thanks to Davide Spriano for writing papers, for inviting me to several meals and for some great moments singing with a guitar in Paris, with Thomas Ng and Shaked Bader in the chorus. Thanks to Eduardo Silva, Francisco Cardona, Jerónimo García, George Shaji, Ismael Morales, and Rodrigo de Pool, for getting me in the mood during conferences. Thanks to Harry Petyt and Marco Linton, for inviting me to speak in their seminars and for building the foundations for more serious math discussions. Special thanks to Jacob Russell for encouraging me to buy climbing shoes. That was a point of inflection. Thanks also for all the mathematical career advice you gave me, you are a very good person Jacob. Thanks to Lawk Mineh for being so peaceful in the conferences and to Suraj Krishna for being so peaceful in the mountains. Thanks to the laugh of Monika Kudlinska, the best sound and visual effects. Thanks Monika for coming to Rennes with your superpower. Thanks to Sam Hughes for being the cause and consequence of some great evenings and activities. Thanks to Shaked Bader for her ability for organisation, listening, sense of responsibility and initiative. I was very lucky to meet you. Thanks to Paige Helms for the random walks and bouldering in Paris; e a Pilar Páez por aparecer en Viena de xeito inesperado. It is not easy to move to another city every single year and start over again and again – especially if you are a foreigner. I would like to express my gratitude to all the fantastic people I met in Rennes during the last years. You made all my stays fun and plenty of random situations product of impromptus decisions. I wish I got to know some of you better than I did. Thanks to Hadrián and Toky for making French easier to learn when I was producing noise instead of pronouncing vowels. To Andrea for being in the O’Connells and detecting my Galician accent at the right time, at the right spot. To Zenab for looking after me when I was sick. To my first neighbours Guillem, Riccardo, Rosalía, Olga, Xiaoling, Zhenyi, Guo Heyu for rescuing me from isolation with all your soirées and traditional cooking. In Rennes I had the opportunity to meet María Cumplido for the first time, my favourite Master teacher and mathematical hero. You gave me the hope, 10 strength and motivation to write a thesis in cotutelle. Thanks María. I would like to thank my second neighbours Roxane, Eline, Lucy, Alanah, Irene, Fabian, Leo, Héctor, Kevin, Sofía, Stefano, Dominik and Ronja for the BBQs v.6 and v.7. Many thanks to my third neighbours Phillipe, Rubén, Jeannie, Claire, Maryam and Maimi for rebooting my life after the lockdown with the Penny Lane and the trips to Saint Malo and Le Boël. I remember having some fun trying to learn sign language with Claire. Thanks to Casa Antonio for the pizzas and to the Bar’hic for the punk. Thanks to my “fourth neighbours” Félix, Valentin, Raquel and Gonzalo. Félix, you are so generous and noble – you can’t imagine. I am especially grateful to Marta Herschel. I met Marta in a bus stop years ago, while I was eavesdropping her conversation with another person. She turned out to be my friend and neighbour, the kitchen rockstar. Now she is famous and plays with a drummer called Camille, who I would also like to acknowledge here for her good vibes. Thanks Marta, you made some of my years in Rennes amazing. Thank you very much to all of the people who made the IRMAR a wonderful place during these years. Thanks to all the PhD students that made it an amazing experience: Axel Péneau, Marie Trin, Victor Lagrandmaison, Lucien Grillet, Emeline Luirard, Max- ence Phalempin, Fabien Narbonne, Marc Abboud, Alice Bouillet, Adrien Abgrall, Jérôme, Sergio Herrero, Matilde Maccan, Mattia, Yago, etc. Thanks for the coffee, the cakes, the discussions, the Dehn’s Army, the movies, the salsa, the swing, the soirées, the bouldering, the evenings at Gayeulles, the concerts, the Fest-Noz, etc. Axel, thanks for listening, for camping in the beach and for being a good friend during the last year. Marie, I wish you good luck, I know that you will be the one pulling the strings in the next year. Thanks to Françoise Dal’Bo, Serge Cantat, Vincent Guirardel, Ludovic Marquis, and the rest of the members of the équipe de théorie ergodique. We had great séminaires and groupes de travail, but of course the best happened during our stays at Saint-Jacut-de-la-Mer. Thanks to Léo Bigorgne and Miguel Rodrigues for kicking the ball smoothly during the matches of the IRMAR F.C. Also thanks to Marie-Aude and all the members of the secretary, the true bosses of the IRMAR. Ahora le toca a los de Madrid. Gracias a todos los miembros del equipo de teoría de grupos del ICMAT: Andrei Jaikin, Leo Margolis, Alejandra Garrido, Javier Aramayona, Catarina Campagnolo, Dominik Francoeur, Rodrigo de Pool, Sergio García, Henrique 11 Mendes, Jan Boscheigden, Iván Chércoles y Pablo Sánchez. Gracias por todos los grupos de lectura, las preguntas y en general las ganas de hacer cosas. A pesar de la pandemia y el estrés de la tesis, me lo he pasado pipa. Gracias a los que se sentaban a la mesa en el IMDEA durante mi primer año de tesis: Marcin, Cristina, Manuel Lainz, Javi, Hang Lu y Manuel de León. Estoy especialmente agradecido a Manuel Lainz. Manuel fue el primer doctorando en aparecer tras llegar a Madrid e hizo que el ICMAT fuese “menos hosco” al principio. También me ayudó a ver que Madrid no es tan mal sitio como creía y que está muy lejos de ser Mordor. Gracias Manuel. También tengo que dar las gracias a mis primeros compañeros de piso: Dani, Ismael, Silva, Auri, Jorge, Kush, a mi segunda compañera de piso Gloria, y a mis terceros compañeros de piso Darcin y Hugo. Ogallá quedase a vivir con Darcin e Hugo, iso era outro nivel. Estoy profundamente agradecido a Florentino Moreno y Borja Manero por los cursos de retórica y oratoria de la escuela de doctorado; y sobretodo a Raúl y Socorro por acogerme en Réplika Teatro. Sin duda, uno de los mayores descubrimientos que hice durante la tesis fue saber que hacer teatro y disfrutarlo es tan posible como maravilloso. Grazas aos meus mellores amigos: Manoel, Nuria, Darcin, Sanchiño, Panchi, Adrián, Nacho, Chouza. A algúns coñézovos dende os 5 anos. Sempre que volvo por Noia todavía parece que temos 5 anos. Por favor, non cambiedes. Grazas a Manoel por sacarme da casa con insistencia, nervio e método sempre que estou atormentado. O resto xa vén despois. Finalmente, grazas a toda á miña familia. Algúns merecen mención especial. Aos meus pais por todas as súas preocupacións e chamadas telefónicas. Á miña nai, pola súa capacidade de aturar a miña neura e xestionar a súa ao mesmo tempo. Grazas por animarme a volver Madrid no medio da pandemia. Ao meu pai, por levarme de excursión a sitios descoñecidos e non insistirme tanto en que hai que ir a Fazouro. Ao meu tío Moncho e ao meu primo Fernando, polos seus diversos debates e digresións á hora da cea. Á miña tía Lola, por ofrecerme sempre aloxamento en Madrid, por alimentarme estes últimos meses e ter unha gran variedade de novelas negras no salón. E á miña prima Carmen, por visitarme, levarme á praia, e tamén de festa. 12 TABLE OF CONTENTS Resumen en español 15 Résumé en français 17 0.1 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0.2 Croissance des sous-groupes quasi-convexes . . . . . . . . . . . . . . . . . . 18 0.3 Croissance exponentielle uniforme uniforme . . . . . . . . . . . . . . . . . . 20 Introduction 25 0.4 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 0.5 Growth of quasi-convex subgroups . . . . . . . . . . . . . . . . . . . . . . . 26 0.6 Uniform uniform exponential growth . . . . . . . . . . . . . . . . . . . . . 36 Notation 43 1 Growth of quasi-convex subgroups in groups with a constricting element 45 1.1 Path system geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.2 Growth estimation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.3 Buffering sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.4 Quasi-convexity in the Intersection–Image property . . . . . . . . . . . . . 55 1.5 Finding a quasi-convex element . . . . . . . . . . . . . . . . . . . . . . . . 56 1.6 Constricting elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.6.1 A G-invariant family . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.6.2 Finding a constricting element . . . . . . . . . . . . . . . . . . . . . 64 1.6.3 Elementary closures . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.6.4 Forcing a geometric separation . . . . . . . . . . . . . . . . . . . . . 67 1.7 Growth of quasi-convex subgroups . . . . . . . . . . . . . . . . . . . . . . . 70 2 Uniform uniform exponential growth in small cancellation groups 77 2.1 Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.1.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 13 TABLE OF CONTENTS 14 2.1.2 Quasi-convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.3 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.1.4 Group action on a δ-hyperbolic space . . . . . . . . . . . . . . . . . 81 2.1.5 Small cancellation theory . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2 Reduced subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.1 Broken geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.2 Quasi-isometric embedding of a free group . . . . . . . . . . . . . . 91 2.2.3 Geodesic extension property . . . . . . . . . . . . . . . . . . . . . . 92 2.3 Growth in groups acting on a δ-hyperbolic space . . . . . . . . . . . . . . . 95 2.3.1 Growth of maximal loxodromic subgroups. . . . . . . . . . . . . . . 95 2.3.2 Producing reduced subsets . . . . . . . . . . . . . . . . . . . . . . . 99 2.3.3 Growth trichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.4 Shortening and shortening-free words . . . . . . . . . . . . . . . . . . . . . 105 2.4.1 Shortening words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.4.2 The growth of shortening-free words . . . . . . . . . . . . . . . . . . 112 2.4.3 The injection of shortening-free words . . . . . . . . . . . . . . . . . 117 2.5 Growth in small cancellation groups . . . . . . . . . . . . . . . . . . . . . . 120 A Properties of constricting subsets 128 A.1 Standard properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.2 Behrstock inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.3 Morseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Bibliography 139 RESUMEN EN ESPAÑOL A una isla del caribe He tenido que emigrar Y trabajar de camarero Lejos lejos de mi hogar De mi hogar. Miña Terra Galega, de Siniestro Total Esta tesis se centra en preguntas que comparan números fáciles de definir pero no fáciles de calcular. La acción de un grupo G sobre un espacio métrico X se dice propia si para cada r > 0, y para cada x ∈ X, el número de elementos u ∈ G que mueven x a distancia a lo sumo r es finito. Sea G un grupo actuando mediante isometrías y propiamente sobre un espacio métrico X. La tasa de crecimiento exponencial relativa de la acción de un subconjunto U ⊂ G sobre X es el número ω(U,X) = lim sup r→∞ 1 r log |{u ∈ U : |ux− x| ⩽ r }|, cuyo valor es independiente del punto x ∈ X. Si G es el grupo fundamental de una variedad hiperbólica cerrada M que actúa sobre el espacio recubridor universal X, entonces ω(G,X) tiene numerosas interpretaciones. Coincide con la entropía de volumen de la variedad M , [71, 62]; el exponente crítico de la serie de Poincaré de G, [67, 75]; la entropía topológica del flujo geodésico en el fibrado tangente unitario de M , [60]; la dimensión de Hausdorff del conjunto límite radial de G, [28], etc. En este contexto, el número ω(G,X) es la piedra angular que une grupos, geometría y dinámica. La discreción de la órbita de G y la curvatura negativa de M juegan un papel determinante en este fenómeno. El objetivo de esta tesis es cuantificar el crecimiento en grupos a partir de sus acciones mediante isometrías sobre espacios métricos. El enfoque consiste en observar desde un punto de vista muy lejano. La hazaña está en la finitud y las condiciones de curvatura negativa o nula de acciones y espacios. Sean δ, κ, N > 0. La acción de un grupo G sobre un espacio δ-hiperbólico X se dice (κ,N)-acilíndrica, [72, 18, 66, 38], si para cada par de puntos x, y ∈ X que distan al menos κ, el número de elementos u ∈ G que mueve 15 TABLE OF CONTENTS cada uno de los puntos x, y a una distancia de a lo sumo 100δ está acotado superiormente por N . La última década ha estado enfocada en el estudio de grupos que admiten una acción acilíndrica sobre un espacio hiperbólico en el sentido de Gromov, [66]. Esta es una familia muy amplia de grupos que incluye grupos relativamente hiperbólicos, grupos de cancelación pequeña clásica infinitamente presentados, grupos modulares de superficies, grupos de automorfismos exteriores de grupos libres, grupos de Artin de ángulo recto, etc. Resulta que la mayoría de las veces los grupos que actúan acilíndricamente sobre un espacio hiperbólico también admiten acciones propias sobre otros espacios que no son necesariamente hiperbólicos, pero que contienen isometrías que se comportan como las isometrías loxodrómicas de un espacio hiperbólico: elementos constrictor, [74], bajo la terminología de [8]. De hecho, el recíproco siempre es cierto. La moraleja de la tesis recoge la siguiente idea de M. Gromov: bajo un punto de vista global curvado de forma negativa o nula, todavía es posible producir resultados sólidos para un grupo típico, lo que a veces puede aproximar nuestra comprensión de los grupos monstruo. Estudiaremos dos problemas diferentes usando argumentos de baja tecnología que involucran la desigualdad triangular. El primero versará sobre el crecimiento de subgrupos cuasi-convexos en grupos actuando propiamente con un elemento constrictor. A mayores, hemos añadido un apéndice en dónde se describen algunas características elementales de la propiedad de constricción. El segundo versará sobre el crecimiento uniforme en cocientes de cancelación pequeña sobre grupos que actúan acilíndricamente sobre un espacio hiperbólico. Los Capítulos 1 y 2 corresponden respectivamente a los siguientes artículos: ▶ X. Legaspi. Constricting elements and the growth of quasi-convex subgroups, 2022. URL: https://orcid.org/0000-0002-1497-6448. ▶ X. Legaspi and M. Steenbock. Uniform uniform exponential growth in small cancel- lation groups, 2023. URL: https://orcid.org/0000-0002-1497-6448. 16 https://orcid.org/0000-0002-1497-6448 https://orcid.org/0000-0002-1497-6448 INTRODUCTION EN FRANÇAIS L’absurdité est surtout le divorce de l’homme et du monde. L’étranger, d’Albert Camus 0.1 Résumé Cette thèse est centré au tour des questions qui comparent des nombres faciles à définir mais pas faciles à calculer. L’action d’un groupe G sur un espace métrique X est propre si pour tout r > 0, et pour tout x ∈ X, le nombre d’éléments u ∈ G qui déplacent x à distance au plus r est fini. Soit G un groupe agissant par isométries et proprement sur un espace métrique X. Le taux de croissance exponentiel relatif de l’action d’un sous-ensemble U ⊂ G sur X est le nombre ω(U,X) = lim sup r→∞ 1 r log |{u ∈ U : |ux− x| ⩽ r }|, dont la valeur est indépendante du point x ∈ X. Si G est le groupe fondamental d’une variété hyperbolique fermée M agissant sur le revêtement universel X, alors ω(G,X) a de nombreuses interprétations. Elle correspond à l’entropie de volume de la variété M , [71, 62] ; l’exposant critique de la série de Poincaré de G, [67, 75] ; l’entropie topologique du flot géodésique dans le fibré unitaire tangent de M , [60] ; la dimension Hausdorff de l’ensemble radial limite de G, [28], etc. Dans ce contexte, le nombre ω(G,X) est la pierre angulaire qui unit les groupes, la géométrie et la dynamique. L’orbite discrète de G et la courbure négative de M jouent un rôle déterminant dans ce phénomène. L’objectif de cette thèse est de quantifier la croissance des groupes à partir de leurs actions par isométries sur des espaces métriques. L’approche consiste à observer d’un point de vue très éloigné. L’exploit est dans la finitude et les conditions de courbure négative ou nulle des actions et des espaces. Soient δ, κ, N > 0. L’action d’un groupe G sur un espace δ-hyperbolique X est (κ,N)-acylindrique, [72, 18, 66, 38], si pour chaque paire de points x, y ∈ X distants d’au moins κ, le nombre d’éléments u ∈ G qui déplacent chacun des points x, y d’une distance d’au plus 100δ est borné au-dessus par N . La dernière décennie 17 TABLE OF CONTENTS a été consacrée à l’étude des groupes qui admettent une action acylindrique sur un espace Gromov hyperbolique, [66]. Il s’agit d’une très large famille de groupes qui comprend des groupes relativement hyperboliques, des groupes de petite simplification à présentation infinie, des groupes modulaires de surfaces, des groupes d’automorphismes extérieurs de groupes libres, des groupes d’Artin à angle droit, etc. Il s’avère que la plupart du temps les groupes qui agissent de manière acylindrique sur un espace hyperbolique admettent aussi des actions propres sur d’autres espaces qui ne sont pas forcément hyperboliques, mais qui contiennent des isométries qui se comportent comme les isométries loxodromiques d’un espace hyperbolique : éléments constricteurs , [74], sous la terminologie de [8]. En fait, la reciproque est toujours vrai. La morale de la thèse reprend l’idée suivante de M. Gromov : sous un point de vue global courbé de façon négative ou nulle, il est encore possible de produire des résultats robustes pour un groupe typique, ce qui peut parfois rapprocher notre compréhension à des groupes monstre. Nous étudierons deux problèmes différents en utilisant des arguments de basse technologie impliquant l’inégalité triangulaire. Le premier traitera de la croissance de sous-groupes quasi-convexes dans les groupes agissant proprement avec un élément constricteur. De plus, nous avons ajouté une annexe décrivant quelques conséquences élémentaires de la propriété de constriction. Le second traitera de la croissance uniforme dans les quotients à petit simplification sur des groupes agissant de manière acylindrique sur un espace hyperbolique. Les Chapitres 1 et 2 correspondent respectivement aux articles suivants : ▶ X. Legaspi. Constricting elements and the growth of quasi-convex subgroups, 2022. URL: https://orcid.org/0000-0002-1497-6448. ▶ X. Legaspi and M. Steenbock. Uniform uniform exponential growth in small cancel- lation groups, 2023. URL: https://orcid.org/0000-0002-1497-6448. 0.2 Croissance des sous-groupes quasi-convexes Il existe une quantité importante d’informations codées dans la géométrie des sous- groupes quasi-convexes d’un groupe. Par exemple, certains groupes hyperboliques bénéfi- cient de la propriété qu’un sous-groupe est quasi-convexe si et seulement s’il est de type fini. Cependant, dans d’autres contextes, c’est loin d’être vrai. Dans le Chapitre 2 nous explorons la croissance de sous-groupes quasi-convexes au-delà du cas hyperbolique. 18 https://orcid.org/0000-0002-1497-6448 https://orcid.org/0000-0002-1497-6448 TABLE OF CONTENTS Nous donnons quelques définitions. Soit G un groupe agissant par isométries sur un espace métrique. Afin de définir des notions très générales de courbure négative ou nulle et de cocompacité convexe, nous utilisons des systèmes de chemins, introduits par A. Sisto dans [74]. En gros, un système de chemins P de X est une collection appropriée de quasi-géodésiques uniformes joignant chaque paire de points de X. Par exemple, les groupes modulaires des surfaces sont accompagnés de chemins hiérarchiques, une famille de quasi-géodésiques spéciales codant des informations substantielles sur la géométrie de l’espace et plus faciles à utiliser que l’ensemble de toutes les (quasi-)géodésiques. Soit P un système de chemins de X. Soit δ ⩾ 0. On dit qu’un sous-ensemble Y de X est δ-constricteur s’il existe une projection à large échelle au point le plus proche de X sur A avec la propriété que tout γ ∈ P joignant n’importe quelle paire de points x, y ∈ X dont les projections p et q sont δ-loin passe par les δ-voisinages de p et q. Un élément g de G est δ-constricteur s’il est d’ordre infini et s’il existe une orbite δ-constricteur du sous-groupe cyclique engendré par g. Soit η ⩾ 0. Un sous-ensemble Y de X est η-quasi-convexe si tout γ ∈ P avec des extrémités dans Y est contenu dans la η-voisinage de Y . Un sous-groupe H de G est η-quasi-convexe s’il existe une orbite η-quasi-convexe de H. Example 0.2.1. — Un espace métrique X est δ-hyperbolique s’il est géodésique et si tout segment géodésique de X est δ-constricteur par rapport au système de chemins constitué de tous les segments géodésiques de X, [61]. En particulier, l’axe des isométries loxodromiques des espaces δ-hyperboliques est constricteur : cette propriété est en fait équivalente à la quasi-convexité dans les espaces δ-hyperboliques, [26], mais pas en général. Par exemple, une géodésique dans le plan euclidien. Example 0.2.2. — Voici des exemples de groupes agissant avec un élément constricteur sur chaqu’un de leurs graphes de Cayley localement finis, voir par exemple [8, 57] et les références qui s’y trouvent. (i) Groupes relativement hyperboliques. (ii) Groupes modulaires des surfaces. (iii) Groupes CAT(0) avec éléments Morse. (iv) Grroupes à petit simplification graphique Gr′(1/6). Nous mentionnerons maintenant deux résultats. Le premier est une généralisation de [4] (voir aussi [47]) et étudie les taux de croissance exponentiels relatifs associés aux graphes de Schreier. 19 TABLE OF CONTENTS Theorem 0.2.3. — Soit G un groupe agissant proprement sur un espace métrique X. Soit P un système de chemins de X. Supposons que G contienne un élément constricteur par rapport à P. Soit H un sous-groupe quasi-convexe d’indice infini de G par rapport à P. Il existe x0 ∈ X avec la propriété suivante. Soit GH un ensemble de représentants de G/H tel que |gx0 − x0| = infh∈H |ghx0 − x0|, pour tout g ∈ GH . Alors ω(GH , X) = ω(G,X). Le deuxième résultat est une généralisation de [37] (voir aussi [27]) et étudie les taux de croissance exponentiels relatifs associés aux sous-groupes. On dit que l’action propre des isométries d’un sous-ensemble Λ sur un espace métrique X est divergente lorsque la série de Poincaré P(s) = ∑ λ∈Λ e −s|λx0−x0| diverge à son exposant critique ω(Λ, X). Ce comportement est indépendant de x0 ∈ X. Theorem 0.2.4. — Soit G un groupe agissant proprement sur un espace métrique X. Soit P un système de chemins de X. Supposons que G contienne un élément constricteur par rapport à P. Soit H un sous-groupe quasi-convexe d’indice infini par rapport à P. Si G n’est pas virtuellement cyclique et l’action de H sur X est divergente, alors ω(H,X) < ω(G,X). Dans le théorème précédent, il existe de nombreuses situations dans lesquelles l’action de H sur X est divergente. Par exemple, si P est le système de chemins de X composé de tous les segments géodésiques, alors H est quasi-convexe au sens classique. Dans cette situation, la fonction de croissance relative de H est sous-multiplicative, et par conséquent l’action de H sur X est divergente [39] (à condition que H soit infini). Une autre situation dans laquelle la fonction de croissance relative est sous-multiplicative est lorsque H a la propriété Morse ou lorsque la fonction de croissance relative est purement une fonction exponentielle, sans autre hypothèse sur P . Cela permet d’appliquer le résultat aux groupes modulaires des surfaces de type fini et leurs sous-groupes convexes cocompacts ou à certains stabilisateurs de multicourbes. Ici, le rôle du système de chemins est joué par les chemins hiérarchiques. 0.3 Croissance exponentielle uniforme uniforme Une question ouverte demande si chaque groupe agissant de manière acylindrique sur un espace hyperbolique a une croissance exponentielle uniforme. Dans le Chapitre 2, on montre que la classe des groupes de croissance exponentielle uniforme uniforme agissant de manière acylindrique sur un espace hyperbolique est fermée en prenant quotients à 20 TABLE OF CONTENTS petit simplification géométrique C ′′(λ, ε) dans le sens de [38, Définition 6.22]. Encore une fois, nous commençons par quelques définitions. Soit G un groupe. Soit U un sous-ensemble symétrique fini de G, notons H le sous- groupe engendré par U , et soit XU le graphe de Cayley correspondant. Le taux de croissance exponentiel de U est le nombre ω(H,U) := ω(H,XU). Soit ξ > 0. On dit que G a croissance exponentielle ξ-uniforme s’il est de type fini et pour tout ensemble générateur symétrique fini U de G, on a ω(G,U) > ξ. On dit que G a croissance exponentielle ξ-uniforme uniforme si chaque sous-groupe de type fini est soit virtuellement nilpotent, soit a croissance exponentielle ξ-uniforme. Example 0.3.1. — Voici des familles de groupes à croissance exponentielle uniforme uni- forme agissant de manière acylindrique sur des espaces hyperboliques : (i) Groupes hyperboliques. (ii) Produits libres de familles dénombrables de groupes à croissance exponentielle ξ-uniforme uniforme. (iii) Quelques groupes cubiques CAT(0). (iv) Groupes modulaires des surfaces. En général, le paramètre de croissance uniforme dépend du groupe. Vers une théorie géométrique des petite simplification. Soit G un groupe agissant par isométries sur un espace δ-hyperbolique X. Une famille de movement – ou ensemble de relations – est un ensemble de la forme Q = { ( ⟨grg−1⟩, gYr ) ∣∣∣ r ∈ R, g ∈ G } , où R ⊂ G est un ensemble d’isométries loxodromiques r – les relateurs – stabilisant leur axe quasi-convexe Yr ⊂ X. Une pièce est une intersection de n’importe quelle paire d’un tel axe. Le rôle des paramètres λ ∈ (0, 1) et ε > 0 dans la condition de petite simplification géométrique C ′′(λ, ε) sur Q est le suivant: ▶ La fraction de la longueur de la pièce la plus long avec la longueur de translation la plus courte des relations r ∈ R est au plus λ. 21 TABLE OF CONTENTS ▶ La longueur de translation la plus courte des relations r ∈ R est au moins εδ. Soit K la closure normale dans G des sous-groupes relation H dans Q. La condition de petite simplification géométrique C ′′(λ, ε) permet d’obtenir des informations substantielles sur le quotient à C ′′(λ, ε)-petite simplification géométrique Ḡ = G/K : par exemple K est un produit libre de sous-groupes relation, Ḡ ressemble localement à G et toute action acylindrique de G sur X induit une autre action acylindrique de Ḡ sur un quotient δ0-espace hyperbolique X̄ dont la constante d’hyperbolicité δ0 est universelle. Le résultat principal du Chapitre 2 est le suivant: Theorem 0.3.2 (Theorem 2.5.5 & Theorem 2.5.6). — Il existe λ ∈ (0, 1) tel que pour chaque N > 0 et ε > 1010N , ce qui suit est vrai. Soient δ > 0, κ ⩾ δ et soit G un groupe agissant (κ,N)-acylindriquement sur un espace δ-hyperbolique X. (i) Si G est à croissance exponentielle ξ-uniforme uniforme, alors chaque quotient à C ′′(λ, ε)-petite simplification géométrique de G est à croissance exponentielle ξ′- uniforme uniforme croissance. La constante ξ′ ne dépend que de ξ et N . (ii) S’il existe un quotient à C ′′(λ, ε)-petite simplification géométrique de G qui est à croissance exponentielle ξ-uniforme uniforme, alors G est à croissance exponentielle ξ′-uniforme uniforme. La constante ξ′ ne dépend que de ξ. Au-delà de la propriété du élément loxodromique court. La stratégie standard pour étudier la croissance exponentielle uniforme dans les groupes hyperboliques exploite le fait que leurs sous-ensembles générateurs ont la propriété du élément loxodromique court : chaque n-ième puissance Un d’un sous-ensemble générateur fini contient une isométrie loxodromique, pour un certain nombre n qui ne dépend pas de l’ensemble U . En général, on ne sait pas si chaque groupe de type fini agissant de manière acylindrique sur un espace hyperbolique a une croissance exponentielle uniforme. L’action acylindrique sur un espace hyperbolique donne une croissance exponentielle uniforme pour des sous- ensembles générateurs finis avec une longue isométrie loxodromique. La propriété du élément loxodromique court permet de prendre des grandes puissances uniformes pour pouvoir exploiter cette autre situation. Cependant, il y a un quotient à petite simplification combinatoire/gradué avec une action acylindrique sur un espace hyperbolique mais sans la propriété du élément loxodromique courte, [63]. Notre résultat principal ne fait pas usage de la propriété du élément loxodromique court. La morale de notre travail est que nous pouvons traiter ce genre de monstre tant que ce soit des quotients à petite simplification de 22 TABLE OF CONTENTS groupes de croissance exponentielle uniforme uniforme agissant de manière acylindrique sur un espace hyperbolique. Cependant, le monstre mentionné est un quotient du produit libre de tous les groupes hyperboliques. On ne sait pas s’il existe une borne inférieure universelle pour le taux de croissance uniforme dans la classe de tous les groupes hyperboliques, indépendante de la constante d’hyperbolicité, c’est une autre question ouverte. Ça équivaut à la croissance exponentielle uniforme uniforme du produit libre de toutes les groupes hyperboliques. 23 INTRODUCTION Kill the boy, Jon Snow. Winter is almost upon us. Kill the boy and let the man be born. from A Dance with Dragons of George R. R. Martin, spoken by Maester Aemon Contents 0.4 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 0.5 Growth of quasi-convex subgroups . . . . . . . . . . . . . . . . 26 0.6 Uniform uniform exponential growth . . . . . . . . . . . . . . 36 0.4 Abstract The focus of this thesis is on questions that compare numbers easy to define but not easy to compute. The action of a group G on a metric space X is called proper if for every r > 0, and for every x ∈ X, the number of elements u ∈ G moving x at distance at most r is finite. Let G be a group acting properly by isometries on a metric space X. The relative exponential growth rate of the action of a subset U ⊂ G on X is the number ω(U,X) = lim sup r→∞ 1 r log |{u ∈ U : |ux− x| ⩽ r }|, whose value is independent of the point x ∈ X. If G is the fundamental group of a closed hyperbolic manifold M acting on the universal cover X, then ω(G,X) has numerous interpretations. It coincides with the volume entropy of the manifold M , [71, 62]; the critical exponent of the Poincaré series of G, [67, 75]; the topological entropy of the geodesic flow on the unit tangent bundle of M , [60]; the Hausdorff dimension of the radial limit set of G, [28], etc. In this context, the number ω(G,X) is the cornerstone bringing together groups, geometry and dynamics. The discreteness of the orbit of G and the negative curvature of M play a major role in this phenomenon. 25 TABLE OF CONTENTS The aim of this thesis is to quantify growth in groups from their actions by isometries on metric spaces. The approach is to observe from far away. The exploit is on the finiteness and non-positive curvature conditions of actions and spaces. Let δ, κ, N > 0. The action of a group G on a δ-hyperbolic space X is called (κ,N)-acylindrical, [72, 18, 66, 38], if for every pair of points x, y ∈ X at distance at least κ, the number of elements u ∈ G moving each of the points x, y at distance at most 100δ is bounded above by N . In the last decade, the focus has been put on groups that admit an acylindrical action on a Gromov hyperbolic space, [66]. This is a vast family of groups that includes relatively hyperbolic groups, infinitely presented classical small cancellation groups, mapping class groups of surfaces, outer automorphism groups of free groups, right angled Artin groups, etc. It turns out that most of the time groups acting acylindrically on a hyperbolic space admit remarkable proper actions on other spaces that are not necessarily hyperbolic, but contain isometries that behave as the loxodromic isometries of a hyperbolic space: constricting elements, [74], under the terminology of [8]. In fact, the converse is always true. The moral of the thesis draws the following idea of M. Gromov: under a non-positively curved global viewpoint, it is still possible to produce strong results for a typical group, which can sometimes approximate our understanding to monster groups. We are going to study two different problems using low tech arguments involving the triangle inequality. The first one will be about the growth of quasi-convex subgroups in groups acting properly with a constricting element. In addition, we have added an appendix describing some elementary consequences of the constriction property. The second will be about the uniform growth in small cancellation quotients over groups acting acylindrically on a hyperbolic space. Chapters 1 and 2 correspond respectively to the following articles: ▶ X. Legaspi. Constricting elements and the growth of quasi-convex subgroups, 2022. URL: https://orcid.org/0000-0002-1497-6448. ▶ X. Legaspi and M. Steenbock. Uniform uniform exponential growth in small cancel- lation groups, 2023. URL: https://orcid.org/0000-0002-1497-6448. 0.5 Growth of quasi-convex subgroups Let G be a group acting by isometries on a metric space X. Let x ∈ X. Let H be a subgroup of G. Let HL and HR be respectively left and right transversals of H such that 26 https://orcid.org/0000-0002-1497-6448 https://orcid.org/0000-0002-1497-6448 TABLE OF CONTENTS for every u ∈ HL and v ∈ HR, |ux− x| = inf h∈H |uhx− x|, and |vx− x| = inf h∈H |hvx− x|. In Chapter 1 we study the numbers ω(H) := ω(H,X), ω(G/H) := ω(HL, X), and ω(H\G) := ω(HR, X). Consider the following general problem. When do G and H determine a solution to to the system of equations below?  ω(H) < ω(G), ω(G/H) = ω(G), ω(H\G) = ω(G). We see from the definitions that ω(H/G) = ω(H\G), and 0 ⩽ max {ω(H), ω(G/H)} ⩽ ω(G). In the extreme case in which H has finite index in G, one can easily prove that ω(H) = ω(G), ω(G/H) = 0. In general, it is a hard problem to obtain precise estimations of relative exponential growth rates of infinite index subgroups. However, it is known, [37, 4, 47], that if G is a non-virtually cyclic group acting geometrically on a hyperbolic space X and H is an infinite index quasi-convex subgroup of G, then ω(H) < ω(G), ω(G/H) = ω(G). The arguments of [37, 4] are based on automatic structures and regular languages, with influence of the works of J. Cannon [22, 23]. This fact also influenced other authors that partially extended the hyperbolic case result, [27]. In Chapter 1 we go beyond the hyperbolic case and we obtain two main results (Theorem 0.5.8 and Theorem 0.5.13) with elementary 27 TABLE OF CONTENTS proofs that do not require the theory of regular languages and automata. We will be interested in groups acting properly on metric spaces conditioned by a very general notion of “non-positive curvature” introduced by A. Sisto in [74] — containing a constricting element with respect to a path system — while the infinite index subgroups object of our study will satisfy a very general notion of “convex cocompactness” — quasi-convexity with respect to a path system. The remaining of this section is structured as follows. First of all, we will mention two applications. Later we will give an informal explanation of our general setting as the result of a natural generalisation of these applications. We expect that this will be enough to understand our main theorems stated right after that. We will give another application at the end. Groups acting properly with a strongly contracting element. Members of this class contain elements that “behave like” a loxodromic isometry in a hyperbolic space – in a strong sense. Let δ ⩾ 0. A subset A of X is δ-strongly contracting if the diameter of the nearest-point projection on A of any metric ball of X not intersecting A is less than δ. An element g of G is δ-strongly contracting if it has infinite order and there exists an orbit of the cyclic subgroup generated by g that is δ-strongly contracting. In his seminal paper M. Gromov introduced the concept of δ-hyperbolic space, [49]. He observed that most of the large scale features of negative curvature can be described in terms of thin triangles. Nowadays, there are plenty of reformulations of the δ-hyperbolicity. In particular, H. Masur and Y. Minsky gave one by describing geodesics in terms of strong contraction: Example 0.5.1. — A geodesic metric space X is hyperbolic if and only if there exists δ ⩾ 0 such that any geodesic segment of X is δ-strongly contracting, [61, Theorem 2.3]. The following are some subclasses of groups acting properly with a strongly contracting element: (i) H = “G is a group acting properly with a loxodromic element on a hyperbolic space X.” In H, an element is loxodromic if and only if it is strongly contracting. See [26]. (ii) RH = “G is a relatively hyperbolic group acting with a hyperbolic element on a locally finite Cayley graph X of G.” In RH, hyperbolic elements are strongly contracting. See [65, Corollary 1.7] and [73, Theorem 2.14]. (iii) CAT0 = “G is a group acting properly with a rank-one element on a proper CAT (0) space X.” In CAT0, rank-one elements are strongly contracting. See [16, Theorem 5.4] 28 TABLE OF CONTENTS and [24]. (iv) ModT = “G is the mapping class group of an orientable surface of genus g and p marked points of complexity 3g + p − 4 > 0 acting on its Teichmüller space endowed with the Teichmüller metric.” In ModT, pseudo-Anosov elements are strongly contracting. See [64] and [61, Proposition 4.6]. (v) GSC = “G is an infinite graphical small cancellation group associated to a Gr′(1/6)- labeled graph with finite components labeled by a finite set S, acting on the Cayley graph X of G with respect to S.” In GSC, loxodromic WPD elements for the action of G on the hyperbolic coned-off Cayley graph constructed by D. Gruber and A. Sisto in [51] are strongly contracting. See [7, Theorem 5.1]. (vi) Gar = “G is the quotient of a ∆-pure Garside group of finite type by its center, acting with a Morse element on the Cayley graph X of G with respect to the Garside generating set.” In Gar, Morse elements are strongly contracting. See [21, Theorem 5.5]. An appropriate notion of convex cocompactness in this setting is just the usual quasi- convexity. Let η ⩾ 0. A subset Y of X is η-quasi-convex if any geodesic of X with endpoints in Y is contained in the η-neighbourhood of Y . A subgroup H of G is η-quasi-convex if there exists an orbit of H that is η-quasi-convex. Our theorem below generalises [77, Theorem 4.8] and [37, Theorems 1.1 and 1.3]: Theorem 0.5.2. — If G is a non-virtually cyclic group acting properly with a strongly contracting element on a geodesic metric space X, and H is an infinite index quasi-convex subgroup of G, then ω(H) < ω(G), ω(G/H) = ω(G). Hierarchically hyperbolic groups. Let Mod(Σg,p) be the mapping class group of an orientable surface Σg,p of genus g and p marked points of complexity 3g + p− 4 > 0. We would like to apply Theorem 0.5.2 to Mod(Σg,p) with respect to the word metric. However, we do not know whether Mod(Σg,p) acts with a strongly contracting element on any of its locally finite Cayley graphs or not. Maybe the candidates that come to mind are the pseudo-Anosov elements, and evidence suggests that not all of them are strongly contracting: K. Rafi and Y. Verberne constructed a generating set U of Mod(Σ0,5) and 29 TABLE OF CONTENTS a pseudo-Anosov element which is not strongly contracting for the action of Mod(Σ0,5) on the Cayley graph of Mod(Σ0,5) with respect to U , [68, Theorem 1.3]. We were able to avoid this setback by looking into the class of hierarchically hyperbolic groups, introduced by J. Behrstock, M. Hagen and A.Sisto in [12, 13] as a generalisation of the Masur and Minsky hierarchy machinery of mapping class groups. Below we provide some examples of hierarchically hyperbolic groups. The reader should note that the metric space where they act with a hierarchically hyperbolic structure is any of their locally finite Cayley graphs: (i) Mapping class groups of finite type surfaces, [13]. (ii) Right-angled Artin groups, [12]. (iii) Right-angled Coxeter groups, [12]. (iv) Fundamental groups of 3-manifolds without NIL or SOL components, [13]. Now consider the following notion of convex cocompactness. A subset Y of X is Morse if for every κ ⩾ 1, λ ⩾ 0, there exists σ ⩾ 0 such that any (κ, l)-quasi-geodesic of X with endpoints in Y is contained in the σ-neighbourhood of Y . A subgroup H of G is Morse if there exists an orbit of H that is Morse. An element g of G is Morse if it has infinite order and the cyclic subgroup generated by g is Morse. We have obtained the next result, partially generalising [27, Theorem A]: Theorem 0.5.3. — If G is a non-virtually cyclic hierarchically hyperbolic group acting on a locally finite Cayley graph X of G with a Morse element, and H is an infinite index Morse subgroup of G, then ω(H) < ω(G), ω(G/H) = ω(G). We know that pseudo-Anosov elements of mapping class groups are Morse with respect to any word metric, [11], and that the infinite index Morse subgroups of the mapping class group are precisely the convex cocompact subgroups in the sense of mapping class groups, [55, Theorem A], which allows us to obtain a more concrete statement: Corollary 0.5.4. — If G is the mapping class group of a surface of genus g and p marked points such that 3g + p− 4 > 0 acting on a locally finite Cayley graph X of G, and H is a 30 TABLE OF CONTENTS convex cocompact subgroup of G, then ω(H) < ω(G), ω(G/H) = ω(G). Remark 0.5.5. — Under the hypothesis of the previous corollary, we remark that the inequality ω(H) < ω(G) was also obtained independently in [27, Corollary C]. Main results. Now that we gave the big picture, we will give a technical definition that encapsulates the classes discussed so far. In order to do so, we make two observations. On the one hand, the strong contraction property can be reformulated in the following way. A subset A of X is strongly contracting if and only if any geodesic segment of X joining any pair of points x, y ∈ X whose projections p and q via a nearest-point projection are far away passes next to p and q, [8, Proposition 2.9]. On the other hand, mapping class groups – or more generally, hierarchically hyperbolic groups – come with hierarchy paths, a family of special quasi-geodesics encoding substantial information about the geometry of the space and easier to work with than the set of all (quasi-)geodesics. For these reasons, in order to define very general notions of non-positive curvature and convex cocompactness, we will be considering path systems, introduced by A. Sisto in [74]: Definition 0.5.6 (Path system group). — Let µ ⩾ 1, ν ⩾ 0. A (µ, ν)-path system group (G,X,P) is a group G acting properly on a geodesic metric space X together with a G-invariant collection P of paths of X satisfying: (PS1) P is closed under taking subpaths. (PS2) For every x, y ∈ X, there exists γ ∈ P joining x to y. (PS3) Every element of P is a (µ, ν)-quasi-geodesic. We refer to P as (µ, ν)-path system. We fix µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system group (G,X,P) for the following definitions. Let δ ⩾ 0. We say that a subset Y of X is δ-constricting if there exist a coarse nearest-point projection of X on A with the property that any γ ∈ P joining any two pair of points x, y ∈ X whose projections p and q are δ-far away passes through the δ-neighbourhoods of p and q (Definition 1.1.8). An element g of G is δ-constricting if it has infinite order and there exists a δ-constricting orbit of the cyclic subgroup generated by g. Let η ⩾ 0. A subgroup Y of X is η-quasi-convex if any γ ∈ P with endpoints in 31 TABLE OF CONTENTS Y is contained in the η-neighbourhood of Y (Definition 1.1.7). A subgroup H of G is η-quasi-convex if there exist an η-quasi-convex orbit of H. Example 0.5.7. — (i) Assume that the metric space X is geodesic. An infinite order element of G is strongly contracting if and only if it is constricting with respect to the set of all the geodesic segments of X, [8, Proposition 2.9]. (ii) Assume that the group G is hierarchically hyperbolic. An infinite order element g of G is Morse if and only if for every κ ⩾ 1, there exists δ ⩾ 0 such that g is δ-constricting with respect to the set of all the κ-hierarchy paths. See [69, Theorem E] and [14, Lemma 1.27]. Finally, we state the main results of Chapter 1. Theorem 0.5.2 and Theorem 0.5.3 are special cases. Our first result generalises work of W. Yang, [77, Theorem 4.8], and F. Dahmani - D. Futer - D. Wise, [37, Theorems 1.1 and 1.3]. The Poincaré series PU(s) based at x ∈ X of a subset U of G is defined as ∀ s ⩾ 0, PU(s) = ∑ u∈U e−s|ux−x| and modifies its behaviour at the relative exponential growth rate ω(U,X): the series diverges if s < ω(U,X) and converges if s > ω(U,X). At s = ω(U,X) the series can converge or diverge depending on the nature of U . This behaviour is independent of the point x ∈ X. We say that the action of U on X is divergent if PU(s) diverges at s = ω(U,X). Theorem 0.5.8. — Let (G,X,P) be a path system group. Assume that G contains a constricting element. Let H be an infinite index subgroup of G. Assume that the following conditions are true: (i) ω(H) < ∞. (ii) The action of H on X is divergent. (iii) H is quasi-convex. Then ω(H) < ω(G). Remark 0.5.9. — Under the hypothesis of Theorem 0.5.8, one may ask if there is a growth gap, i.e, if sup H ω(H) < ω(G), 32 TABLE OF CONTENTS where the supremum is taken among the infinite index subgroups H of G satisfying (i), (ii) and (iii). In our context, the answer is yes: there is a growth gap when G is a hyperbolic group with Kazhdan’s Property (T), [34, Theorem 1.2]. However, one can show that there is no growth gap among free groups, [37, Theorem 9.4], or fundamental groups of compact special cube complexes, [58, Theorem 1.5]. The answer to our context could be different if one studied semigroups instead of subgroups, [77, Theorem A]. In [49, 5.3.C], M. Gromov stated that in a torsion-free hyperbolic group G, any infinite index quasi-convex subgroup H is a free factor of a larger quasi-convex subgroup. Gromov’s ideas were later developed by G. N. Arzhantseva in [6, Theorem 1]. More recently, J. Russell, D. Spriano and H. C. Tran generalised her result to the context of groups with the “Morse local-to-global property”, [70, Corollary 3.5]. Further, the problem seems connected to the “PNaive property” studied by C. Abbott and F. Dahmani in the context of groups acting acylindrically on a hyperbolic space, [1]. In our context, we have obtained the following, in which there is no torsion-free assumption. We will see that Theorem 0.5.8 is, in part, a consequence of this result: Theorem 0.5.10. — Let (G,X,P) be a path system group. Assume that G contains a constricting element g0. Let H be an infinite index quasi-convex subgroup of G. There exist an element g ∈ G conjugate to a large power of g0 and a finite extension E of ⟨g⟩ such that the intersection H ∩E is finite and the natural morphism H ∗H∩E ⟨g,H ∩E⟩ → G is injective. According to Proposition 1.1.5 (6), the subgroup generated by a constricting element is always Morse, and in particular quasi-convex. Hence we obtain the following alternative: Corollary 0.5.11. — Let (G,X,P) be a path system group. Assume that G contains a constricting element. Then, either G is virtually cyclic or contains a free subgroup of rank two. Remark 0.5.12. — To the best of our knowledge, the previous corollary has not been recorded for the class of groups acting properly with a strongly contracting element. The Tits alternative is known for hierarchically hyperbolic groups [43, Theorem 9.15], which is a much stronger result. In our second result we generalise work of Y. Antolín, [4, Theorem 3], and R. Gitik - E. Rips, [47, Theorem 2]: 33 TABLE OF CONTENTS Theorem 0.5.13. — Let (G,X,P) be a path system group. Assume that G contains a constricting element. Let H be an infinite index quasi-convex subgroup of G. Then ω(G/H) = ω(G). Note that the study of [47, Theorem 2] concerns double cosets in the hyperbolic group case. We remark that in [40, VII D 39], P. de la Harpe says about the growth of double cosets: “this theme has not received yet too much attention, but probably should”. In our context, for sake of simplicity, we decided to study single cosets instead, but one could possibly extend our result. Further, we remark that our result is connected to the study of I. Kapovich on the hyperbolicity and amenability of the Schreier graphs of infinite index quasi-convex subgroups of hyperbolic groups, [53, 54]. Now we are going to record a joint corollary to Theorem 0.5.8 and Theorem 0.5.13. In general, it is not easy to decide whether the action of a groups is divergent or not. However, the following is a well-known consequence of Fekete’s Subadditivity Lemma: Lemma 0.5.14 ([39, Proposition 4.1 (1)]). — Let G be a group acting properly on a geodesic metric space X. Let x ∈ X. Let H ⩽ G be a quasi-convex subgroup (in the classical sense). Then ω(H) = inf n⩾1 1 n log |{h ∈ H : |hx− x| ⩽ n }| = lim n→∞ 1 n log |{h ∈ H : |hx− x| ⩽ n }|. In particular ω(H) < ∞. If in addition H is infinite, then the action of H on X is divergent. In combination with Corollary 0.5.11, we obtain: Corollary 0.5.15. — Let (G,X,P) be a path system group. Assume that G is non- virtually cyclic and contains a constricting element. (i) If P is the set of all the geodesic segments of X, then for every infinite index quasi-convex subgroup H of G, we have ω(H) < ω(G), ω(G/H) = ω(G). 34 TABLE OF CONTENTS (ii) For every infinite index Morse subgroup H of G, we have ω(H) < ω(G), ω(G/H) = ω(G). Remark 0.5.16. — One can prove that the class of groups acting properly with a constricting element with respect to a path system is invariant under equivariant quasi-isometries. How- ever, strongly contracting elements are not preserved under equivariant quasi-isometries, [7, Theorem 4.19]. In particular, Corollary 0.5.15 applies for instance to the action on a locally finite Cayley graph of any group acting geometrically on a CAT (0) space with a rank-one element. Remark 0.5.17. — The proofs of Theorem 0.5.2 and Theorem 0.5.3 now follow from our main results (Theorem 0.5.8 and Theorem 0.5.13) in view of Example 0.5.7. Hierarchical quasi-convexity. In hierarchically hyperbolic groups there is a notion of convex cocompactness more natural than Morseness. Let G be a hierarchically hyperbolic group. A subgroup H of G is hierarchically quasi-convex if and only if for every κ ⩾ 1, there exists η ⩾ 0 such that H is η-quasi-convex with respect to the set of all the κ- hierarchy paths of G, [69, Proposition 5.7]. Finally, we deduce two more applications from Theorem 0.5.8 and Theorem 0.5.13: Theorem 0.5.18. — If G is a hierarchically hyperbolic group acting on a locally finite Cayley graph X of G with a Morse element, and H is an infinite index subgroup of G satisfying: (i) The action of H on X is divergent. (ii) H is hierarchically quasi-convex. Then ω(H) < ω(G). Theorem 0.5.19. — If G is a hierarchically hyperbolic group acting on a locally finite Cayley graph X of G with a Morse element, and H is an infinite index hierarchically quasi-convex subgroup of G, then ω(G/H) = ω(G). 35 TABLE OF CONTENTS 0.6 Uniform uniform exponential growth Let G be a group with finite symmetric generating set U . Denote by XU the corre- sponding Cayley graph. In Chapter 2 we study the number ω(U) := ω(G,XU). The n-th product set Un is the collection of elements u1·...·un ∈ G such that u1, · · · , un ∈ U . The role of ω(U) is to give us information about the exponential behaviour of |Un| as n increases. The generating sets of virtually nilpotent groups have vanishing exponential growth rate, since a celebrated theorem of M. Gromov shows that those are exactly the groups of polynomial growth, [48]. Let ξ > 0. The group G has ξ-uniform exponential growth if for every finite symmetric generating set U of G, we have ω(U) > ξ. A group has ξ-uniform uniform exponential growth if every finitely generated subgroup is either virtually nilpotent or has ξ-uniform exponential growth. Uniform uniform exponential growth is particularly well-studied in groups of non- positive curvature. Indeed, groups of uniform uniform exponential growth include hyperbolic groups, [56, 9, 19], free products of countable families of groups with ξ-uniform uniform exponential growth (folklore), mapping class groups, [3, 59, 2], or cocompactly special cubulated CAT(0) groups, [44, 2]. It is unknown whether the outer automorphism group of the free group of rank ⩾ 2 has uniform uniform exponential growth, [15]. All of the groups in this list admit non-elementary acylindrical actions on Gromov hyperbolic spaces, [72, 17, 38]. Geometric small cancellation quotients. The main goal of Chapter 2 is to prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking geometric C ′′(λ, ε)-small cancellation quotients in the sense of [38, Definition 6.22]. This result is Theorem 0.6.2 below. Before stating the theorem, we are going to give some definitions. Let δ > 0. Let G be a group acting by isometries on a δ-hyperbolic space X. Acylindricity. Let κ, N > 0. The action of G on X is (κ,N)-acylindrical if for every pair of points x, y ∈ X at distance at least κ, the number of elements u ∈ G moving each of the points x, y at distance at most 100δ is bounded above by N . In practice, the number N has two meanings for us: (1) The largest size of the finite subgroups of virtually cyclic subgroups in G containing 36 TABLE OF CONTENTS a loxodromic isometry. (2) The fraction ∆(g) ∥g∥ of the longest intersection ∆(g) between the axis of any pair of conjugates of an arbitrary loxodromic isometry g of G, with the translation length ∥g∥ of g, whenever this translation is larger than 100δ. Geometric small cancellation theory. A loxodromic moving family – or set of relations – is a set of the form Q = { ( ⟨grg−1⟩ , gYr) ∣∣∣ r ∈ R, g ∈ G } , where R ⊂ G is a set of loxodromic isometries r – the relators – stabilizing their quasi- convex axis Yr ⊂ X. A piece is an intersection of any pair of such axis. The role of the parameters λ ∈ (0, 1) and ε > 0 in the geometric C ′′(λ, ε)-small cancellation condition on Q is the following: ▶ The fraction of the length of the longest piece with the shortest translation length of the relators r ∈ R is at most λ. ▶ The shortest translation length of the relators r ∈ R is at least εδ. Let K be the normal closure in G of the relator subgroups H in Q. The geometric C ′′(λ, ε)- small cancellation condition permits to obtain substantial information of the geometric C ′′(λ, ε)-small cancellation quotient Ḡ = G/K: for instance K is a free product of relator subgroups, Ḡ locally looks like G and any acylindrical action of G on X induces another acylindrical action of Ḡ on a quotient δ̄-hyperbolic space X̄ whose hyperbolicity constant δ̄ is universal. Main theorem. The following corollary captures the essence of the main theorem. Corollary 0.6.1. — There exists a universal constant λ > 0 such that for every group G acting acylindrically on a hyperbolic space X, there exist ε > 0 depending only on the acylindricity and hyperbolicity constants such that the following statements are equivalent. (i) G has uniform uniform exponential growth. (ii) Every geometric C ′′(λ, ε)-small cancellation quotient of G has uniform uniform exponential growth. (iii) There exist a geometric C ′′(λ, ε)-small cancellation quotient of G that has uniform uniform exponential growth. The main theorem of Chapter 2 is the following. 37 TABLE OF CONTENTS Theorem 0.6.2 (Theorem 2.5.5 & Theorem 2.5.6). — There exists λ > 0 such that for every N > 0 the following holds. Let δ > 0, κ ⩾ δ, and ε ⩾ 1010 max{N, κ/δ}. Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. (i) If G has ξ-uniform uniform exponential growth, then every geometric C ′′(λ, ε)-small cancellation quotient of G has ξ′-uniform uniform exponential growth. The constant ξ′ depends only on ξ and N . (ii) If there exist a geometric C ′′(λ, ε)-small cancellation quotient of G that has ξ-uniform uniform exponential growth, then G has ξ′-uniform uniform exponential growth. The constant ξ′ depends only on ξ. Remark 0.6.3. — The dependence of ε on κ, N and δ is not a strong condition. In fact, the intersection of the axis of two loxodromic elements in a group acting acylindrically on a hyperbolic space is controled in terms of κ, N , δ and the translation length of the loxodromic elements. Thus to prove that a set of relators satisfies the geometric C ′′(λ, ε)-condition, one usually considers relators of sufficient length compared to κ, N and δ anyway. Beyond short loxodromics. The standard strategy to study uniform exponential growth in hyperbolic groups exploits the fact that their finite symmetric generating sets have the short loxodromic property: every n-th power Un of a finite symmetric generating set contains a loxodromic isometry, for some number n that does not depend on the set U . In general, it is unknown whether every finitely generated group acting acylindrically on a hyperbolic space has uniform exponential growth. The acylindrical action on a hyperbolic space yields uniform exponential growth for finite symmetric generating sets with a long loxodromic isometry. The short loxodromic property permits to take uniform large powers so that we can exploit this other situation. However, there is a finitely generated (combinatorial/graded) small cancellation quotient with an acylindrical action on a hyperbolic space but without the short loxodromic property, [63]. Our main result does not make use of the short loxodromic property. The moral of our work is that we can deal with this kind of monster as long as these are small cancellation quotients of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space. However, the aforementioned monster is a quotient of the free product of all hyperbolic groups. It is unkown whether this free product has uniform uniform exponential growth, owing to it is unkown whether there is a universal lower bound for the uniform growth rate of all hyperbolic groups, independent of the hyperbolicity constant, [19, Section 14, 38 TABLE OF CONTENTS Question 2]. The following example shows that the short loxodromic property plays no role in the proof of Theorem 0.6.2. Example 0.6.4. — There are infinite families of geometric small cancellation quotients that are hyperbolic groups containing arbitrarily large torsion balls. These groups act acylindrically with uniform acylindricity parameters and have ξ-uniform uniform expo- nential growth, for some uniform growth exponent ξ > 0, see [36]. The uniform uniform exponential growth rate of the small cancellation quotient in Theorem 0.6.2 (i) does not depend on the cardinality of large torsion balls, nor does it depend on the hyperbolicity constant. Classical small cancellation groups We now discuss groups given by a presentation that satisfies the classical C ′′(λ)-small cancellation condition. We refer to a group admiting such a presentation as classical C ′′(λ)-small cancellation group. These are exacly the geometric small cancellation quotients over free groups. In this situation, the geometric small cancellation condition involving the parameter ε becomes trivial. A classical C ′′(λ)- small cancellation group is always finitely presented, hence, hyperbolic. Thus it has uniform uniform exponential growth by [49, 56]. However, in that approach the uniform uniform exponential growth rate depends on λ. The following is a consequence of Theorem 0.6.2 for the free group case. Corollary 0.6.5. — There exist λ > 0 and ξ > 0 such that every classical C ′′(λ)-small cancellation group has ξ-uniform uniform exponential growth. Note that there is a generic class of classical C ′′(1/6)-small cancellation groups such that every 2-generated subgroup is free, [10]. This immediately implies Corollary 0.6.5 for this generic class of classical C ′′(1/6)-small cancellation groups, [40]. Remark 0.6.6. — The classical C ′′(λ)-small cancellation condition in Corollary 0.6.5 is reminiscent of our proof that uses geometric small cancellation theory. To this date, geometric small cancellation theory has not been developed under a geometric C ′(λ, ε)- small cancellation condition. We expect, however, that this is possible, and thus that our results hold for classical C ′(λ)-small cancellation groups - finitely and infinitely presented. Strategy of proof. To prove Theorem 0.6.2 (i), we need to discuss the growth of finite symmetric subsets of sufficiently large energy in groups acting acylindrically on a 39 TABLE OF CONTENTS hyperbolic space X. If G acts by isometries on X, the ℓ∞-energy L(U) of a finite subset U ⊂ G is defined by L(U) = inf x∈X max u∈U |ux− x|. If U = {g}, the ℓ∞-energy coincides with the translation length of g. The following example explains why the energy is important in the study of uniform exponential growth. Example 0.6.7. — When G is the fundamental group of a compact hyperbolic manifold, there exists a constant µ > 0 – the Margulis constant – such that if U ⊂ G is a finite set with L(U) < µ, then the subgroup of G generated by U is virtually nilpotent. If T denotes the injectivity radius of the action of G on the universal cover and is smaller than the Margulis constant µ, then the acylindricity constant κ is about 1/T, [42]. Definition 0.6.8 (Definition 2.2.1). — Let α > 0. We say that a finite subset U ⊂ G is α-reduced at p ∈ X if U ∩ U−1 = ∅ and for every pair of distinct u1, u2 ∈ U ⊔ U−1, the Gromov product satisfies (u1p, u2p)p < 1 2 min{|u1p− p|, |u2p− p|} − α− 2δ. Remark 0.6.9. — Roughly speaking, if a set U ⊂ G is reduced then the orbit map from the free group generated by U to X is a quasi-isometric embedding. The following is a well-known theorem of [56, 9], see also [45]. Theorem 0.6.10 (Theorem 2.3.8). — For every κ, N > 0, there exist an integer c > 1 with the following property. Let δ, α > 0. Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Then one of the following conditions holds: (i) L(U) ⩽ 104 max {κ, δ, α}. (ii) The subgroup ⟨U⟩ is virtually cyclic and contains a loxodromic element. (iii) There exists an α-reduced subset S ⊂ U c such that |S| ⩾ max { 2, 1 c |U | } . Moreover, ω(U) ⩾ 1 c log |U |. 40 TABLE OF CONTENTS Our main contribution to Theorem 2.3.8 is the dependence of the involved constants: for our purpose it is important that the number c only depends on the acylindricity parameters κ and N . Remark 0.6.11. — If the injectivity radius of the action of G on X is large, then every finite symmetric subset of G satisfies either (ii) or (iii). In general this is however not the case. We will later use uniform uniform exponential growth of G in order to apply Theorem 2.3.8 to some power of an arbitrary symmetric subset U in G. Theorem 2.3.8 with Fekete’s Subadditive Lemma and the fact that ω(Un) = nω(U) implies the following corollary. It is a weak form of purely exponential growth, [25, 77]. Corollary 0.6.12. — For every κ, N > 0, there exists ξ > 1 with the following property. Let δ > 0 and κ ⩾ δ. Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity of energy L(U) > 104κ that does not generate a virtually cyclic subgroup. Then, for every n ⩾ 0, enω(U) ⩽ |Un| ⩽ eξnω(U). To prove Theorem 0.6.2 (i), we follow a strategy of [36] that estimates product set growth in Burnside groups. In particular, we use the viewpoint of geometric small cancellation theory. As previously mentioned, the Small Cancellation Theorem gives a universal constant δ̄ > 0 such that any geometric small cancellation quotient Ḡ of a group G acting acylindrically on a δ-hyperbolic space X, for appropriate choice of the small cancellation parameters, acts acylindrically on a δ̄-hyperbolic space X̄. Let Ū ⊂ Ḡ be a finite symmetric generating set containing the identity that is not contained in an elliptic or virtually cyclic subgroup. If the energy of Ū is larger than 104δ̄, then the exponential growth rate of Ū is bounded below by a universal strictly positive constant (Lemma 2.1.23). Otherwise, we fix a pre-image U of Ū in G of minimal energy for the action of G on X (Lemma 2.1.32). Such a pre-image may not have large energy > 104δ. Indeed, it may consist entirely of torsion- elements and thus have small energy < 104δ. However, our pre-image U is not contained in any elliptic subgroup. Thus some power of U contains a loxodromic element, hence, for some exponent n, we have L(Un) > 104δ. We stress that the exponent n depends on the set U . We now apply Theorem 0.6.10 to Un. Since U is not contained in any virtually cyclic subgroup, we obtain a reduced subset S in U cn, which freely generates a free subgroup. Next, we adapt the counting argument of [29, 36] to prove that for every r ⩾ 1, the 41 TABLE OF CONTENTS proportion of elements in Sr that contain a large part of a relator is small compared to |Sr| (Proposition 2.4.9). A combination of a consequence of Greendlinger’s Lemma (Proposition 2.4.16) and Fekete’s Subadditive Lemma then implies that the exponential growth rate of Ū satisfies ω(Ū) ⩾ β · ω(U). for β = sup θ∈(0,1) inf { θ · log 3 2 log (2c) , 1 − θ } · 1 c . Finally, assume that G has ξ-uniform uniform exponential growth. A combination of this fact with the previous inequality yields Theorem 0.6.2 (i). The proof of Theorem 0.6.2 (ii) is similar and we postpone its discussion. 42 NOTATION Let X be a metric space. Given two points x, x′ ∈ X, we write |x− x′| for the distance between them. The ball of X of center x ∈ X and radius r > 0 is BX(x, r) = { y ∈ X : |x− y| ⩽ r }. The distance between a point x ∈ X and a subset Y ⊂ X is d(x, Y ) = inf { |x− y| : y ∈ Y }. Let η ⩾ 0. The η-neighbourhood of a subset Y ⊂ X is Y +η = {x ∈ X : d(x, Y ) ⩽ η }. The distance between two subsets Y, Z ⊂ X is d(Y, Z) = inf { |y − z| : y ∈ Y, z ∈ Z }. The Hausdorff distance between two subsets Y, Z ⊂ X is dHaus(Y, Z) = inf { ε ⩾ 0 : Y ⊂ Z+ε and Z ⊂ Y +ε }. A path is a continuous map α : [a, b] → X. The initial and terminal points of α are α(a) and α(b), respectively. We denote by α− and α+ the initial and terminal points of α, respectively. They form the endpoints of α. We will frequently identify a path and its image. A subpath of α is a restriction of α to a subinterval of [a, b]. The path α joins the point x ∈ X to the point y ∈ X if α− = x and α+ = y. Note that for every x, y ∈ α there may be more than one subpath of α joining x to y, unless the points are given by the parametrisation of α. If x, y ∈ α are given by the parametrisation, we denote by [x, y]α the parametrised subpath of α joining x to y. The length of a path α is denoted by ℓ(α). If α joins a point x ∈ X to a point y ∈ Y of a closed subset Y ⊂ X, the entrance point of 43 TABLE OF CONTENTS α in Y is the point y′ ∈ α satisfying ℓ([x, y′]α) = inf z∈α∩Y ℓ([x, z]α). Unless otherwise stated a path is a rectifiable path parametrised by arc length. Let κ ⩾ 1, l ⩾ 0. A path α : [a, b] → X is a (κ, l)-quasi-geodesic if for every t, t′ ∈ [a, b], |α(t) − α(t′)| ⩽ |t− t′| ⩽ κ|α(t) − α(t′)| + l. Note that that ℓ(α|[t,t′]) = |t− t′|. Let L ⩾ 0. We say that α is a L-local (κ, l)-quasi-geodesic if any subpath of α whose length is at most L is a (κ, l)-quasi-geodesic. A geodesic is a (1, 0)-quasi-geodesic. The metric space X is geodesic if for every pair of points x, x′ ∈ X there exists a geodesic of X joining x to x′. We write [x, x′] for a geodesic joining them. Recall that there may be multiple geodesics joining two points. 44 Chapter 1 GROWTH OF QUASI-CONVEX SUBGROUPS IN GROUPS WITH A CONSTRICTING ELEMENT THE GIANT: The first thing I will tell you is: “There’s a man in a smiling bag”. The second thing is: “The owls are not what they seem”. The third thing is: “Without chemicals, he points”. COOPER: What do these things mean? THE GIANT: This is all I am permitted to say. from Twin Peaks, created by David Lynch and Mark Frost Contents 1.1 Path system geometry . . . . . . . . . . . . . . . . . . . . . . . 46 1.2 Growth estimation criteria . . . . . . . . . . . . . . . . . . . . . 49 1.3 Buffering sequences . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.4 Quasi-convexity in the Intersection–Image property . . . . . 55 1.5 Finding a quasi-convex element . . . . . . . . . . . . . . . . . . 56 1.6 Constricting elements . . . . . . . . . . . . . . . . . . . . . . . . 63 1.6.1 A G-invariant family . . . . . . . . . . . . . . . . . . . . . . . . 63 1.6.2 Finding a constricting element . . . . . . . . . . . . . . . . . . 64 1.6.3 Elementary closures . . . . . . . . . . . . . . . . . . . . . . . . 65 1.6.4 Forcing a geometric separation . . . . . . . . . . . . . . . . . . 67 1.7 Growth of quasi-convex subgroups . . . . . . . . . . . . . . . . 70 The results of this chapter correspond to the following article: ▶ X. Legaspi. Constricting elements and the growth of quasi-convex subgroups, 2022. URL: https://orcid.org/0000-0002-1497-6448. 45 https://orcid.org/0000-0002-1497-6448 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element In Section 1.1 we will introduce the definitions of path system group, quasi-convex subgroup and constricting element. We also state some standard properties (Proposi- tion A.1.1) that will be proven in the Appendix A. In Section 1.2 we will explain the two criteria that we will use to estimate the growth of quasi-convex subgroups. The rest of the chapter is devoted to the development of our geometric framework so that we can apply these criteria. In Section 1.3 we will prove a version of the Bounded Geodesic Image Property of hyperbolic spaces, but for quasi-convex subsets insetad of geodesics. In Section 1.4 we will introduce the notion of buffering sequence and we will give a version of Behrstock’s inequality. In Section 1.5, given an infinite index quasi-convex subgroup and a quasi-convex element, we will produce another quasi-convex element whose orbit is “transversal” to the given subgroup. The proofs of both of our main results (Theorem 0.5.8 and Theorem 0.5.13) share this argument. In Section 1.6 we will study the elementary closures of constricting elements and also some geometric separation properties. Finally, in Section 1.7 we will prove our main results (including Theorem 0.5.10) by constructing an appropriate buffering sequence in each situation. 1.1 Path system geometry This section is devoted to present the notations and vocabulary of the main geometric objects of this chapter. We formalise our notions of “convex cocompactness” and “non- positive curvature”. Path system spaces. Definition 1.1.1 (Path system space). — Let µ ⩾ 1, ν ⩾ 0. A (µ, ν)-path system space (X,P) is a metric space X together with a collection P of paths of X satisfying: (PS1) P is closed under taking subpaths. (PS2) For every x, y ∈ X, there exists γ ∈ P joining x to y. (PS3) Every element of P is a (µ, ν)-quasi-geodesic. We refer to P as (µ, ν)-path system. We fix µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system space (X,P). Definition 1.1.2 (Quasi-convex subset). — Let η ⩾ 0. A subset Y ⊂ X is η-quasi-convex if every γ ∈ P with endpoints in Y is contained in the η-neighbourhood of Y . 46 1.1. Path system geometry Definition 1.1.3 (Constricting subset). — Let δ ⩾ 0. A subset A ⊂ X is δ-constricting if there exists a map πA : X → A satisfying: (CS1) Coarse retraction. For every x ∈ A, we have |x− πA(x)| ⩽ δ. (CS2) Constriction. For every x, y ∈ X and for every γ ∈ P joining x to y, if we have |πA(x)−πA(y)| > δ, then γ ∩BX(πA(x), δ) ̸= ∅ and γ ∩BX(πA(y), δ) ̸= ∅. We refer to πA : X → A as δ-constricting map. x πA(x) πA(y) y γ A ⩽ δ > δ ⩽ δ Figure 1.1 – The constriction property. Notation 1.1.4. — Let πA : X → A be a map between X and a subset A ⊂ X. For every x, y ∈ X, we denote |x − y|A = |πA(x) − πA(y)|. For every subset Y ⊂ X, we denote diamA(Y ) = diam(πA(Y )). For every x ∈ X and for every pair of subsets Y, Z ⊂ X, we denote dA(x, Y ) = d(πA(x), πA(Y )) and dA(Y, Z) = d(πA(Y ), πA(Z)). Note that dA may not be a distance over the collection of subsets of X: it may not satisfy the triangle inequality. We will keep this notation for the rest of the paper. The following are some standard properties: Proposition 1.1.5. — For every δ ⩾ 0, there exist a constant θ ⩾ 0 and a pair of maps, σ : R⩾1 × R⩾0 → R⩾0 and ζ : R⩾0 → R⩾0, such that any δ-constricting map πA : X → A satisfies the following properties: 47 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element (1) Coarse nearest-point projection. For every x ∈ X, we have |x− πA(x)| ⩽ µd(x,A) + θ. (2) Coarse equivariance. Let H be a group acting by isometries on X such that A and P are H-invariant. Then for every h ∈ H and for every x ∈ X, we have |πA(hx) − hπA(x)| ⩽ θ. (3) Coarse Lipschitz map. For every x, y ∈ X, we have |x− y|A ⩽ µ|x− y| + θ. (4) Intersection–Image. For every γ ∈ P, we have | diam(A+δ ∩ γ) − diamA(γ)| ⩽ θ. (5) Behrstock inequality. Let πB : X → B be a δ-constricting map. Then for every x ∈ X, we have min {dA(x,B), dB(x,A)} ⩽ θ. (6) Morseness. Let κ ⩾ 1, l ⩾ 0. Let α be a (κ, l)-quasi-geodesic of X with endpoints in A. Then α ⊂ A+σ(κ,l). (7) Coarse invariance. Let ε ⩾ 0. Let B ⊂ X be a subset such that dHaus(A,B) ⩽ ε. Then B is ζ(ε)- constricting. Proof. — We give some references. For (1), (3) and (4), see [74, Lemma 2.4]. For (5), see [74, Lemma 2.5]. For (6), see [74, Lemma 2.8 (1)]. We leave the proof of the properties (2) and (7) as an exercise. Path system groups. Let G be a group acting by isometries on a metric space X. The quasi-stabilizer G of x ∈ X of radius r > 0 is defined as StabG(x, r) = {g ∈ G : |x− gx| ⩽ r}. The action of G on X is proper if for every x ∈ X and for every r > 0, we have | StabG(x, r)| < ∞. Let η ⩾ 0. The action of G on X is η-cobounded if for every x, x′ ∈ X, there exists g ∈ G such that |x− gx′| ⩽ η. 48 1.2. Growth estimation criteria Definition 1.1.6 (Path system group). — Let µ ⩾ 1, ν ⩾ 0. A (µ, ν)-path system group (G,X,P) is a group G acting properly on a metric space X together with a G-invariant collection P of paths of X such that (X,P) is a (µ, ν)-path system space. We fix µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system group (G,X,P). Definition 1.1.7 (Quasi-convex subgroup). — A subgroup H ⩽ G is η-quasi-convex if there exists an H-invariant η-quasi-convex subset Y ⊂ X such that the action of H on Y is η-cobounded. We will write (H, Y ) when we need to stress the η-quasi-convex subset Y that H is preserving. Definition 1.1.8 (Constricting element).— Let δ ⩾ 0. An element g ∈ G is δ-constricting if the following holds: (CE1) g has infinite order. (CE2) There exists a ⟨g⟩-invariant δ-constricting subset A ⊂ X so that the action of ⟨g⟩ on A is δ-cobounded. We will write (g, A) when we need to stress the δ-constricting subset A that ⟨g⟩ is preserving. 1.2 Growth estimation criteria In this section, we fix a group G acting properly on a metric space X and a subgroup H ⩽ G. The goal is to establish simple criteria so that we can check if H is a solution to the system of equations ω(H) < ω(G), ω(G/H) = ω(G). Our criterion to estimate the relative exponential growth rate is basically [39, Crite- rion 2.4]. The statement that we actually need is more specific, so we will give a proof for the convenience of the reader. Recall that the action of a subgroup H ⩽ G on X is divergent if its Poincaré series PH(s) diverges at s = ω(H). Proposition 1.2.1 ([39, Criterion 2.4]). — Assume that the following conditions are true: (i) ω(H) < ∞. (ii) The action of H on X is divergent. 49 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element (iii) There exist subgroups K ⩽ G and F ⩽ H ∩K so that F is a proper finite subgroup of K and the natural homomorphism ϕ : H ∗F K → G is injective. Then ω(H) < ω(G). Remark 1.2.2. — In the proof below, note that the relative exponential growth rate makes sense for any subset of G, as it does the notion of Poincaré series. Proof. — Since the action of H on X is divergent, in particular H is infinite and hence H − F is non-empty. Since F is a proper subgroup of K, there exists k ∈ K − F . Denote by U the set of elements of H ∗F K that can be written as words that alternate elements of H − F and k, always with an element of H − F at the beginning and with a k at the end. The inequality ω(ϕ(U)) ⩽ ω(G) can be deduced from the definition. It is enough to prove that there exists s0 ⩾ 0 such that ω(H) < s0 ⩽ ω(ϕ(U)). Let o ∈ X. Since ω(H) < ∞, the interval (ω(H),∞) is non-empty. Since the action of H on X is divergent, there exists s0 ∈ (ω(H),∞) such that ∑h∈H−F e −s0|o−hko| > 1; otherwise one obtains a contradiction with the divergence of the action of H on X. In order to obtain the inequality s0 ⩽ ω(ϕ(U)), it suffices to show that the Poincaré series Pϕ(U)(s) = ∑ g∈ϕ(U) e −s|o−go| diverges at s = s0. Since ϕ : H ∗F K → G is injective, we have Pϕ(U)(s) ⩾ ∑ m⩾1 ∑ h1,··· ,hm∈H−F e−s|o−h1kh2k···hmko|. By the triangle inequality, for every m ⩾ 1 and for every h1, · · · , hm ∈ H − F , we have |o− h1kh2k · · ·hmko| ⩽ ∑m i=1 |o− hiko|. Thus, ∑ h1,··· ,hm∈H−F e−s|o−h1kh2k···hmko| ⩾  ∑ h∈H−F e−s|o−hko| m . We see that PH(s0) = ∞ follows from the claim. Our criterion to estimate the quotient exponential growth rate is the following: Definition 1.2.3. — Let ϕ : G → G. We say that G is ϕ-coarsely G/H if there exist θ ⩾ 0 and x ∈ X satisfying the following conditions: (CQ1) For every u, v ∈ G, if ϕ(u)H = ϕ(v)H, then |ϕ(u)x− ϕ(v)x| ⩽ θ. (CQ2) For every u ∈ G, |ux− ϕ(u)x| ⩽ θ. Proposition 1.2.4. — If there exist ϕ : G → G such that G is ϕ-coarsely G/H, then ω(G) = ω(G/H). 50 1.2. Growth estimation criteria Proof. — The inequality ω(G/H) ⩽ ω(G) can be deduced from the defintion. Assume that there exist ϕ : G → G such that G is ϕ-coarsely G/H for x ∈ X and θ ⩾ 0. Claim 1.2.5. — There exist κ ⩾ 1 such that for every r > 0, | StabG(x, r)| ⩽ κ|p(StabG(x, r + θ))|. Let κ = | StabG(x, 3θ)|. Let r > 0. Let p : G ↠ G/H be the natural projection. Let q : G → G/H the map that sends u to ϕ(u)H. Note that the quasi-stabilizer StabG(x, r) can be decomposed as the disjoint union of the sets q−1(q(u)) such that q(u) ∈ q(StabG(x, r)). Hence, | StabG(x, r)| ⩽ ∑ q(u)∈q(StabG(x,r)) |q−1(q(u))|. It suffices to estimate the size of q(StabG(x, r)) and the size of q−1(q(u)), for every u ∈ G. First we prove that |q(StabG(x, r))| ⩽ |p(StabG(x, r + θ))|. Let u ∈ StabG(x, r). By the triangle inequality, |x− ϕ(u)x| ⩽ |x− ux| + |ux− ϕ(u)x|. By the hypothesis (CQ2), we have |ux− ϕ(u)x| ⩽ θ. Hence |x− ϕ(u)x| ⩽ r + θ. Conse- quently, q(StabG(x, r)) ⊂ p(StabG(x, r + θ)). Now we prove that for every u ∈ G, we have |q−1(q(u))| ⩽ κ. Let u ∈ G. Since |u StabG(x, 3θ)| = | StabG(x, 3θ)| = κ, it is enough to prove that u−1q−1(q(u)) ⊂ StabG(x, 3θ). Let v ∈ q−1(q(u)). By the triangle inequality, |x− u−1vx| = |ux− vx| ⩽ |ux− ϕ(u)x| + |ϕ(u)x− ϕ(v)x| + |ϕ(v)x− vx|. Since q(u) = q(v), we have that ϕ(u)H = ϕ(v)H. It follows from the hypothesis (CQ1) that |ϕ(u)x−ϕ(v)x| ⩽ θ. By the hypothesis (CQ2), we have max{|ux−ϕ(u)x|, |vx−ϕ(v)x|} ⩽ θ. Thus, |x− u−1vx| ⩽ 3θ. This proves the claim. Consequently, ω(G) ⩽ lim sup r→∞ 1 r log |p(StabG(x, r + θ))|. Finally, observe that lim sup r→∞ 1 r log |p(StabG(x, r + θ))| = lim sup r→∞ r + θ r 1 r + θ log |p(StabG(x, r + θ))|. Hence ω(G) ⩽ ω(G/H). 51 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element 1.3 Buffering sequences In this section, we fix constants µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system space (X,P). Despite the fact that our space X does not carry any global geometric condition, we still can obtain some control through constricting subsets. We could ignore the “wild regions” if, for instance, we were able to “jump” from one constricting subset to another. The buffering sequences below encapsulate this idea. In fact, the proofs of our main results consist essentially in building up some particular buffering sequences. W. Yang had already introduced this concept for piece-wise geodesics in [77]. Definition 1.3.1. — Let δ, ε, L ⩾ 0. Let A be a collection of subsets of X. A finite sequence of subsets Y0, A1, Y1, · · · , An, Yn ⊂ X where Y0 and Yn are the only possible empty sets is (δ, ε, L)-buffering on A if for every i ∈ J1, nK the set Ai belongs to A and there exists a δ-constricting map πAi : X → Ai with the following properties whenever Yi and Yi−1 are non-empty: (BS1) max{diamAi (Ai+1), diamAi+1(Ai)} ⩽ ε if i ̸= n. (BS2) max{diamAi (Yi−1), diamAi (Yi)} ⩽ ε. (BS3) max{d(Ai, Yi−1), d(Ai, Yi)} ⩽ ε. (BS4) dAi (Yi−1, Yi) ⩾ L. What makes buffering sequences remarkable is that they satisfy a variant of Behrstock inequality. We will find a direct application of the following inequality later in the study of the quotient exponential growth rates: Proposition 1.3.2. — For every δ, ε ⩾ 0, there exists θ ⩾ 0 with the following property. Let A, Y,B ⊂ X be a (δ, ε, 0)-buffering sequence on {A,B}. Then for every x ∈ X, min {dA(x, Y ), dB(x, Y )} ⩽ θ. Proof. — Let δ, ε ⩾ 0. Let θ0 = θ0(δ) ⩾ 0 be the constant of Proposition 1.1.5. Let θ > θ0 + 1. Its exact value will be precised below. Let A, Y,B ⊂ X be a (δ, ε, 0)-buffering sequence on {A,B}. Let x ∈ X. By symmetry, it suffices to show that if dA(x, Y ) > θ, then dB(x, Y ) ⩽ θ. Assume that dA(x, Y ) > θ. Let a ∈ A such that |x−a|B ⩽ dB(x,A)+1. Let b ∈ B. Let y ∈ Y . By (BS3), we have max{d(A, Y ), d(B, Y )} ⩽ ε; hence there exist p ∈ A+ε+1 ∩ Y and q ∈ B+ε+1 ∩ Y . It follows from the definition of buffering sequence that max {|b− πB(q)|A, |q − p|A, |a− πA(p)|B, |p− y|B} ⩽ ε. 52 1.3. Buffering sequences Y1 Y2 Y3 A1 A2 A3 A4 ≥ L ≥ L ≥ L ≥ L Figure 1.2 – An example of a buffering sequence in the Poincaré disk model. In this example, the sets Ai are subpaths of length ⩾ L of a given bi-infinite geodesic α. Each set Yi is the collection of geodesics that are orthogonal to the geodesic segment of α that is between Ai and Ai+1. In particular, the sets Yi are quasi- convex. For more intuition, one could interpret this picture on a tree. 53 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Applying together Proposition 1.1.5 (1) Coarse nearest-point projection and (3) Coarse Lipschitz map, we obtain max{|πB(q) − q|A, |πA(p) − p|B} ⩽ µ2(ε+ 1) + µθ0 + θ0. Claim 1.3.3. — dA(x,B) > θ0 By the triangle inequality, |x− b|A ⩾ |x− p|A − |b− πB(q)|A − |πB(q) − q|A − |q − p|A. Moreover, |x− p|A ⩾ dA(x, Y ). Since the element b is arbitrary and we have dA(x, Y ) > θ0 + 1, we obtain dA(x,B) > θ0. This proves the claim. Finally, we are going to estimate dB(x, Y ). By the triangle inequality, |x− y|B ⩽ |x− a|B + |a− πA(p)|B + |πA(p) − p|B + |p− y|B. Since dA(x,B) > θ0, it follows from Proposition 1.1.5 (5) Behrstock inequality and the definition of a that |x−a|B ⩽ θ0+1. Since the element y is arbitrary, we obtain dB(x, Y ) ⩽ θ for θ = 2θ0 + 1 + 2ε+ µ2(ε+ 1) + µθ0. The corollary below will be applied to the study of the relative exponential growth rates: Corollary 1.3.4. — For every δ, ε, θ ⩾ 0 there exists L ⩾ 0 with the following property. Let Y0, A1, Y1, · · · , An, Yn ⊂ X be an (δ, ε, L)-buffering sequence on {Ai}. Then for every i ∈ J1, nK, dAi (Y0, Yi) > θ. Proof. — Let δ, ε, θ ⩾ 0. Let θ0 = θ0(δ, ε) ⩾ 0 be the constant of Proposition 1.3.2. We put L = θ + θ0 + 1. Let y0 ∈ Y0. Let i ∈ J1, nK. Claim 1.3.5. — dAi (y0, Yi) ⩾ dAi (Yi−1, Yi) − dAi (y0, Yi−1). Let yi−1 ∈ Yi−1 and yi ∈ Yi. By the triangle inequality, |y0 − yi|Ai ⩾ |yi−1 − yi|Ai − |y0 − yi−1|Ai . Note that |yi−1 − yi|Ai ⩾ dAi (Yi−1, Yi). Since the elements yi−1, yi are arbitrary, this proves the claim. 54 1.4. Quasi-convexity in the Intersection–Image property Finally, we prove by induction on i ∈ J1, nK that, dAi (Y0, Yi) > θ. If i = 1, then dA1(Y0, Y1) > θ follows from (BS4), since L > θ. Assume that i ∈ J1, n−1K and dAi (Y0, Yi) > θ. Then dAi (y0, Yi) > θ0. It follows from Proposition 1.3.2 that dAi+1(y0, Yi) ⩽ θ0. By (BS4), dAi+1(Yi, Yi+1) ⩾ L. Applying the previous claim, we obtain dAi+1(y0, Yi+1) > θ. Since the element y0 is arbitrary, dAi+1(Y0, Yi+1) > θ. This concludes the inductive step. 1.4 Quasi-convexity in the Intersection–Image prop- erty In this section, we fix constants µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system space (X,P). In this section, we prove a variant of Proposition 1.1.5 (4) Intersection–Image. Basically, we will be exchanging paths of P for quasi-convex subsets of X, further thickening the involved sets. Proposition 1.4.1. — For every δ, η ⩾ 0, there exist θ ⩾ 0 and ζ : R⩾0 × R⩾0 → R⩾0 with the following property. Let πA : X → A be a δ-constricting map. Let Y be an η-quasi-convex subset of X. Let ε1 ⩾ 0, ε2 ⩾ 0. Then | diam(A+θ+ε1 ∩ Y +ε2) − diamA(Y )| ⩽ ζ(ε1, ε2). Proof. — Let δ, η ⩾ 0. Let θ0 = θ0(δ) ⩾ 0 be the constant of Proposition 1.1.5. We put θ = δ+ η+ 1. Let ζ : R⩾0 × R⩾0 → R⩾0 depending on δ, η. Its exact value will be precised below. Let πA : X → A be a δ-constricting map. Let Y be an η-quasi-convex subset of X. Let ε1 ⩾ 0, ε2 ⩾ 0. First we prove that diamA(Y ) ⩽ diam(A+θ+ε1 ∩ Y +ε2) + ζ(ε1, ε2). Let x, y ∈ Y . It suffices to assume that |x − y|A > δ. Let γ ∈ P joining x to y. By (CS2), there exist p, q ∈ γ such that max{|πA(x) − p|, |πA(y) − q|} ⩽ δ. Since the subset Y is η-quasi-convex, there exist p′, q′ ∈ Y such that max{|p− p′|, |q − q′|} ⩽ η + 1. By the triangle inequality, |x− y|A ⩽ |πA(x) − p| + |p− p′| + |p′ − q′| + |q′ − q| + |q − πA(y)|. 55 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Since p′, q′ ∈ A+θ+ε1 ∩ Y +ε2 , we have |p′ − q′| ⩽ diam(A+θ+ε1 ∩ Y +ε2). Hence, |x− y|A ⩽ diam(A+θ+ε1 ∩ Y +ε2) + 2δ + 2η + 1. Now we prove that diam(A+θ+ε1 ∩ Y +ε2) ⩽ diamA(Y ) + ζ(ε1, ε2). Let x, y ∈ A+θ+ε1 ∩ Y +ε2 . Since x, y ∈ Y +ε2 , there exist x′, y′ ∈ Y such that max{|x − x′|, |y − y′|} ⩽ ε2 + 1. By the triangle inequality, |x− y| ⩽ |x− πA(x)| + |x− x′| + |x′ − y′|A + |y′ − y|A + |πA(y) − y|. Since x, y ∈ A+θ+ε1 , it follows from Proposition 1.1.5 (1) Coarse nearest-point projection that max{|x− πA(x)|, |y − πA(y)|} ⩽ µ(θ + ε1) + θ0. It follows from Proposition 1.1.5 (3) Coarse Lipschitz Map that, max{|x− x′|A, |y − y′|A} ⩽ µ(ε2 + 1) + θ0. Since πA(x′), πA(y′) ∈ πA(Y ), we have |x′ − y′|A ⩽ diamA(Y ). Hence, |x− y| ⩽ diamA(Y ) + 2µ(θ + ε1) + 2µ(ε2 + 1) + 4θ0. Finally, we put ζ(ε1, ε2) = max{2δ + 2η + 1, 2µ(θ + ε1) + 2µ(ε2 + 1) + 4θ0}. Applying the symmetry of Proposition 1.4.1 in combination with Proposition 1.1.5 (6) Morseness and (7) Coarse invariance, we deduce: Corollary 1.4.2. — For every δ ⩾ 0, there exists θ ⩾ 0 with the following property. Let πA : X → A and πB : X → B be δ-constricting maps. Then: | diamA(B) − diamB(A)| ⩽ θ. 1.5 Finding a quasi-convex element Given a torsion-free hyperbolic group G containing a loxodromic element g0 and an infinite index quasi-convex subgroup H, one can find another loxodromic element g ∈ G conjugate to g0 so that H has trivial intersection with ⟨g⟩ [6, Theorem 1]. The goal of this section is to reimplement this fact in our setting, using a “quasi-convex element” instead 56 1.5. Finding a quasi-convex element of a loxodromic element. In this section, we fix constants µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system group (G,X,P). Definition 1.5.1 (Quasi-convex element). — Let η ⩾ 0. An element g ∈ G is η-quasi- convex if the following holds: (QE1) g has infinite order. (QE2) ⟨g⟩ is an η-quasi-convex subgroup of G. We will write (g, A) when we need to stress the η-quasi-convex subset A that ⟨g⟩ is preserving. The main result of this section is the following. Proposition 1.5.2. — Let η ⩾ 0. Assume that G contains an η-quasi-convex element (g, A). There exists θ = θ(η, g, A) ⩾ 1 satisfying the following. Let (H, Y ) be an η-quasi- convex subgroup of G. Then: (i) For every u ∈ G, if diam(uA ∩ Y ) > θ, then uA ⊂ Y +θ. (ii) Let H ⩽ K ⩽ G. If [K : H] > θ, then there exist k ∈ K such that diam(kA∩Y ) ⩽ θ. Remark 1.5.3. — Under the notation of (ii), when K = G, the element kgk−1 has the desired property that we were looking for. Note that (kgk−1, kA) is quasi-convex since P is G-invariant. The rest of the section is devoted to the proof of Proposition 1.5.2. Definition 1.5.4. — Let κ ⩾ 1, l ⩾ 0. A map ϕ : (Y, dY ) → (Z, dZ) between two metric spaces is a (κ, l)-quasi-isometric embedding if for every y, y′ ∈ Y , 1 κ dY (y, y′) − l ⩽ dZ(ϕ(y), ϕ(y′)) ⩽ κdY (y, y′) + l. We start with a variant of Milnor-Schwarz Theorem. If U is a generating set of a group H, we denote by dU the word metric of H with respect to U . Lemma 1.5.5. — For every η ⩾ 0, there exist θ ⩾ 1 with the following property. Let (H, Y ) be an η-quasi-convex subgroup of G. For every y ∈ Y , there exists a finite generating set U of H such that the orbit map (H, dU ) → X, h 7→ hy is a (θ, θ)-quasi-isometric embedding. 57 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element For the proof, one can use the same kind of argument as that of Milnor-Schwarz Theorem, but bearing in mind that Y might not be a length metric space, which is required by the original statement. The only difference here is that one uses the paths of P with endpoints in Y . They are enough for the proof since they approximate sufficiently well the distances, at least in this situation. Lemma 1.5.6. — Let η ⩾ 0. Let H ⩽ G be an abelian subgroup. Let Y ⊂ X be an H-invariant subset so that the action of H on Y is η-cobounded. Then, for every h ∈ H and for every y, z ∈ Y , ∣∣∣|y − hy| − |z − hz| ∣∣∣ ⩽ 2η. Proof. — Let h ∈ H. Let y, z ∈ Y . Since the action of H on Y is η-cobounded, there exists k ∈ H such that |z − ky| ⩽ η. By the triangle inequality, |y − hy| ⩽ |ky − khy| ⩽ |ky − z| + |z − hz| + |hz − khy|. Since the subgroup H is abelian, |hz − khy| = |z − ky|. Thus, |y − hy| ⩽ |z − hz| + 2η. Finally, exchanging the roles of y and z, we obtain |y − hy| ⩾ |z − hz| − 2η. Next, we are going to check that we can obtain uniform quasi-isometric embeddings of Z in X via the orbit maps of quasi-convex elements of G that share the same constant. For this reason, we introduce the following definition: Definition 1.5.7. — Let g ∈ G. Let x ∈ X. The stable translation length of g is ∥g∥∞ = lim sup m→∞ 1 m |gmx− x|. Note that ∥g∥∞ does not depend on the choice of the point x ∈ X. Remark 1.5.8. — Let g ∈ G. By subadditivity, for every x ∈ X, we have ∥g∥∞ = inf m⩾1 1 m |gmx− x| = lim m→∞ 1 m |gmx− x|. Lemma 1.5.9. — Let η ⩾ 0. Let g ∈ G. Let A ⊂ X be a ⟨g⟩-invariant subset so that the action of ⟨g⟩ on A is η-cobounded. The following statements are equivalent: (i) There exists x ∈ X such that the orbit map Z → X, m 7→ gmx is a quasi-isometric embedding. (ii) ∥g∥∞ > 0. 58 1.5. Finding a quasi-convex element (iii) There exists θ = θ(η, g, A) ⩾ 1 such that for every a ∈ A, the orbit map Z → X, m 7→ gma is a (θ, 0)-quasi-isometric embedding. Proof. — The implication (iii) ⇒ (i) already holds. (i) ⇒ (ii). Assume that there exists x ∈ X such that the orbit map Z → X, m 7→ gmx is a quasi-isometric embedding. Then there exist κ ⩾ 1, l ⩾ 0 such that for every m ⩾ 1, 1 κ − l m ⩽ 1 m |x− gmx| ⩽ κ+ l m . Therefore, ∥g∥∞ ⩾ 1 κ > 0. (ii) ⇒ (iii). Assume that ∥g∥∞ > 0. Let ∥g∥A = infa∈A |a − ga|. Then we can define θ = max { ∥g∥A + 2η, 1 ∥g∥∞ , 1 } . Let a ∈ A. Applying the triangle inequality we obtain that for every m ∈ Z, |a− gma| ⩽ |a− ga||m|. It follows from Lemma 1.5.6 that |a−ga| ⩽ ∥g∥A +2η. Since ∥g∥∞ = infn∈Z−{0} 1 |n| |a−g|n|a|, we obtain that for every m ∈ Z, |a− gma| ⩾ ∥g∥∞ |m|. Hence the orbit map Z → X, m 7→ gma is a (θ, 0)-quasi-isometric embedding. Lemma 1.5.10. — Let η ⩾ 0. Let (g, A) be an η-quasi-convex element of G. There exists θ = θ(η, g, A) ⩾ 1 such that for every a ∈ A, the orbit map Z → X, m 7→ gma is a (θ, 0)-quasi-isometric embedding. Moreover, ∥g∥∞ > 0. Proof. — We are going to apply Lemma 1.5.5 and Lemma 1.5.9. Let a ∈ A. According to Lemma 1.5.5, there exist a finite generating set U of ⟨g⟩ such that the orbit map ϕ : (⟨g⟩, dU) → X, h 7→ ha is a quasi-isometric embedding. Furthermore, since g has infinite order, the map χ : Z → ⟨g⟩, m 7→ gm is an isomorphism. Let V = χ−1(U). In particular χ : (Z, dV ) → (⟨g⟩, dU) is an isometry. Morover, the map ψ : Z → (Z, dV ) is a quasi-isometric embedding. Hence, the composition ϕ◦χ◦ψ is a quasi-isometric embedding. Now both of the statements of the lemma follow from Lemma 1.5.9. We continue by upper bounding the length of a quasi-geodesic of X by the number of points of an orbit of a subgroup H of G that fall inside a precise neighbourhood of this quasi-geodesic, whenever the quasi-geodesic falls also inside a neighbourhood of that orbit. Lemma 1.5.11. — For every η ⩾ 0, κ ⩾ 1, l ⩾ 0, there exists θ ⩾ 1 with the following property. Let H ⩽ G. Let Y ⊂ X be an H-invariant subset such that the action of H on Y is η-cobounded. Let y ∈ Y . Let γ be a (κ, l)-quasi-geodesic of X such that γ ⊂ Y +η. 59 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Let U = {u ∈ H : uy ∈ γ+2η+1}. Then ℓ(γ) ⩽ θ|U |. Proof. — Let η ⩾ 0, κ ⩾ 1, l ⩾ 0. Let θ = θ(η, κ, l) ⩾ 1. Its exact value will be precised below. Let H, Y , y, γ : [0, L] → X and U as in the statement. Let m = ⌊ L θ ⌋ + 1. We fix a partition 0 = t0 ⩽ t1 ⩽ · · · ⩽ tm = L of [0, L] such that |tm−1 − tm| ⩽ θ and such that if m ⩾ 2, then for every i ∈ J0,m − 2K, we have |ti − ti+1| = θ. Hence ℓ(γ) = L ⩽ θm. We prove that m ⩽ |U |. Let i ∈ J0,m− 1K. Denote xi = γ(ti). Since the action of H on Y is η-cobounded and γ ⊂ Y +η, for every i ∈ J0,m − 1K, there exists hi ∈ H such that |xi − hiy| ⩽ 2η + 1. In particular, hi ∈ U . From now on we may assume that m ⩾ 2, otherwise there is nothing to show. Let i, j ∈ J0,m− 1K such that i ̸= j. We claim that hi ̸= hj . The claim will follow when we show that |hiy−hjy| > 0. By the triangle inequality, |hiy − hjy| ⩾ |xi − xj| − |xi − hiy| − |xj − hjy|. Since γ is a (κ, l)-quasi-geodesic, |xi − xj| ⩾ 1 κ |ti − tj| − l κ . Since i, j ∈ J0,m− 1K, we have that |ti − tj| ⩾ θ. To sum up, |hiy − hjy| ⩾ θ κ − l κ − 4η − 2. Finally, we put θ = κ ( l κ + 4η + 2 ) + 1. Hence, |hiy − hjy| > 0. In particular, we obtain m ⩽ |U |. The following fact is a direct consequence of the triangle inequality: Lemma 1.5.12. — Let η ⩾ 0. Let H ⩽ G. Let Y ⊂ X be an H-invariant subset so that the action of H on Y is η-cobounded. Then, for every y, z ∈ Y , there exists h ∈ H such that for every r > 0, h−1 StabG(y, r)h ⊂ StabG(z, r + 2η). Finally, we show that there is a uniform threshold that ensures the existence of a uniformly short element in the intersection of any pair of quasi-convex subgroups of G that share the same constant. 60 1.5. Finding a quasi-convex element Lemma 1.5.13. — For every η ⩾ 0, there exists θ ⩾ 1 with the following property. Let (H,Y ) and (K,Z) be η-quasi-convex subgroups of G. If diam(Y ∩Z) > θ, then there exist y ∈ Y ∩ Z and h ∈ H ∩K ∩ StabG(y, θ) − {1G}. Proof. — Let η ⩾ 0. Let θ0 = θ0(η, µ, ν) ⩾ 1 be the constant of Lemma 1.5.11. Let o ∈ Y . We denote W = StabG(o, 6η + 2). Let θ1 = θ0|W | + θ0. Note that the constant θ1 is finite since the action of G on X is proper. We put θ = 2θ1 + 4η + 2. Let (H, Y ) and (K,Z) be η-quasi-convex subgroups of G. Assume that diam(Y ∩ Z) > θ. Since diam(Y ∩ Z) > θ1, there exist y, z ∈ Y ∩ Z such that |y − z| > θ1. Let β ∈ P joining y to z. Since ℓ(β) > θ1, there exist z′ ∈ β and a subpath γ of β joining y to z′ such that ℓ(γ) = θ1. We denote U = {u ∈ H : uy ∈ γ+2η+1} and V = StabG(y, 4η + 2). The first step is to construct a map ϕ : U → V . Let u ∈ U . By definition of U , there exists x ∈ γ such that |uy − x| ⩽ 2η + 1. Since the subgroup (K,Z) is η-quasi-convex, there exists ku ∈ K such that |x− kuy| ⩽ 2η + 1. By the triangle inequality, |uy − kuy| ⩽ |uy − x| + |x− kuy|. Consequently, |u−1kuy − y| ⩽ 4η + 2. Hence, u−1ku ∈ V . We define ϕ : U → V to be the map that sends every u ∈ U to u−1ku ∈ V . Next, we show that the map ϕ : U → V is not injective. Since Y is η-quasi-convex, we have that γ ⊂ β ⊂ Y +η. It follows from Lemma 1.5.11 that |U | ⩾ 1 θ0 ℓ(γ). By hypothesis, ℓ(γ) = θ0|W |+θ0. Since the action of H on Y is η-cobounded, it follows from Lemma 1.5.12 that there exists h ∈ H such that h−1V h ⊂ W and hence |W | ⩾ |h−1V h| = |V |. Consequently, |U | > |V |. Therefore, the map ϕ : U → V is not injective. Now we claim that U ⊂ StabG(y, θ1 + 2η + 1). Let u ∈ U . By definition of U , there exists x ∈ γ such that d|x− uy| ⩽ 2η + 1. By the triangle inequality, |y − uy| ⩽ |y − x| + |x− uy|. Moreover, |y − x| ⩽ ℓ(γ) = θ1. Hence |y − uy| ⩽ θ1 + 2η + 1. Finally, since the map ϕ : U → V is not injective, there exist u1, u2 ∈ U such that u1 ≠ u2 and u−1 1 ku1 = u−1 2 ku2 . In particular, u2u −1 1 ∈ H ∩K − {1G}. Further, according to the triangle inequality, |y − u2u −1 1 y| ⩽ |y − u2y| + |u2y − u2u −1 1 y|. 61 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element It follows from the claim above that |y − u2u −1 1 y| ⩽ θ. Therefore, u2u −1 1 ∈ H ∩ K ∩ StabG(y, θ) − {1G}. We are ready to prove the proposition: Proof of Proposition 1.5.2. — Let η ⩾ 0. Assume that G contains an η-quasi-convex element (g, A). We are going to determine the value of θ = θ(η, g, A) ⩾ 1. By Lemma 1.5.10, there exists θ0 = θ0(η, g, A) ⩾ 1 such that for every a ∈ A, the orbit map Z → X, m 7→ gma is a (θ0, 0)-quasi-isometric embedding. Let θ1 = θ1(η) ⩾ 1 be the constant of Lemma 1.5.13. Let θ2 = η + θ2 0θ1. Let o ∈ A. We denote U = StabG(o, 2θ2 + η + 1). Let θ = max{θ2, |U |}. Note that the constant θ is finite since the action of G on X is proper. Let (H,Y ) be an η-quasi-convex subgroup of G. (i) Let u ∈ G. Assume that diam(uA ∩ Y ) > θ. Let a ∈ A. We prove that ua ∈ Y +θ2 . Since P is G-invariant, the element (ugu−1, uA) is η-quasi-convex. Since diam(uA∩ Y ) > θ1, according to Lemma 1.5.13, there exist b ∈ A and M ∈ Z − {0} such that ub ∈ uA ∩ Y and ugMu−1 ∈ H ∩ StabG(ub, θ1). Since the action of ⟨g⟩ on A is η-cobounded, there exists m ∈ Z such that |a− gmb| ⩽ η. By Euclid’s division Lemma, there exist q, r ∈ Z such that m = qM + r and 0 ⩽ r ⩽ |M | − 1. By the triangle inequality, d(ua, Y ) ⩽ |ua− ugqMb| ⩽ |ua− ugmb| + |ugmb− ugqMb|. Note that |ua− ugmb| = |a− gmb| ⩽ η. Moreover, it follows from Lemma 1.5.10 that |ugmb− ugqMb| = |grb− b| ⩽ θ0|r|. Note also that |r| ⩽ |M |. Applying again Lemma 1.5.10, we obtain that |M | ⩽ θ0|gMb− b|. By Lemma 1.5.13, |gMb− b| = |ugMu−1ub− ub| ⩽ θ1. Hence, d(ua, Y ) ⩽ θ2 ⩽ θ. (ii) Let H ⩽ K ⩽ G. We argue by contraposition. Assume that for every k ∈ K, we have diam(kA ∩ Y ) > θ. We prove that [K : H] ⩽ |U |. It follows from (i) that KA ⊂ Y +θ2 . Then there exists y ∈ Y such that |o − y| ⩽ θ2 + 1. Since the action of H on Y is η-cobounded, we have that Y ⊂ (Hy)+η. Hence Ko ⊂ (Hy)+θ2+η. In particular, for every k ∈ K, there exists hk ∈ H such that |ko− hky| ⩽ θ2 + η. Let 62 1.6. Constricting elements K ′ be a set of representatives of the set H\K of right cosets of H. Then the set K ′′ = {h−1 k k : k ∈ K ′} is a set of representatives of H\K. We claim that K ′′ ⊂ U . Let k ∈ K ′. By the triangle inequality, |h−1 k ko− o| = |ko− hko| ⩽ |ko− hky| + |hky − hko|. Thus, |h−1 k ko− o| ⩽ 2θ2 + η + 1. This proves the claim. Consequently, [K : H] ⩽ |K ′′| ⩽ |U | ⩽ θ. 1.6 Constricting elements Hypothesis and conventions for this section. We fix: ▶ Constants µ ⩾ 1 and ν, δ ⩾ 0. ▶ A (µ, ν)-path system group (G,X,P). ▶ A δ-constricting element (g, A). ▶ A δ-constricting map πA : X → A. 1.6.1 A G-invariant family The set of G-translates of A is a G-invariant family of δ-constricting subsets. Indeed, consider the stabilizer Stab(A) of A and fix a set Rg of representatives of G/ Stab(A). Let u ∈ G and u0 ∈ Rg such that uA = u0A. The map πuA : X → uA defined as ∀x ∈ X, πuA(x) = u0πA(u−1 0 x). is then δ-constricting since P is G-invariant. Moreover, the element (ugu−1, uA) is δ- constricting. To cope with the possible lack of ⟨ugu−1⟩-equivariance of the map πuA : X → uA, we make the following observation: Proposition 1.6.1. — There exists θ ⩾ 0 satisfying the following. Let u ∈ G. Then: (i) For every x ∈ X, we have |πuA(x) − uπA(u−1x)| ⩽ δ. (ii) For every Y ⊂ X, we have | diamuA(Y ) − diam(uπA(u−1Y ))| ⩽ θ. 63 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Proof. — Let θ0 = θ0(δ) ⩾ 0 be the constant of Proposition 1.1.5. We put θ = 2θ0. Let u ∈ G. (i) Let x ∈ X. Denote y = u−1x. Let u0 ∈ Rg such that uA = u0A. We see that, |πuA(x) − uπA(u−1x)| = |u0πA(u−1 0 x) − uπA(u−1x)| = |πA(u−1 0 uy) − u−1 0 uπA(y)|. Since u−1 0 u ∈ Stab(A), it follows from Proposition 1.1.5 (2) Coarse equivariance that |πuA(x) − uπA(u−1x)| ⩽ θ0. (ii) Let Y ⊂ X. Let y, y′ ∈ Y . By the triangle inequality, ∣∣∣|πuA(y) − πuA(y′)|−|uπA(u−1y) − uπA(u−1y′)| ∣∣∣ ⩽ |πuA(y) − uπA(u−1y)| + |uπA(u−1y′) − πuA(y′)|. It follows from (i) that max { |uπuA(y) − uπA(u−1y)|, |uπA(u−1y′) − πuA(y′)| } ⩽ θ0. Hence, we have | diamuA(Y ) − diam(uπA(u−1Y ))| ⩽ 2θ0. 1.6.2 Finding a constricting element The goal of this subsection is to combine Proposition 1.5.2 and Proposition 1.4.1. We suggest to compare (ii) below with the property (BS2) of the buffering sequences. Proposition 1.6.2. — Let η ⩾ 0. There exists θ ⩾ 1 satisfying the following. Let (H,Y ) be an η-quasi-convex subgroup of G. Then: (i) For every u ∈ G, if diamuA(Y ) > θ, then uA ⊂ Y +θ. (ii) Let H ⩽ K ⩽ G. If [K : H] > θ, then there exists k ∈ K such that diamkA(Y ) ⩽ θ. Proof. — Let η ⩾ 0. Let θ = θ(η) ⩾ 1. Its exact value will be precised below. It follows from Proposition 1.1.5 (6) Morseness and (7) Coarse invariance that there exists θ0 ⩾ 0 such that the element (g, A) is θ0-quasi-convex. Let θ1 = max{η, θ0}. By Proposition 1.4.1, there exist θ2 ⩾ 0, ζ ⩾ 0 depending both on θ1 such that for every u ∈ G and for every θ1-quasi-convex subset Y ⊂ X, we have diamuA(Y ) − ζ ⩽ diam(uA+θ2 ∩ Y ) ⩽ diamuA(Y ) + ζ. 64 1.6. Constricting elements According to Proposition 1.1.5 (6) Morseness and (7) Coarse invariance, there exist θ3 = θ3(θ2) ⩾ 0 such that the element (g, A+θ2) is θ3-quasi-convex. Let θ4 = max{η, θ3}. Let θ5 = θ5(θ4, g, A) ⩾ 1 be the constant of Proposition 1.5.2. Finally, we put θ = θ5 + ζ. Let (H, Y ) be an η-quasi-convex subgroup of G. (i) Let u ∈ G. Assume that diamuA(Y ) > θ. According to Proposition 1.4.1, we have diam(uA+θ2 ∩ Y ) > θ5 and according to Proposition 1.5.2 (i) this implies that uA ⊂ Y +θ5 ⊂ Y +θ. (ii) Let H ⩽ K ⩽ G. We argue by contraposition. Assume that for every k ∈ K, we have diamkA(Y ) > θ. According to Proposition 1.4.1, for every k ∈ K, we have diam(kA+θ2 ∩ Y ) > θ5 and according to Proposition 1.5.2 (ii) this implies that [K : H] ⩽ θ5 ⩽ θ. 1.6.3 Elementary closures The elementary closure of (g, A) could be thought as the set of elements u ∈ G such that uA is “parallel” to A: Definition 1.6.3. — The elementary closure of (g, A) in G is defined as E(g, A) = {u ∈ G : dHaus(uA,A) < ∞}. Observe that E(g, A) is a subgroup of G since dHaus is a pseudo-distance. This subsection is devoted to provide a further description E(g, A). We suggest to compare the proposition below with the property (BS1) of the buffering sequences. Proposition 1.6.4. — There exists θ ⩾ 1 satisfying the following: (i) For every u ∈ G, we have max{diamuA(A), diamA(uA)} > θ ⇐⇒ dHaus(uA,A) ⩽ θ. (ii) E(g, A) = {u ∈ G : dHaus(uA,A) ⩽ θ}. (iii) [E(g, A) : ⟨g⟩] ⩽ θ. 65 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Proof. — Let θ0 ⩾ 0 be the constant of Proposition 1.6.1. According to Proposition 1.1.5 (6) Morseness, there exists θ1 ⩾ 0 such that the element (g, A) is θ1-quasi-convex. Let θ2 = θ2(θ1) ⩾ 1 be the constant of Proposition 1.6.2. We put θ = θ0 + θ2. Claim 1.6.5. — Let u ∈ G. If dHaus(uA,A) < ∞, then diamuA(A) = ∞. Let u ∈ G. Assumme that dHaus(uA,A) < ∞ and denote ε = dHaus(uA,A) + 1. By Proposition 1.4.1, there exist θ3, ζ ⩾ 0 such that for every u ∈ G we have diamuA(A) − ζ ⩽ diam(uA+θ3 ∩ A+ε) ⩽ diamuA(A) + ζ. Note that uA ⊂ uA+θ3 ∩ A+ε and diam(uA) = diam(A). Since the action of G on X is proper and since the element g has infinite order, we have that diam(A) = ∞. Consequently, we have diam(uA+θ3 ∩ A+ε) = ∞. Finally, it follows from Proposition 1.4.1 that diamuA(A) = ∞. This proves the claim. (i) Let u ∈ G. Assume that max{diamuA(A), diamA(uA)} > θ. By Proposition 1.6.1, diamu−1A(A) ⩾ diamA(u−1πA(uA)) − θ0. Hence, diamu−1A(A) > θ2. It follows from Proposition 1.6.2 (i) that uA ⊂ A+θ and u−1A ⊂ A+θ. Hence dHaus(uA,A) ⩽ θ. The converse follows from the claim above. (ii) This follows from (i) and the claim above. (iii) This follows from (i), (ii) and Proposition 1.6.2 (ii). Finally, we obtain an algebraic description of E(g, A). Corollary 1.6.6. — There exist θ ⩾ 1 and M ∈ J1, θK such that for every u ∈ G, the following statements are equivalent: (i) u ∈ E(g, A). (ii) There exists p ∈ {−1, 1} such that ugMu−1 = gpM . (iii) There exist m,n ∈ Z − {0} such that ugmu−1 = gn. Further, let E+(g, A) = {u ∈ G : ugMu−1 = gM}. Then [E(g, A) : E+(g, A)] ⩽ 2. Proof. — By Proposition 1.6.4 (ii), there exists θ0 ⩾ 1 such that [E(g, A) : ⟨g⟩] ⩽ θ0. Let θ = θ0! We construct M ∈ J1, θK. First, we claim that there exists a subgroup 66 1.6. Constricting elements K ⩽ ⟨g⟩ such that K ⊴ E(g, A) and [E(g, A) : K] ⩽ θ. Consider the natural action of E(g, A) by right multiplication on the set ⟨g⟩\E(g, A) of right cosets of ⟨g⟩. This gives an homomorphism ϕ : E(g, A) → Sym(⟨g⟩\E(g, A)). Choose K = Ker(ϕ). Note that ⟨g⟩ = {h ∈ E(g, A) : ϕ(h)(⟨g⟩)} = ⟨g⟩. Thus, K ⩽ ⟨g⟩. Morover, K ⊴ E(g, A). Further, we have that |Sym(⟨g⟩\E(g, A))| = [E(g, A) : ⟨g⟩]! and hence [E(g, A) : K] divides [E(g, A) : ⟨g⟩]! Therefore, [E(g, A) : K] ⩽ θ. This proves the claim. Now, since the element g has infinite order, the subgroup E(g, A) is infinite. Hence, since [E(g, A) : K] < ∞ there exists M ⩾ 1 such that K = ⟨gM⟩. Finally, we remark that M is equal to the order of the element ϕ(g). Hence, M ⩽ θ. Let u ∈ G. The implication (ii) ⇒ (iii) already holds. (i) ⇒ (ii). Assume that u ∈ E(g, A). Since the subgroup ⟨gM⟩ is normal in E(g, A), there exists p ∈ Z such that ugMu−1 = gpM . In particular, ⟨gM⟩ = u⟨gM⟩u−1 = ⟨ugMu−1⟩ = ⟨gpM⟩. Hence, if p ̸∈ {−1,+1}, then ⟨gM⟩ ̸⊂ ⟨gpM⟩. Contradiction. (iii) ⇒ (i). Assume that there exist m,n ∈ Z − {0} such that ugmu−1 = gn. Since both ⟨gm⟩ and ⟨gn⟩ have finite index in ⟨g⟩, there exist ζ ⩾ 0 the actions of ⟨ugmu−1⟩ on uA and of ⟨gn⟩ on A are both ζ-cobounded. Let x ∈ uA and y ∈ A. We obtain dHaus(uA,A) ⩽ ζ + |x− y|. Hence dHaus(uA,A) < ∞. Finally, let E+(g, A) = {u ∈ G : ugMu−1 = gM}. We prove that [E(g, A) : E+(g, A)] ⩽ 2. It is enough to assume that E(g, A) ̸= E+(g, A). Let u, v ∈ E(g, A) − E+(g, A). We show that v−1u ∈ E+(g, A). Since ugMu−1 = vgMv−1 = g−M , we have v−1ugMu−1v = v−1g−Mv = gM and therefore v−1u ∈ E+(g, A). Hence [E(g, A) : E+(g, A)] = 2 1.6.4 Forcing a geometric separation In this subsection, we build large powers of our constricting element (g, A) to produce a translate Y ′ of a subset Y so that the distance between their projections to a preferred G-translate of A is large. We will do it in two different ways. We will apply these results to verify (BS4) in the construction of buffering sequences. Our main tool will be: Lemma 1.6.7. — There exists θ ⩾ 0 such that for every x, x′ ∈ X and for every m ∈ Z, |x− gmx′|A ⩾ |m| ∥g∥∞ − |x− x′|A − θ. 67 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element Proof. — Let θ = θ(δ) ⩾ 0 be the constant of Proposition 1.1.5. Let x, x′ ∈ X. Let m ∈ Z. If m = 0, then there is nothing to do. Assume that m ̸= 0. By the triangle inequality, |x− gmx′|A ⩾ |πA(x) − gmπA(x)| − |x− x′|A − |gmπA(x′) − πA(gmx′)|. Note that 1 |m| |πA(x) − gmπA(x)| ⩾ inf n⩾1 1 n |πA(x) − gnπA(x)| = ∥g∥∞ . By Proposition 1.1.5 (2) Coarse equivariance, we have |gmπA(x′)−πA(gmx′)| ⩽ θ. Therefore, we have |x− gmx′|A ⩾ |m| ∥g∥∞ − |x− x′|A − θ. The first way of forcing a geometric separation will be applied to the study of the relative exponential growth rates: Proposition 1.6.8. — For every ε, θ ⩾ 0, there exists M ⩾ 1 with the following property. Let H ⩽ G be a subgroup. Let Y ⊂ X be an H-invariant subset. If diamA(Y ) ⩽ ε, then for every u ∈ ⟨gM , H ∩ E(g, A)⟩ −H ∩ E(g, A), we have dA(Y, uY ) > θ. Proof. — Let ε, θ ⩾ 0. Let θ0 ⩾ 0 be the constant of Proposition 1.1.5. By Lemma 1.6.7, there exists θ1 ⩾ 0 such that for every x, x′ ∈ X and for every m ∈ Z, |x− gmx′|A ⩾ |m| ∥g∥∞ − |x− x′|A − θ1. Combining Lemma 1.5.10 and Proposition 1.1.5 (6) Morseness, we obtain ∥g∥∞ > 0. According to Corollary 1.6.6, there exists M0 ⩾ 1 such that E(g, A) = { u ∈ G : ∃ p ∈ {−1,+1}ugM0u−1 = gpM0 } . Let m0 > θ−2ε−2θ0−θ1 M0∥g∥∞ . We put M = M0m0. Let H ⩽ G be a subgroup. Let Y ⊂ X be an H-invariant subset. Assume that diamA(Y ) ⩽ ε. Let u ∈ ⟨gM , H ∩ E(g, A)⟩ − H ∩ E(g, A) and y, y′ ∈ Y . It follows from Corollary 1.6.6 that there exists n ∈ Z multiple of M and f ∈ H ∩ E(g, A) such that u = gnf . By the triangle inequality, |y − gnfy′|A ⩾ |y − gny′|A − |πA(gny′) − gnπA(y′)| − |y′ − fy′|A − |gnπA(fy′) − πA(gnfy′)|. 68 1.6. Constricting elements By Lemma 1.6.7, |y − gny′|A ⩾ |n| ∥g∥∞ − |y − y′|A − θ1 Note that u ̸∈ H∩E(g, A) implies n ̸= 0. Hence |n| ⩾ |M |. Since f ∈ H and diamA(Y ) ⩽ ε, max{|y − y′|A, |y′ − fy′|A} ⩽ ε. By Proposition 1.1.5 (2) Coarse equivariance, max{|πA(gny′) − gnπA(y′)|, |gnπA(fy′) − πA(gnfy′)|} ⩽ θ0. Since the elements y, y′ are arbitrary, we obtain dA(Y, uY ) > θ. The second way of forcing a geometric separation will be applied to the study of the quotient exponential growth rates: Proposition 1.6.9. — For every ε, θ ⩾ 0, there exist M ⩾ 1 and f : G×X → {1G, g M} with the following property. Let Y ⊂ X be subset. If diamA(Y ) ⩽ ε, then for every u ∈ G and for every y ∈ Y , we have duA(y, uf(u, y)Y ) > θ. Proof. — Let ε, θ ⩾ 0. Let θ0 ⩾ 0 be the constant of Proposition 1.6.1. By Lemma 1.6.7, there exists θ1 ⩾ 0 such that for every x, x′ ∈ X and for every m ∈ Z, |x− gmx′|A ⩾ |m| ∥g∥∞ − |x− x′|A − θ1. Combining Lemma 1.5.10 and Proposition 1.1.5 (6) Morseness, we obtain ∥g∥∞ > 0. We put M > 2θ+2ε+8θ0+θ1 ∥g∥∞ . Then, for every u ∈ G and for every x ∈ X, there exists f(u, x) ∈ {1G, g M} such that |u−1x−f(u, x)|A > θ+ε+4θ0: if |u−1x−x|A > θ+ε+4θ0, we choose f(u, x) = 1G, otherwise we choose f(u, x) = gM . This defines f : G×X → {1G, g M}. Let Y ⊂ X be a subset. Assume that diamA(Y ) ⩽ ε. Let u ∈ G. Let y, y′ ∈ Y . By abuse of notation, we write f instead of f(u, y). By the triangle inequality, |y − ufy′|uA ⩾ |y − ufy|uA − |ufy − ufy′|uA, |y − ufy|uA ⩾ |u−1y − fy|A − |πuA(y) − uπA(u−1y)| − |πuA(ufy) − uπA(fy)|, |ufy − ufy′|uA ⩽ |πuA(ufy) − ufπA(y)| + |y − y′|A + |ufπA(y′) − πuA(ufy′)|. By hypothesis, |u−1y − fy|A > θ + ε + 4θ0 and |y − y′|A ⩽ diamA(Y ) ⩽ ε. By Proposi- 69 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element tion 1.6.1, max{|πuA(y) − uπA(u−1y)|, |πuA(ufy) − uπA(fy)|} ⩽ θ0. max{|πuA(ufy) − ufπA(y)|, |ufπA(y′) − πuA(ufy′)|} ⩽ θ0. Since the element y′ is arbitrary, we obtain duA(y, ufY ) > θ. 1.7 Growth of quasi-convex subgroups In this section, our first goal is to prove Theorem 0.5.8. This result can be deduced from Proposition 1.2.1 and Proposition 1.7.1 below. Our second goal is to prove Theorem 0.5.13. This result can be deduced from Proposition 1.2.4 and Proposition 1.7.3 below. Hypothesis and conventions for this section. We fix: ▶ Constants µ ⩾ 1 and ν, δ, η ⩾ 0. ▶ A (µ, ν)-path system group (G,X,P). ▶ A δ-constricting element (g0, A0). ▶ An infinite index η-quasi-convex subgroup (H, Y ) of G. We are going to replace the axis A0 for A′ 0 = E(g0, A0)A0. As a consequence of Proposition 1.6.4 (ii), we have dHaus(A0, A ′ 0) < ∞. Up to replacing δ for a larger constant, it follows from Proposition 1.1.5 (7) Coarse invariance and Corollary 1.6.6 that the element (g0, A ′ 0) is δ-constricting. By abuse of notation, we still denote A0 = A′ 0. In this new setting, for every k ∈ E(g0, A0), we have kA0 = A0. Let θ0 = θ0(δ, η) ⩾ 1 be the constant of Proposition 1.6.2. Since [G : H] = ∞, it follows from Proposition 1.6.2 (ii) that there exist u ∈ G such that diamuA0(Y ) ⩽ θ0. We denote (g, A) = (ug0u −1, uA0). Proposition 1.7.1 (Theorem 0.5.10). — There exist M ⩾ 1 such that the natural homo- morphism H ∗H∩E(g,A) ⟨gM , H ∩ E(g, A)⟩ → G is injective. Remark 1.7.2. — It follows from Proposition 1.4.1 and Proposition 1.6.4 that the subgroup H ∩ E(g, A) is finite. By Proposition 1.6.4, the subgroup E(g, A) is a finite extension of ⟨g⟩. Hence the proposition proves Theorem 0.5.10. Since g has infinite order, the finite subgroup H ∩ E(g, A) is a proper subgroup of ⟨gM , H ∩ E(g, A)⟩. Hence we can apply Proposition 1.2.1 to deduce Theorem 0.5.8. 70 1.7. Growth of quasi-convex subgroups Proof. — Let θ1 = θ1(δ) ⩾ 0 be the constant of Proposition 1.6.1. Let ε = max{θ0 + 2θ1, d(A, Y )}. Let L = L(δ, ε, 0) ⩾ 0 be the constant of Corollary 1.3.4. By Proposition 1.6.8, there exists M ⩾ 1 such that for every u ∈ ⟨gM , H ∩ E(g, A)⟩ − H ∩ E(g, A), we have dA(Y, uY ) > L− 2θ1. Let ϕ : H ∗H∩E(g,A) ⟨gM , H ∩ E(g, A)⟩ → G be the natural homomorphism. Let w ∈ H ∗H∩E(g,A) ⟨gM , H ∩E(g, A)⟩ such that w ̸= 1. We are going to prove that ϕ(w) ̸= 1. Note that the homomorphisms ϕ|H and ϕ|⟨gM ,H∩E(g,A)⟩ are injective. If w ∈ H∪⟨gM , H∩E(g, A)⟩, then ϕ(w) ̸= 1. Assume that w ̸∈ H ∪ ⟨gM , H ∩ E(g, A)⟩. Note that if there exists a conjugate w′ of w such that ϕ(w′) ̸= 1, then ϕ(w) ̸= 1. Up to replacing w by a cyclic conjugate, there exist n ⩾ 1 and a sequence h1, k1, · · · , hn, kn ∈ G such that w = h1k1 · · ·hnkn and such that for every i ∈ {1, · · · , n} we have hi ∈ H − H ∩ E(g, A) and ki ∈ ⟨gM , H ∩E(g, A)⟩ −H ∩E(g, A). For every i ∈ J1, nK, we denote ui = h1k1 · · ·hi and vi = h1k1 · · ·hiki. We also denote v0 = 1G. We are going to prove that the sequence v0Y, u1A, v1Y, · · · , unA, vnY is (δ, ε, L)- buffering on {uiA} and then apply Corollary 1.3.4. Let i ∈ J1, nK. Let us prove (BS1). Assume for a moment that i ̸= n. Since we had modified the axis A0 above, for every j ∈ J1, nK, we have kjA = A. Hence πuiA(ui+1A) = πviA(ui+1A), πui+1A(uiA) = πui+1A(viA). By Proposition 1.6.1, diamviA(ui+1A) ⩽ diam(viπA(hiA)) + θ1, diamui+1A(viA) ⩽ diam(ui+1πA(h−1 i A)) + θ1, diamA(h−1 i A) ⩽ diamhiA(A) + θ1. By Proposition 1.6.4 (i) and (ii), for every u ̸∈ E(g, A), we have max{diamA(uA), diamuA(A)} ⩽ θ0. Consequently, max{diamuiA(ui+1A), diamui+1A(uiA)} ⩽ θ0 + 2θ1 ⩽ ε. Let us prove (BS2). Note that, πuiA(vi−1Y ) = πuiA(uiY ), 71 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element πuiA(viY ) = πviA(viY ). By Proposition 1.6.1, diamuiA(uiY ) ⩽ diam(uiπA(Y )) + θ1, diamviA(viY ) ⩽ diam(viπA(Y )) + θ1. Since diamA(Y ) ⩽ θ0, we obtain max{diamuiA(vi−1Y ), diamuiA(viY )} ⩽ θ0 + θ1 ⩽ ε. Let us prove (BS3). We have, max{d(uiA, vi−1Y ), d(uiA, viY )} = max{d(uiA, uiY ), d(viA, viY )} ⩽ d(A, Y ) ⩽ ε. Let us prove (BS4). It follows from Proposition 1.6.1 (i) that, duiA(vi−1Y, viY ) ⩾ dA(Y, kiY ) − 2θ1. By the choice of M , we have dA(Y, kiY ) > L+ 2θ1. Hence, we have duiA(vi−1Y, viY ) ⩾ L. This proves that the sequence v0Y, u1A, v1Y, · · · , unA, vnY is (δ, ε, L)-buffering on {uiA}. It follows from Corollary 1.3.4 that dunA(Y, ϕ(w)Y ) > 0. Hence, ϕ(w) ̸= 1. Recall that given ϕ : G → G, we say that G is ϕ-coarsely G/H if there exist θ ⩾ 0, x ∈ X satisfying the following conditions: (CQ1) For every u, v ∈ G, if ϕ(u)H = ϕ(v)H, then |ϕ(u)x− ϕ(v)x| ⩽ θ. (CQ2) For every u ∈ G, |ux− ϕ(u)x| ⩽ θ. Proposition 1.7.3. — There exist M ⩾ 1 and a map f : G → {1G, g M} with the following property. Let ϕ : G → G, u 7→ ufu. Then G is ϕ-coarsely G/H. We prove some preliminar lemmas. Lemma 1.7.4. — There exists θ ⩾ 0 such that for every m ∈ Z, we have diamA(gmY ) ⩽ θ. Proof. — Let θ1 ⩾ 0 be the constant of Proposition 1.1.5. We put θ = θ0 + 2θ1. Let m ∈ Z. 72 1.7. Growth of quasi-convex subgroups Let x, x′ ∈ Y . By the triangle inequality, |gmx− gmx′|A ⩽ |πA(gmx) − gmπA(x)| + |x− x′|A + |gmπA(x′) − πA(gmx′)|. By Proposition 1.1.5 (2) Coarse equivariance, max{|πA(gmx) − gmπA(x)|, |gmπA(x′) − πA(gmx′)|} ⩽ θ1. Moreover, we have |x − x′|A ⩽ diamA(Y ) ⩽ θ0. Since x, x′ are arbitrary, we obtain diamA(gmY ) ⩽ θ0 + 2θ1. Lemma 1.7.5. — For every ε ⩾ 0, there exists θ ⩾ 0 with the following property. Let A1, A2 ⊂ X be δ-constricting subsets such that dHaus(A1, A2) ⩽ ε. Let x ∈ A+ε 1 and y ∈ A+ε 2 such that |x− y|A1 ⩽ ε. Then |x− y| ⩽ θ. Proof. — Let θ1 ⩾ 0 be the constant of Proposition 1.1.5. Let ε ⩾ 0. Let θ ⩾ 0. Its exact value will be precised below. Let A1, A2 ⊂ X be δ-constricting subsets such that dHaus(A1, A2) ⩽ ε. Let x ∈ A+ε 1 and y ∈ A+ε 2 such that |x − y|A1 ⩽ ε. By the triangle inequality, |x− y| ⩽ |x− πA1(x)| + |x− y|A1 + |πA1(y) − y|. Since x, y ∈ A+2ε+1 1 , it follows from Proposition 1.1.5 (1) Coarse nearest-point projection that max{|x− πA1(x)|, |πA1(y) − y|} ⩽ µ(2ε+ 1) + θ1. Finally, we put θ = ε+ 2µ(2ε+ 1) + 2θ1. We are ready to prove Proposition 1.7.3: Proof of Proposition 1.7.3. — Let θ1 ⩾ 0 be the constant of Proposition 1.6.1. Let θ2 ⩾ 0 be the constant of Proposition 1.6.4. Let θ3 ⩾ 0 be the constant of Lemma 1.7.4. Let ε = max{θ2 + 2θ1, θ1 + θ3, d(A, Y ) + 1}. In particular, there exists y ∈ A+ε ∩ Y . Let θ4 = θ4(δ, ε) ⩾ 0 be the constant of Proposition 1.3.2. By Proposition 1.6.9, there exist M ⩾ 1 and f : G → {1G, g M} such that for every u ∈ G, we have duA(y, uf(u)Y ) > θ4. For every u ∈ G, we denote fu = f(u) and we put ϕ : G → G, u 7→ ufu. Let θ5 = θ5(ε) ⩾ 0 be the constant of Lemma 1.7.5. We put θ = max{|y − gMy|, θ5}. We are going to prove that G is ϕ-coarsely G/H with respect to y and θ. 73 Chapter 1 – Growth of quasi-convex subgroups in groups with a constricting element In order to prove (CQ1), we just need to observe that for every u ∈ G, we have |uy − ufuy| = |y − fuy| ⩽ |y − gMy| ⩽ θ. Let us prove (CQ2). Let u, v ∈ G. Assume that ufuH = vfvH. We claim that dHaus(uA, vA) ⩽ θ2. By Proposition 1.6.4 (i), it suffices to prove that max{diamv−1uA(A), diamA(v−1uA)} > θ2. We argue by contradiction. Assume instead that max{diamv−1uA(A), diamA(v−1uA)} ⩽ θ2. We are going to prove that the sequence uA, ufuY, vA is (δ, ε, 0)-buffering on {uA, vA} and then apply Proposition 1.3.2. Note that the condition (BS4) is void in this case. Let us prove (BS1). By Proposition 1.6.1, diamuA(vA) ⩽ diam(uπA(u−1vA)) + θ1, diamvA(uA) ⩽ diam(vπA(v−1uA)) + θ1, diamA(u−1vA) ⩽ diamv−1uA(A) + θ1. Hence, max{diamuA(vA), diamvA(uA)} ⩽ θ2 + 2θ1 ⩽ ε. Let us prove (BS2). By Proposition 1.6.1, diamuA(ufuY ) ⩽ diam(uπA(fuY )) + θ1, diamvA(vfvY ) ⩽ diam(vπA(fvY )) + θ1. By Lemma 1.7.4, we have max{diamA(fuY ), diamA(fvY )} ⩽ θ3. Hence, max{diamuA(ufuY ), diamvA(vfvY )} ⩽ θ1 + θ3 ⩽ ε. Let us prove (BS3). The hypothesis ufuH = vfvH implies ufuY = vfvY and therefore max{d(uA, ufuY ), d(vA, ufuY )} = max{d(uA, ufuY ), d(vA, vfvY )} = d(A, Y ) ⩽ ε. Hence, the sequence uA, ufuY, vA is (δ, ε, 0)-buffering on {uA, vA}. It follows from Propo- sition 1.3.2 that min {duA(y, ufuY ), dvA(y, ufuY )} ⩽ θ4. 74 1.7. Growth of quasi-convex subgroups However, by construction, min {duA(y, ufuY ), dvA(y, ufuY )} > θ4. Contradiction. Therefore, dHaus(uA, vA) ⩽ θ2. This proves the claim. In particular, dHaus(uA, vA) ⩽ ε. Since y ∈ A+ε, we have ufuy ∈ uA+ε and vfvy ∈ vA+ε. Since ufuy, vfvy ∈ ufuY , we have |ufuy−vfvy|uA ⩽ diamuA(ufuY ) ⩽ ε. According to Lemma 1.7.5, |ufuy − vfvy| ⩽ θ. This proves (CQ2). 75 Chapter 2 UNIFORM UNIFORM EXPONENTIAL GROWTH IN SMALL CANCELLATION GROUPS Words are pale shadows of forgotten names. As names have power, words have power. Words can light fires in the minds of men. Words can wring tears from the hardest hearts. There are seven words that will make a person love you. There are ten words that will break a strong man’s will. But a word is nothing but a painting of a fire. A name is the fire itself. from The Name of the Wind, of Patrick Rothfus Contents 2.1 Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . 78 2.1.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.1.2 Quasi-convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.3 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.1.4 Group action on a δ-hyperbolic space . . . . . . . . . . . . . . 81 2.1.5 Small cancellation theory . . . . . . . . . . . . . . . . . . . . . 85 2.2 Reduced subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.1 Broken geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.2 Quasi-isometric embedding of a free group . . . . . . . . . . . . 91 2.2.3 Geodesic extension property . . . . . . . . . . . . . . . . . . . . 92 2.3 Growth in groups acting on a δ-hyperbolic space . . . . . . . 95 2.3.1 Growth of maximal loxodromic subgroups. . . . . . . . . . . . 95 2.3.2 Producing reduced subsets . . . . . . . . . . . . . . . . . . . . 99 2.3.3 Growth trichotomy . . . . . . . . . . . . . . . . . . . . . . . . . 102 77 Chapter 2 – Uniform uniform exponential growth in small cancellation groups 2.4 Shortening and shortening-free words . . . . . . . . . . . . . . 105 2.4.1 Shortening words . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.4.2 The growth of shortening-free words . . . . . . . . . . . . . . . 112 2.4.3 The injection of shortening-free words . . . . . . . . . . . . . . 117 2.5 Growth in small cancellation groups . . . . . . . . . . . . . . . 120 The results of this chapter correspond to the following article: — X. Legaspi and M. Steenbock. Uniform uniform exponential growth in small cancel- lation groups, 2023. URL: https://orcid.org/0000-0002-1497-6448. In Section 2.1 we will overview Gromov hyperbolic spaces, acylindricity and geometric small cancellation theory. In Section 2.2 we will see that reduced subsets generate free subgroups with the Geodesic Extension Property. This property will be relevant to the counting argument of Section 2.4.2. In Section 2.3 we generalise work of M. Koubi [56] and G. Arzhantseva - I. Lysenok, [9]. The goal is to produce reduced subsets inside uniform powers of other subsets of isometries. In Section 2.4 we study the subsets of shortening-free words of a free subgroup generated by a reduced subset. These are infinite subsets, each depending on a geometric small cancellation family, such that (i) their elements are not killed when taking the geometric small cancellation quotient and (ii) their relative growth rate does not decrease too much when taking the geometric small cancellation quotient. We will prove (i) and (ii) in Section 2.4.2 and Section 2.4.3, respectively. Finally, Section 2.5 is devoted to the proof of our main theorem (Theorem 0.6.2). 2.1 Hyperbolic geometry We collect some facts on hyperbolic geometry in the sense of Gromov, [49], including its version of small cancellation theory, [50, 41]. See also [26, 46, 52, 32]. 2.1.1 Hyperbolicity Let X be a metric space. The Gromov product of three points x, y, z ∈ X is defined by (x, y)z = 1 2{|x− z| + |y − z| − |x− y|}. 78 https://orcid.org/0000-0002-1497-6448 2.1. Hyperbolic geometry Definition 2.1.1. — Let δ ⩾ 0. The metric space X is δ-hyperbolic if it is geodesic and for every x, y, z and t ∈ X, the four point inequality holds, that is (x, z)t ⩾ min {(x, y)t, (y, z)t} − δ. Convention 2.1.2. — Let δ ⩾ 0. For the remainder of this section, we assume that the space X is δ-hyperbolic. If δ = 0, then it can be isometrically embedded in an R-tree, [46, Chapitre 2, Proposition 6]. Note that X is δ′-hyperbolic for every δ′ ⩾ δ. In this chapter we always assume for convenience that the hyperbolicity constant δ is positive. We write ∂X for the Gromov boundary of X. We can use the boundary defined with sequences converging at infinity, [26, Chapitre 2, Définition 1.1]. Note that we did not assume the space X to be proper, thus we use the boundary defined with sequences converging at infinity, [26, Chapitre 2, Définition 1.1]. Hyperbolicity has the following consequences. Lemma 2.1.3 ([42, Lemmas 2.3 and 2.4]). — Let x, y, z ∈ X. Then (x, y)z ⩽ d(z, [x, y]) ⩽ (x, y)z + 4δ. Lemma 2.1.4 ([9, Lemma 2]). — Let i ∈ J1, 2K. Let xi, yi ∈ X. Then |x1 − y1| + |x2 − y2| ⩽ |x1 − x2| + |y1 − y2| + 2 diam([x1, y1]+8δ ∩ [x2, y2]+8δ). 2.1.2 Quasi-convexity Let η ⩾ 0. A subset Y ⊂ X is η-quasi-convex if every geodesic joining two points of Y is contained in Y +η. For instance, geodesics are 2δ-quasi-convex. A subset Y ⊂ X is strongly quasi-convex if it is 2δ-quasi-convex and for every y, y′ ∈ Y , the induced path metric | · |Y on Y satisfies |y − y′|X ⩽ |y − y′|Y ⩽ |y − y′|X + 8δ. Quasi-convexity in hyperbolic spaces has the following consequences. Lemma 2.1.5 ([26, Chapitre 1, Proposition 3.1];[42, Lemma 2.4]). — Let η ⩾ 0. Let 79 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Y ⊂ X be an η-quasi-convex subset. Then for every x ∈ X and for every y, y′ ∈ Y , d(x, Y ) ⩽ (y, y′)x + η + 3δ. Given a point x ∈ X and a subset Y ⊂ X, then y ∈ Y is a projection of x on Y if |x− y| ⩽ d(x, Y ) + δ. Lemma 2.1.6 ([26, Chapitre 2, Proposition 2.1];[30, Lemma 2.12]). — Let η ⩾ 0. Let Y ⊂ X be an η-quasi-convex subset. (i) Let x ∈ X. Let y be a projection of x on Y . Then for every y′ ∈ Y , (x, y′)y ⩽ η + δ. (ii) Let i ∈ J1, 2K. Let xi ∈ X. Let yi be a projection of xi on Y . Then, |y1 − y2| ⩽ max {|x1 − x2| − |x1 − y1| − |x2 − y2| + 2ε, ε}, where ε = 2η + 3δ. Lemma 2.1.7 ([26, Chapitre 10, Proposition 1.2]; [30, Lemma 2.13]). — Let η ⩾ 0. Let Y ⊂ X be an η-quasi-convex subset. Then for every ε ⩾ η, the subset Y +ε is 2δ-quasi- convex. Lemma 2.1.8 ([41, Lemma 2.2.2 (2)]; [30, Lemma 2.16]). — Let i ∈ J1, 2K. Let ηi ⩾ 0. Let Yi ⊂ X be an ηi-quasi-convex subset. Then for every ε ⩾ 0, diam(Y +ε 1 ∩ Y +ε 2 ) ⩽ diam(Y +η1+3δ 1 ∩ Y +η2+3δ 2 ) + 2ε+ 4δ. 2.1.3 Isometries Let G be a group acting by isometries on X. Let x ∈ X be a point. Classification of isometries. Recall that an isometry g ∈ G is either elliptic, i.e. the orbit ⟨g⟩ · x is bounded, loxodromic, i.e. the map Z → X sending m to gmx is a quasi- isometric embedding or parabolic, i.e. it is neither loxodromic or elliptic, [26, Chapitre 9, Théorème 2.1]. Note that these definitions do not depend on the point x. 80 2.1. Hyperbolic geometry Translation lengths. To measure the action of an isometry g ∈ G on X we define the translation length and the stable translation length as ∥g∥ = inf x∈X |gx− x|, and ∥g∥∞ = lim n→+∞ 1 n |gnx− x|. Note that the definition of ∥g∥∞ does not depend on the point x. These two lengths are related as follows, [26, Chapitre 10, Proposition 6.4]. ∥g∥∞ ⩽ ∥g∥ ⩽ ∥g∥∞ + 16δ. (2.1.1) The isometry g is loxodromic if, and only if, its stable translation length is positive, [26, Ch. 10, Prop. 6.3]. Axis. The axis of g ∈ G is the set Ag = {x ∈ X : |gx− x| ⩽ ∥g∥ + 8δ }. Lemma 2.1.9 ([41, Proposition 2.3.3];[30, Proposition 2.28]). — Let g ∈ G. Then Ag is 10δ-quasi-convex and ⟨g⟩-invariant. Moreover, for every x ∈ X, ∥g∥ + 2d(x,Ag) − 10δ ⩽ |gx− x| ⩽ ∥g∥ + 2d(x,Ag) + 10δ. ℓ∞-Energy. To measure the action of a finite subset of isometries U ⊂ G on X we define the ℓ∞-energy of U at x and the ℓ∞-energy of U as L(U, x) = max u∈U |ux− x|, and L(U) = inf x∈X L(U, x). The point x is almost-minimizing the ℓ∞-energy of U if L(U, x) ⩽ L(U) + δ. It is easy to see that the translation length and the ℓ∞-energy are related as follows. For every g ∈ U , ∥g∥ ⩽ L(U). (2.1.2) 2.1.4 Group action on a δ-hyperbolic space Let G be a group acting by isometries on X. 81 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Classification of group actions. We denote by ∂G the set of all accumulation points of an orbit G · x in the boundary ∂X. This set does not depend on the point x. One says that the action of G on X is ▶ elliptic, if ∂G is empty, or equivalently if one (hence any) orbit of G is bounded; ▶ parabolic, if ∂G contains exactly one point; ▶ loxodromic, if ∂G contains exactly two points; ▶ non-elementary, if ∂G contains at least 3 points, or equivalently if ∂G is infinite. If the action of G is elliptic, parabolic or loxodromic, we will say that this action is elementary. In this context, being elliptic (respectively parabolic, loxodromic, etc) refers to the action of G on X. However, if there is no ambiguity we will simply say that G is elliptic (respectively parabolic, loxodromic, etc). Lemma 2.1.10 ([31, Propositon 3.6]). — If |∂G| ⩾ 2, then G contains a loxodromic isometry. Acylindricity. For our purpose we require some properness for this action. We will use an acylindrical action on a metric space, keeping in mind the parameters that appear in the definition, [38, Proposition 5.31]. Recall that we assumed X to be δ-hyperbolic, with δ > 0. Definition 2.1.11 (Acylindrical action). — Let κ, N > 0. The group G acts (κ,N)- acylindrically on the δ-hyperbolic space X if the following holds: for every x, y ∈ X with |x− y| ⩾ κ, the number of elements u ∈ G satisfying |ux− x| ⩽ 100δ and |uy − y| ⩽ 100δ is bounded above by N . Definition 2.1.12 (Global injectivity radius).— The global injectivity radius of the action of G on X is T(G,X) = inf{ ∥g∥∞ : g ∈ G loxodromic }, with the convention inf ∅ = +∞. Lemma 2.1.13 ([18, Lemma 4.2]; c.f. [35, Lemma 3.9]). — Assume that the action of G on X is (κ,N)-acylindrical. Then T(G,X) ⩾ δ N . 82 2.1. Hyperbolic geometry Loxodromic subgroups. Let H ⩽ G be a loxodromic subgroup with limit set ∂H = {ξ, η}. The H-invariant cylinder, denoted by CH , is the open 20δ-neighborhood of all 103δ-local (1, δ)-quasi-geodesics with endpoints ξ and η at infinity. Lemma 2.1.14 (Invariant cylinder; [31, Lemma 3.13]). — Let H ⩽ G be a loxodromic subgroup. Then the subset CH is invariant under the action of H and strongly quasiconvex. Lemma 2.1.15 ([30, Corollary 2.7]). — Let γ : I → X be a 103δ-local (1, δ)-quasi-geodesic. Then: (i) For every t, t′, s ∈ I such that t ⩽ s ⩽ t′, we have (γ(t), γ(t′))γ(s) ⩽ 6δ. (ii) For every x ∈ X and for every y, y′ ∈ γ, we have d(x, γ) ⩽ (y, y′)x + 9δ. The maximal loxodromic subgroup containing H is the stabiliser of the set ∂H. For a loxodromic element g ∈ G, we denote by E(g) the maximal loxodromic subgroup containing g. We define the equivalence relation ∼g on G by u ∼g v if and only if u−1v ∈ E(g), for every u, v ∈ G. The fellow travelling constant of a loxodromic element g ∈ G is ∆(g) = sup{ diam(uA+20δ g ∩ vA+20δ g ) : u, v ∈ G, u ̸∼g v }. Lemma 2.1.16 ([38, Proof of Proposition 6.29]). — Assume that the action of G on X is (κ,N)-acylindrical. Let g ∈ G be a loxodromic element. Then ∆(g) ⩽ κ+ (N + 2)∥g∥∞ + 100δ. Lemma 2.1.17 ([38, Lemma 6.5]). — Assume that the action of G on X is acylindrical. Let g ∈ G be a loxodromic element. Then E(g) is virtually cyclic. The subgroup H+ ⩽ G fixing pointwise ∂H is an at most index 2 subgroup of H. The next corollary is a well-known consequence of Lemma 2.1.10, Lemma 2.1.17 and [76, Lemma 4.1]. Corollary 2.1.18. — Assume that the action of G on X is acylindrical. The set F of all elements of finite order of H+ is a finite normal subgroup of H. Moreover there exists a loxodromic element h ∈ H+ such that the map F ⋊ϕ ⟨h⟩ → H+ that sends (f, g) to fg is an isomorphism, where ϕ : ⟨h⟩ → Aut(F ) is the action by conjugacy of ⟨h⟩ on F . For a loxodromic element g ∈ G, we denote by F (g) the set of all elements of finite order of E+(g). We say that g is primitive if its image in E+(g)/F (g) generates the quotient. 83 Chapter 2 – Uniform uniform exponential growth in small cancellation groups The following lemma permits to produce primitive loxodromic elements uniformly. It will be useful during section section 2.3. Lemma 2.1.19 ([56]; [9]; [45, Lemma 2.7]). — For every κ > 0 and N > 0 there exists a positive integer n0 with the following property. Let U ⊂ G be a finite symmetric subset containing the identity. Assume that the action of G on X is (κ,N)-acylindrical. If L(U) > 50δ, then there exist a primitive loxodromic element g ∈ Un0 such that ∥g∥∞ ⩾ 1 2 L(U). Definition 2.1.20 (Loxodromic wideness). — The loxodromic wideness of the action of G on X is Φ(G,X) = sup{ |F (g)| : g ∈ G loxodromic }, with the convention sup∅ = −∞. Lemma 2.1.21 ([66, Lem. 6.8]). — Assume that the action of G on X is (κ,N)-acylindrical. Then Φ(G,X) ⩽ N. Classification of acylindrical actions. Following the proof of D. Osin [66, Theo- rem 1.1], one gets the following classification. It already appears in [49]. Lemma 2.1.22. — Assume that the action of G on X is acylindrical. Then G satisfies exactly one of the following three conditions. (i) G is elliptic, or equivalently one (hence any) orbit of G is bounded. (ii) G is loxodromic, or equivalently G is virtually cyclic and contains a loxodromic element. (iii) G is non-elementary, or equivalently H contains a free group F2 of rank 2 and one (hence any) orbit of F2 is unbounded. In particular, if the action of G on X is acylindrical, then every isometry g ∈ G is either elliptic or loxodromic, [18]. The following trichotomy is a direct consequence of the previous lemma and [19, Theorem 13.1]. Lemma 2.1.23. — Let G be a group acting acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Then one of the following conditions holds: 84 2.1. Hyperbolic geometry (T’1) L(U) ⩽ 104δ. (T’2) The subgroup ⟨U⟩ is virtually cyclic and contains a loxodromic element. (T’3) ω(U) ⩾ 1 103 log 3. 2.1.5 Small cancellation theory Let G be a group acting by isometries on X. We recall that X is a δ-hyperbolic space. Loxodromic moving family. The following definition generalises the conjugacy closure of a symmetrised set of relations in classical small cancellation theory. Definition 2.1.24 (Loxodromic moving family). — A loxodromic moving family Q is a set of the form Q = { (g ⟨h⟩ g−1, gCh) ∈ Q : g ∈ G, h ∈ L }, where L ⊂ G is a set of loxodromic elements and Ch stands for the ⟨h⟩-invariant cylinder. Let Q be a loxodromic moving family. The fellow travelling constant of Q is ∆(Q, X) = sup{ diam(Y +20δ 1 ∩ Y +20δ 2 ) : (H1, Y1) ̸= (H2, Y2) ∈ Q }. The injectivity radius of Q is T(Q, X) = inf{ ∥h∥ : h ∈ H − {1}, (H,Y ) ∈ Q }. Note that here we require the translation length and not the stable translation length, which was present in the definition of the global injectivity radius T(G,X). We denote K = ⟨⟨H | (H,Y ) ∈ Q⟩⟩ and Ḡ = G/K. We denote by π : G↠ Ḡ the natural projection and write ḡ for π(g) for short, for every g ∈ G. The notation Ū may refer to either a subset of Ḡ or to π(U), for some U ⊂ G. Definition 2.1.25 (Small cancellation condition). — Let λ > 0 and ε > 0. We say that Q satisfies the geometric C ′′(λ, ε)-small cancellation condition if: (SC1) ∆(Q, X) < λT(Q, X), (SC2) T(Q, X) > εδ. In that case we say that Ḡ is a geometric C ′′(λ, ε)-small cancellation quotient. 85 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Cone-off space. Let ρ > 0. We denote by Y the collection of cylinders gCh such that g ∈ G and h ∈ L . Let Y ∈ Y . Note that gCh = Cghg−1 . The cone of radius ρ over Y , denoted by Zρ(Y ), is the quotient of Y × [0, ρ] by the equivalence relation that identifies all the points of the form (y, 0). The apex of the cone Zρ(Y ) is the equivalence class of (y, 0). By abuse of notation, we still write (y, 0) for the equivalence class of (y, 0). We denote by V the collection of apices of the cones over the elements of Y . Let ι : Y ↪→ Zρ(Y ) be the map that sends y to (y, ρ). The cone-off space of radius ρ over X relative to Q, denoted by Ẋρ = Ẋρ(Q, X), is the space obtained by attaching for every Y ∈ Y , the cone Zρ(Y ) on X along Y according to ι : Y ↪→ Zρ(Y ). There is a natural metric on Ẋρ(Q) and an action by isometries of G on Ẋρ. Quotient space. The quotient space of radius ρ over X relative to Q, denoted by X̄ρ = X̄ρ(Q, X), is the orbit space Ẋρ/K. We denote by ζ : Ẋρ ↠ X̄ρ the natural projection and write x̄ for ζ(x) for short. Furthermore, we denote by V̄ the image in X̄ρ of the apices V . We consider X̄ρ as a metric space equipped with the quotient metric, that is for every x, x′ ∈ Ẋρ |x̄− x̄′|X̄ = inf h∈K |hx− x′|Ẋ . We note that the action of G on Ẋρ induces an action by isometries of Ḡ on X̄ρ. Convention 2.1.26. — In what follows, we are going to assume that X is a metric graph whose edges all have the same constant length. This is to ensure that both the cone-off space Ẋρ and the quotient space X̄ρ are geodesic spaces, [20, I.7.19]. This is not a restrictive assumption, as explained in [38, Section 5.3]. The following lemma summarises Proposition 3.15 and Theorem 6.11 of [30]. It will be central in the proof of Theorem 0.6.2. Lemma 2.1.27 (Small Cancellation Theorem [30]). — There exist positive numbers δ0, δ̄, ∆0, ρ0 satisfying the following. Let 0 < δ ⩽ δ0 and ρ > ρ0. Let G be a group acting by isometries on a δ-hyperbolic space X. Let Q be a loxodromic moving family such that ∆(Q, X) ⩽ ∆0 and T(Q, X) > 100π sinh ρ. Then: (i) X̄ρ is a δ̄-hyperbolic space on which Ḡ acts by isometries. (ii) Let r ∈ (0, ρ/20]. If for all v ∈ V , the distance |x − v| ⩾ 2r then the projection ζ : Ẋρ → X̄ρ induces an isometry from B(x, r) onto B(x̄, r). 86 2.1. Hyperbolic geometry (iii) Let (H,Y ) ∈ Q. If v ∈ V stands for the apex of the cone Zρ(Y ), then the natural projection π : G↠ Ḡ induces an isomorphism from Stab(Y )/H onto Stab(v̄). Remark 2.1.28. — It is important to note that in this statement the constants δ0, δ̄, ∆0, ρ0 are independent of G, X, Q or δ. Moreover δ0 and ∆0 (respectively ρ0) can be chosen arbitrarily small (respectively large). We will refer to δ0, δ̄, ∆0, ρ0 as the constants of the Small Cancellation Theorem. For the remainder of this subsection, we choose δ, ρ, G, X, and Q satisfying the hypothesis of the Small Cancellation Theorem (Lemma 2.1.27). The following lemmas are consequence of the Small Cancellation Theorem. Lemma 2.1.29 ([31, Proposition 5.16]). — Let E be an elliptic (respectively loxodromic) subgroup of G for its action on X. Then the image of E through the natural projection π : G↠ Ḡ is elliptic (respectively elementary) for its action on X̄ρ. Lemma 2.1.30 ([31, Proposition 5.17]). — Let E be an elliptic subgroup of G for its action on X. Then the natural projection π : G↠ Ḡ induces an isomorphism from E onto its image. Lemma 2.1.31 ([31, Proposition 5.18]). — Let Ē be an elliptic subgroup of Ḡ for its action on X̄ρ. One of the following holds. (i) There exists an elliptic subgroup E of G for its action on X such that the natural projection π : G↠ Ḡ induces an isomorphism from E onto Ē. (ii) There exists v̄ ∈ V̄ such that Ē ⊂ Stab(v̄). Lemma 2.1.32 ([36, Proposition 9.13]). — Let Ū ⊂ Ḡ be a finite set such that L(Ū) ⩽ ρ/5. If, for every v̄ ∈ V̄ , the set Ū is not contained in Stab(v̄), then there exists a pre-image U ⊂ G of Ū of energy L(U) ⩽ π sinh L(Ū). Lemma 2.1.33 (Greendlinger’s Lemma). — Let x ∈ X. Let g ∈ G. If g ∈ K − {1}, then there exists (H, Y ) ∈ Q with the following property. Let y0 an y1 be the respective projections of x and gx on Y . Then |y0 − y1| > T(H,X) − 2π sinh ρ− 23δ. Remark 2.1.34. — The previous statement is obtained from [33, Theorem 3.5] after applying [33, Proposition 1.11], [30, Proposition 2.4 (2)] and [30, Lemma 2.31]. Note that 87 Chapter 2 – Uniform uniform exponential growth in small cancellation groups in [33, Theorem 3.5] there is an extra assumption saying that the loxodromic moving family is finite up to conjugacy. That assumption is only needed to make sure that the action is co-compact, hence the quotient group hyperbolic. We don’t need it here. Lemma 2.1.35 ([38, Proposition 5.33]). — If the action of G on X is acylindrical, then so is the action of Ḡ on X̄ρ. 2.2 Reduced subsets Let δ ⩾ 0. In this section, we fix a group G acting by isometries on a δ-hyperbolic space X. The set of the inverses in G of the elements of U ⊂ G is represented by U−1. Definition 2.2.1. — Let α > 0. We say that a finite subset U ⊂ G is α-reduced at p ∈ X if U ∩ U−1 = ∅ and for every pair of distinct u1, u2 ∈ U ⊔ U−1, (u1p, u2p)p < 1 2 min{|u1p− p|, |u2p− p|} − α− 2δ. Remark 2.2.2. — If U ⊂ G is α-reduced at p ∈ X, then |up−p| > 2α, for every u ∈ U⊔U−1. We clarify some vocabulary. Let U ⊂ G be a subset. A letter is an element of the alphabet U ⊔U−1. A word over U ⊔U−1 is any finite sequence u1 · · ·un with ui ∈ U ⊔U−1. The number n is called the length of the the given word u1 · · ·un. We denote by |w|U the length of any word w over U ⊔ U−1. We admit the word of length 0, the empty word. We write w1 ≡ w2 to express letter-for-letter equality of words w1 and w2 over U ⊔ U−1. A word u1 · · ·un over U ⊔U−1 is reduced if it does not contain a pair of adjacent letters of the form uiu −1 i or u−1 i ui. The free group F(U) is the set of reduced words over U ⊔ U−1 with the group operation “concatenate and reduce”. The natural homomorphism ψ : F(U) → G is the evaluation of the elements of F(U) on G. 2.2.1 Broken geodesics The next lemma is used to produce quasi-geodesics by concatenating some sequences of points of X with geodesics. Lemma 2.2.3 (Broken Geodesic Lemma [9, Lemma 1]). — Let n ⩾ 2. Let x0, · · · , xn be a sequence of n+ 1 points of X. Assume that (xi−1, xi+1)xi + (xi, xi+2)xi+1 < |xi − xi+1| − 3δ, (2.2.1) 88 2.2. Reduced subsets for every i ∈ J1, n− 2K. Then the following holds. (i) |x0 − xn| ⩾ n−1∑ i=0 |xi − xi+1| − 2 n−1∑ i=1 (xi−1, xi+1)xi − 2(n− 2)δ. (ii) (x0, xn)xj ⩽ (xj−1, xj+1)xj + 2δ, for every j ∈ J1, n− 1K. (iii) The geodesic [x0, xn] lies in the 5δ-neighbourhood of the broken geodesic γ = [x0, x1] ∪ · · · ∪ [xn−1, xn], while γ is contained in the r-neighbourhood of [x0, xn], where r = sup 1⩽i⩽n−1 (xi−1, xi+1)xi + 14δ. Figure 2.1 – A sequence (xi) satisfying Equation 2.2.1. This sequence does not correspond to a reduced word over a reduced subset since for every i, the midpoint mi of the geodesic [xi−1, xi] falls inside the overlap of two consecutive geodesics. 89 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Figure 2.2 – Another sequence (xi) satisfying Equation 2.2.1. This sequence could correspond to a reduced word over an α- reduced subset since for every i, the midpoint mi of the geodesic [xi−1, xi] falls at distance at least α from the the overlap of two consecutive geodesics. The geodesic segments in red have length 2α. In particular, every geodesic [xi−1, xi] that does not fall in any of the two extremes has length at least 2α. We verify the condition of Lemma 2.2.3 permitting to obtain broken geodesics. Proposition 2.2.4. — Let α > 0. Let U ⊂ G be an α-reduced subset at p ∈ X. Let n ⩾ 2. Let w ≡ u1 · · ·un be an element of F(U). Consider the sequence of n+ 1 points x0 = p, x1 = u1p, x2 = u1u2p, · · · , xn = u1 · · ·unp. Then (i) (xi−1, xi+1)xi + (xi, xi+2)xi+1 < |xi − xi+1| − 2(α + 2δ), for every i ∈ J1, n− 2K. (ii) |wp− p| ⩾ 1 2 |u1p− p| + 1 2 |unp− p| + 2(n− 1)(α + δ) + 2δ. Proof. — (i) Let i ∈ J1, n− 2K. We have (xi−1, xi+1)xi = (u−1 i p, ui+1p)p, (xi, xi+2)xi+1 = (u−1 i+1p, ui+2p)p and |xi − xi+1| = |p − ui+1p|. Since w is a reduced word over U ⊔ U−1, we have u−1 i ̸= ui+1 and u−1 i+1 ̸= ui+2. Hence we can apply the fact that the subset U is α-reduced at p, obtaining (u−1 i p, ui+1p)p < 1 2 |ui+1p− p| − α− 2δ, (u−1 i+1p, ui+2p)p < 1 2 |u−1 i+1p− p| − α− 2δ. 90 2.2. Reduced subsets It remains to add the two above inequalities to obtain (xi−1, xi+1)xi + (xi, xi+2)xi+1 < |xi − xi+1| − 2(α + 2δ). (ii) Since n ⩾ 2, applying (i) and Lemma 2.2.3 (i) to the sequence x0, · · · , xn, we obtain |wp− p| ⩾ |u1p− p| + n−1∑ i=2 |uip− p| + |unp− p| − (u−1 1 p, u2p)p − n−1∑ i=2 [(u−1 i p, ui+1p)p + (u−1 i−1p, uip)p] − (u−1 n−1p, unp) − 2(n− 2)δ. Since U is α-reduced at p, n−1∑ i=2 [(u−1 i p, ui+1p)p + (u−1 i−1p, uip)p] < n−1∑ i=2 |uip− p| − 2(n− 2)(α + 2δ). and (u−1 1 p, u2p)p < 1 2 |u1p− p| − α− 2δ, (u−1 n−1p, unp) < 1 2 |unp− p| − α− 2δ. Consequently, |wp− p| ⩾ 1 2 |u1p− p| + 1 2 |unp− p| + 2(n− 1)(α + δ) + 2δ. 2.2.2 Quasi-isometric embedding of a free group Recall that L(U, p) denotes the ℓ∞-energy of U ⊂ G at p ∈ X (subsection 2.1.3). Proposition 2.2.5. — Let α > 0. Let U ⊂ G be an α-reduced subset at p ∈ X. Then, for every w ∈ F(U), we have 2α|w|U ⩽ |wp− p| ⩽ L(U, p)|w|U . In particular, the natural homomorphism ψ : F(U) → G is injective. 91 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Proof. — Let w ≡ u1 · · ·un be an element of F(U). If n = 0, then there is nothing to do. If n = 1, then the result is a direct consequence of the fact that the subset U is α-reduced. Assume that n ⩾ 2. It follows from the triangle inequality that |wp − p| ⩽ L(U, p)n. In regards to the second inequality, we apply Proposition 2.2.4 (ii) to the sequence of n+ 1 points x0 = p, x1 = u1, x2 = u1u2p, · · · , xn = wp = u1 · · ·unp, to obtain |wp− p| ⩾ 1 2 |u1p− p| + 1 2 |unp− p| + 2(n− 1)(α + δ) + 2δ. According to Remark 2.2.2, we have max {|u1p− p|, |unp− p|} ⩾ 2α. Hence, |wp− p| ⩾ 2αn. Finally, if w ∈ F(U) is not the empty word, then |wp − p| ⩾ 2α. By definition, α > 0. Therefore w ̸= 1 in G. Consequently, the natural homomorphism ψ : F(U) → G is injective. 2.2.3 Geodesic extension property This is the main result of this section. Our proof is based on [36, Lemma 3.2]. Proposition 2.2.6. — Let α > 0. Let U ⊂ G be an α-reduced subset at p. Let w ≡ u1 · · ·um and w′ ≡ u′ 1 · · ·u′ m′ be two elements of F(U). Then U satisfies the geodesic extension property, that is, if (p, w′p)wp < 1 2 |ump− p| − δ, then w is a prefix of w′. Remark 2.2.7. — The geodesic extension property has the following meaning: if the geodesic [p, w′p] extends [p, wp] as a path in X, then w′ extends w as a word over U ⊔ U−1. Proof. — The proof is by contrapositive. Assume that w is not a prefix of w′. Let r be the largest integer such that ui = u′ i, for every i ∈ J1, r − 1K. In particular, r ∈ J1,mK. For 92 2.2. Reduced subsets simplicity, denote q = u1 · · ·ur−1p = u′ 1 · · ·u′ r−1p. It follows from the four point inequality that (p, w′p)wp ⩾ min{(p, q)wp, (q, wp′)wp} − δ. (2.2.2) From now on, the focus will be on showing that min{(p, q)wp, (q, wp′)wp} ⩾ 1 2 |ump− p|. Using the definition of Gromov product, (p, q)wp = |wp− q| − (p, wp)q, (q, w′p)wp = |wp− q| − (wp,w′p)q. (2.2.3) We are going to estimate |wp− q|, (p, wp)q, and (wp,w′p)q. Claim 2.2.8. — |wp− q| ⩾ 1 2 |urp− p| + 1 2 |ump− p| + 2(m− r)(α + δ). Proof. — Note that m − r + 1 ⩾ 1. If m − r + 1 = 1, then there is nothing to do. If m− r + 1 ⩾ 2, then we apply Proposition 2.2.4 (ii) to the sequence of m− r + 2 points q = u1 · · ·ur−1p, u1 · · ·urp, u1 · · ·ur+1p, · · · , wp = u1 · · ·ump, and we obtain |wp− q| ⩾ 1 2 |urp− p| + 1 2 |ump− p| + 2(m− r)(α + δ). For simplicity, denote t = u1 · · ·urp and t′ = u′ 1 · · ·u′ rp. Claim 2.2.9. — (p, wp)q < 1 2 |urp− p|. Proof. — Applying Lemma 2.2.3 (ii) and Proposition 2.2.4 (i) to the sequence of m+ 1 points p, u1p, u1u2p, · · · , wp = u1 · · ·ump, 93 Chapter 2 – Uniform uniform exponential growth in small cancellation groups we get (p, wp)q ⩽ (u1 · · ·ur−2p, t)q + 2δ. Since U is α-reduced at p, (u1 · · ·ur−2p, t)q = (u−1 r−1p, urp)p < 1 2 |urp− p| − α− 2δ. Consequently, (p, wp)q < 1 2 |urp− p| − α. This proves our claim. Claim 2.2.10. — (wp,w′p)q < 1 2 |urp− p|. Proof. — If r − 1 = m′, then w′p = q and the claim holds. Hence we can suppose that r − 1 < m′. It follows from the choice of r that ur ̸= u′ r. It follows from the four point inequality that min{(t, wp)q, (wp,w′p)q, (w′p, t′)q} ⩽ (t, t′)q + 2δ. Since U is α-reduced at p, (t, t′)q = (urp, u ′ rp)q < 1 2 min{|urp− p|, |u′ rp− p|} − α− 2δ. Consequently, min{(t, wp)q, (wp,w′p)q, (w′p, t′)q} < 1 2 min{|urp− p|, |u′ rp− p|} − α. (2.2.4) We must prove that the minimum of Equation 2.2.4 is attained by (wp,w′p)q. In order to do so, let’s see first that the minimum of Equation 2.2.4 is not achieved by (t, wp)q. Using the definition of Gromov product, (t, wp)q = |q − t| − (q, wp)t. By definition, |q − t| = |urp− p|. Recall that m− r + 1 ⩾ 1. If m− r + 1 = 1, we have (q, wp)t = (u−1 r p, p)p = 0. 94 2.3. Growth in groups acting on a δ-hyperbolic space If m− r + 1 ⩾ 2, applying Lemma 2.2.3 (ii) and Proposition 2.2.4 (i) to the sequence of m− r + 2 points q = u1 · · ·ur−1p, t = u1 · · ·urp, u1 · · ·ur+1p, · · · , wp = u1 · · ·ump, we obtain (q, wp)t ⩽ (q, u1 · · ·ur+1p)t + 2δ. Since U is α-reduced, (q, u1 · · ·ur+1p)t = (u−1 r p, ur+1p)p < 1 2 |urp− p| − α− 2δ. Consequently, (t, wp)q ⩾ 1 2 |urp− p| > 1 2 |urp− p| − α. Thus, the minimum of Equation 2.2.4 cannot be achieved by (t, wp)q. Similarly, it cannot be achieved by (w′p, t′)q. Therefore, the only possibility is that it is achieved by (wp,w′p)q. This proves our claim. Finally, combining Equation 2.2.2 and Equation 2.2.3 with our three claims, we obtain (p, w′p)wp ⩾ min{(p, q)wp, (q, w′p)wp} − δ > 1 2 |ump− p| − δ. 2.3 Growth in groups acting on a δ-hyperbolic space In this section, we review and adapt some of the techniques of M. Koubi. [56] – further developed by G. Arzhantseva and I. Lysenok, [9]. These techniques permit to study exponential growth rates of finite symmetric subsets in groups acting by isometries on hyperbolic spaces in the sense of M. Gromov. In particular, we clarify what are the involved parameters for acylindrical actions, which permits to obtain Theorem 2.3.8. 2.3.1 Growth of maximal loxodromic subgroups. Let G be a group acting acylindrically on a hyperbolic space X. The goal of this subsection is to prove that the maximal loxodromic subgroups of G have some sort of 95 Chapter 2 – Uniform uniform exponential growth in small cancellation groups uniform linear growth. We adapt an argument that was written for hyperbolic groups in [5, p. 484]. Recall that Φ(G,X) stands by the loxodromic wideness of the action of G on X (Definition 2.1.20). Given a loxodromic element g ∈ G, we denoted by ∥g∥∞ its stable translation length (subsection 2.1.3) and by E(g) the maximal loxodromic subgroup of G containing g (subsection 2.1.4). Proposition 2.3.1. — Let G be a group acting acylindrically on a hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Let g ∈ G be a primitive loxodromic element. Then, for every n ⩾ 1, |Un ∩ E(g)| ⩽ 2Φ(G,X) ( L(U) ∥g∥∞ 4n+ 1 ) . First, we focus on the case of the cyclic group generated by a loxodromic isometry. Lemma 2.3.2. — Let G be a group acting acylindrically on a hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Let g ∈ G be a loxodromic element. Then, for every n ⩾ 1, |Un ∩ ⟨g⟩ | ⩽ L(U) ∥g∥∞ 2n+ 1. Proof. — Let n ⩾ 1. We have, |Un ∩ ⟨g⟩ | = |{ k ∈ Z : gk ∈ Un }|. Since the subset U is symmetric, |{ k ∈ Z : gk ∈ Un }| ⩽ 2|{ k ∈ N − {0} : gk ∈ Un }| + 1. Let k ⩾ 1 such that gk ∈ Un. Since the element g is loxodromic, we have ∥g∥∞ > 0. Observe that k = ∥gk∥∞ ∥g∥∞ . Let x ∈ X. Then ∥gk∥∞ ⩽ ∥gk∥ ⩽ |gkx− x| ⩽ max h∈Un |hx− x| = L(Un, x). Since the point x is arbitrary, we get ∥gk∥∞ ⩽ L(Un). By the triangle inequality, L(Un) ⩽ 96 2.3. Growth in groups acting on a δ-hyperbolic space nL(U). Hence, k ⩽ L(U) ∥g∥∞n. Therefore, |Un ∩ ⟨g⟩ | ⩽ L(U) ∥g∥∞ 2n+ 1. We are ready for the proof of the proposition. Proof of Proposition 2.3.1. — Let F (g) be the set of all elements of finite order of E+(g). Recall that F (g) is a normal subgroup of E+(g). Since the action of G on X is acylindrical and E(g) is a loxodromic subgroup of G, there exists a loxodromic element h ∈ E+(g) such that the map F (g) ⋊ϕ ⟨h⟩ → E+(g), (f, k) 7→ fk is a group isomorphism, where ϕ : ⟨h⟩ → Aut(F (g)) is the action by conjugacy of ⟨h⟩ on F (g) (Corollary 2.1.18). Let n ⩾ 1. Let E0 be a set of representatives of E(g)/⟨h⟩. We have |Un ∩ E(g)| = ∑ r∈E0 |Un ∩ r ⟨h⟩ |. First we are going to estimate |E0|. By definition, [E(g) : E+(g)] ⩽ 2. Since the homomorphism ⟨h⟩ → F (g) ⋊ϕ ⟨h⟩ , k 7→ (1, k) is a split of the exact sequence, 0 F (g) F (g) ⋊ϕ ⟨h⟩ ⟨h⟩ 0ι π we have [E+(g) : ⟨h⟩] = |F (g)| ⩽ Φ(G,X). Consequently, |E0| ⩽ 2Φ(G,X). Since the action of G on X is acylindrical, we have Φ(G,X) < ∞ (Lemma 2.1.21). Now we are going to estimate |Un ∩ r⟨h⟩| for r ∈ E0. We may assume that Un ∩ r⟨h⟩ is non-empty. Then there exist s ∈ Un ∩ r⟨h⟩. In particular r⟨h⟩ = s⟨h⟩. Hence, |Un ∩ r ⟨h⟩ | = |Un ∩ s ⟨h⟩ | = |s(s−1Un ∩ ⟨h⟩)| = |s−1Un ∩ ⟨h⟩ |. 97 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Since U is symmetric, s−1 ∈ Un. Since U contains the identity, s−1Un ⊂ U2n. Therefore, |s−1Un ∩ ⟨h⟩ | ⩽ |U2n ∩ ⟨h⟩ |. According to Lemma 2.3.2, |U2n ∩ ⟨h⟩ | ⩽ L(U) ∥h∥∞ 4n+ 1. Consequently, |Un ∩ r ⟨h⟩ | ⩽ L(U) ∥h∥∞ 4n+ 1. Finally, since the element g is primitive, we have that g ∈ {h, h−1}. It follows from our two estimations above that |Un ∩ E(g)| ⩽ 2Φ(G,X) ( L(U) ∥g∥∞ 4n+ 1 ) . Given a subset U ⊂ G and a loxodromic element g ∈ G, we fix a set of representatives U(g) of the equivalence relation induced on U by ∼g. Recall that the equivalence relation ∼g on G was previously defined by u ∼g v if and only if u−1v ∈ E(g), for every u, v ∈ G (subsection 2.1.4). The reason that makes the set U(g) of interest is that the set of conjugates of g by the elements of U(g) is a set of “independent” loxodromic elements and has the same size as U(g). We obtain the following. Corollary 2.3.3. — Let G be a group acting acylindrically on a hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Let g ∈ G be a primitive loxodromic element. Let a0 = 2Φ(G,X) ( L(U) ∥g∥∞ 8 + 1 ) . Then, |U(g)| ⩾ 1 a0 |U |. Proof. — Consider the surjective map U → U(g) that sends every element of U to its class representative in U(g). We are going to estimate its injectivity. Let u, v ∈ U such that u ∼g v. By definition, u−1v ∈ E(g). Since the subset U is symmetric, u−1v ∈ U2. Therefore, v ∈ u(U2 ∩ E(g)). Note that |u(U2 ∩ E(g))| = |U2 ∩ E(g)|. Consequently, each u ∈ U(g) 98 2.3. Growth in groups acting on a δ-hyperbolic space has at most |U2 ∩ E(g)| elements in its equivalence class. According to Proposition 2.3.1, |U2 ∩ E(g)| ⩽ a0. Therefore, |U(g)| ⩾ 1 a0 |U |. 2.3.2 Producing reduced subsets Recall that given a loxodromic element g ∈ G, we denoted by ∆(g) its fellow travelling constant (subsection 2.1.4). The goal of this subsection is to produce a reduced subset using the conjugates of a loxodromic isometry of large stable translation length. More precisely, we will prove the following. Proposition 2.3.4. — Let δ > 0 and α > 0. Let G be a group acting acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing a loxodromic element g ∈ U such that ∥g∥∞ > 103δ. Let p ∈ X. Let b0 = 200 ∥g∥∞ [∆(g) + L(U, p) + δ + α]. Then for every b ⩾ b0, the set S = {ugbu−1 : u ∈ U(g) } satisfies the following: (i) S ⊂ U b+2. (ii) |S| = |U(g)|. (iii) S is α-reduced at p. Proof. — The conclusions (i) and (ii) are immediate. We are going to prove (iii) S is α-reduced at p (Definition 2.2.1). By construction, S ∩S−1 = ∅. Let i ∈ J1, 2K. Let ui ∈ U . Let εi ∈ {−1, 1}. Assume that the elements u1g ε1bu−1 1 and u2g ε2bu−1 2 are distinct. Case u1 = u2. Since the elements u1g ε1bu−1 1 and u2g ε2bu−1 2 are distinct, we have ε1 = −ε2. Denote h = u1g ε1bu−1 1 . It is enough to prove that (hp, h−1p)p ⩽ b 2 ∥g∥∞ − α− 2δ. Let η− and η+ be the points of ∂X fixed by ⟨h⟩ and γ : R → X be an ⟨h⟩-invariant 103δ-local (1, δ)-quasi-geodesic joining η− to η+. This choice is possible since ∥g∥∞ > 103δ. It follows from Lemma 2.1.15 applied to γ that (hp, h−1p)p ⩽ L(U, p) + 6δ. 99 Chapter 2 – Uniform uniform exponential growth in small cancellation groups It is clear that L(U, p) + 6δ ⩽ b 2 ∥g∥∞ − α− 2δ. ‘ Case u1 ̸= u2. In particular u1 ̸∼g u2, which means that u−1 1 u2 does not belong to E(g). Claim 2.3.5. — d(p,Ag) ⩽ 1 2 L(U, p) + 5δ. Proof. — It follows from Lemma 2.1.9 that d(p,Ag) ⩽ 1 2 |gp− p| + 5δ. Moreover, since g ∈ U , we have |gp− p| ⩽ L(U, p). This proves our claim. Consider the points xi = uip and yi = uig εibp. Claim 2.3.6. — diam([x1, y1]+8δ ∩ [x2, y2]+8δ) ⩽ ∆(g) + L(U, p) + 44δ. Proof. — Denote σ = d(p,Ag) + 10δ. We have, max {d(xi, uiAg), d(yi, uiAg)} ⩽ σ. Recall that the axis Ag is 10δ-quasi-convex (Lemma 2.1.9). Hence, since σ ⩾ 10δ, the subset uiA +σ g is 2δ-quasi-convex (Lemma 2.1.7). Consequently, [xi, yi] ⊂ uiA +σ+2δ g . Therefore, diam([x1, y1]+8δ ∩ [x2, y2]+8δ) ⩽ diam(u1A +σ+10δ g ∩ u2A +σ+10δ g ). According to Lemma 2.1.8, diam(u1A +σ+10δ g ∩ u2A +σ+10δ g ) ⩽ diam(u1A +13δ g ∩ u2A +13δ g ) + 2(σ + 10δ) + 4δ. Moreover, diam(u1A +13δ g ∩ u2A +13δ g ) ⩽ diam(u1A +20δ g ∩ u2A +20δ g ). 100 2.3. Growth in groups acting on a δ-hyperbolic space Since u−1 1 u2 does not belong to E(g), diam(u1A +20δ g ∩ u2A +20δ g ) ⩽ ∆(g). Since the action of G on X is acylindrical, we have ∆(g) < ∞ (Lemma 2.1.16). Combining the above estimations with the previous claim, we obtain diam([x1, y1]+8δ ∩ [x2, y2]+8δ) ⩽ ∆(g) + L(U, p) + 54δ. This proves our claim. Denote si = uig εibu−1 i . Claim 2.3.7. — (s1p, s2p)p ⩽ ∆(g) + 5 L(U, p) + 54δ. Proof. — By definition, (s1p, s2p)p = 1 2(|s1p− p| + |s2p− p| − |s1p− s2p|). By the triangle inequality, |sip− p| ⩽ |xi − yi| + 2|uip− p|, |s1p− s2p| ⩾ |y1 − y2| − |u1p− p| − |u2p− p|. Consequently, (s1p, s2p)p ⩽ 1 2(|x1 − y1| + |x2 − y2| − |y1 − y2|) + 3 2(|u1p− p| + |u2p− p|). Combining the previous claim with Lemma 2.1.4, we obtain |x1 − y1| + |x2 − y2| − |y1 − y2| ⩽ |x1 − x2| + 2(∆(g) + L(U, p) + 44δ). By the triangle inequality, |x1 − x2| ⩽ |u1p− p| + |u2p− p|. Moreover, since ui ∈ U , we have |uip− p| ⩽ L(U, p). Combining the above estimations, we 101 Chapter 2 – Uniform uniform exponential growth in small cancellation groups obtain (s1p, s2p)p ⩽ ∆(g) + 5 L(U, p) + 44δ. This proves our claim. Finally, note that 1 2 min {|s1p− p|, |s2p− p|} − α− 2δ ⩾ b 2 ∥g∥∞ − α− 2δ. Since b ⩾ b0, we obtain b 2 ∥g∥∞ − α− 2δ > ∆(g) + 5 L(U, p) + 54δ. Therefore, the previous claim implies that (s1p, s2p)p < 1 2 min {|s1p− p|, |s2p− p|} − α− 2δ. 2.3.3 Growth trichotomy We are going to combine the two previous subsections in the following result. Theorem 2.3.8 (Theorem 0.6.10). — For every κ > 0 and N > 0, there exist an integer c > 1 with the following property. Let δ > 0 and α > 0. Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Let p ∈ X be a point almost-minimizing the ℓ∞-energy L(U). Then one of the following conditions holds: (T1) L(U) ⩽ 104 max {κ, δ, α}. (T2) The subgroup ⟨U⟩ is virtually cyclic and contains a loxodromic element. (T3) There exist a finite subset S ⊂ G with the following properties: (i) S ⊂ U c, (ii) |S| ⩾ max { 2, 1 c |U | } , (iii) S is α-reduced at p. Moreover, ω(U) ⩾ 1 c log |U |. 102 2.3. Growth in groups acting on a δ-hyperbolic space Proof. — Let κ > 0 and N > 0. Let n0 be the positive integer of Lemma 2.1.19 depending on κ and N . We fix auxiliar parameters a1 = 200Nn0, and b1 = 200(N + 2) + 500n0 + 700. We put c ⩾ max { a1, n0(b1 + 2), 2n0(b1 + 2) log a1 log 2 } . Let δ > 0 and α > 0. Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity. Let p ∈ X be a point almost-minimizing the ℓ∞-energy L(U). Assume that L(U) > 104 max {κ, δ, α}. Since L(U) > 50δ, according to Lemma 2.1.19 there exist a primitive loxodromic element g ∈ Un0 such that ∥g∥∞ ⩾ 1 2 L(U). (2.3.1) In particular ∥g∥∞ ⩾ 103δ. Let H = ⟨U⟩. Note that the loxodromic g belongs to H. Assume in addition that the subgroup H is not virtually cyclic. We prove (T3). We are going to apply Corollary 2.3.3 and Proposition 2.3.4 to Un0 and g. Let a0 = 2Φ(G,X) ( L(Un0) ∥g∥∞ 8 + 1 ) , b0 = 200 ∥g∥∞ [∆(g) + L(Un0 , p) + δ + α]. By the triangle inequality, L(Un0) ⩽ n0 L(U), and L(Un0 , p) ⩽ n0 L(U, p). Since the point p ∈ X is almost-minimizing the ℓ∞-energy L(U), we have L(U, p) ⩽ L(U)+δ. Since the action of G on X is (κ,N)-acylindrical, it follows from Lemma 2.1.21 and Lemma 2.1.16 that Φ(G,X) ⩽ N, and ∆(g) ⩽ κ+ (N + 2) ∥g∥∞ + 100δ. Using the hypothesis L(U) > 104 max {κ, δ, α} and Equation 2.3.1, we obtain, max { L(U) ∥g∥∞ , κ ∥g∥∞ , δ ∥g∥∞ , α ∥g∥∞ } ⩽ 2. Consequently, we obtain a0 ⩽ a1 and b0 ⩽ b1. Let S = {ugb1u−1 : u ∈ Un0(g) }. 103 Chapter 2 – Uniform uniform exponential growth in small cancellation groups The points (i) and (iii) follow from Proposition 2.3.4 (i) and (iii). We are going to prove (ii). According to Proposition 2.3.4 (ii), we have |S| = |Un0(g)|. If |Un0(g)| = 1, then u ∼g g, for every u ∈ Un0 . Hence Un0 is contained in E(g). Since U contains the identity, U ⊂ Un0 . Thus H is virtually cyclic (Lemma 2.1.17). Contradiction. Hence |Un0(g)| ⩾ 2. Further, it follows from Proposition 2.3.1 that |Un0(g)| ⩾ 1 a1 |Un0|. Since U contains the identity, |Un0 | ⩾ |U |. Therefore, |S| ⩾ max { 2, 1 a1 |U | } . This implies our point (ii). Let’s verify the last conclusion about ω(U). Let n ⩾ 1. We have |Un0(b1+2)n| ⩾ |Sn| ⩾ |S|n ⩾ max { 2n, ( 1 a1 |U | )n} , where the first inequality follows from (i); the second from (iii), which implies that the natural homomorphism F(S) → G is injective (Proposition 2.4.16); and the third from (ii). Consequently, ω(U) = lim sup n→∞ 1 n0(b1 + 2)n log |Un0(b1+2)n| ⩾ 1 n0(b1 + 2) max { log 2, log ( 1 a1 |U | )} . Finally, note that 1 a1 |U | ⩾ |U | 1 2 ⇔ log |U | ⩾ 2 log a1. If log |U | ⩾ 2 log a1, we obtain ω(U) ⩾ 1 n0(b1 + 2) log ( 1 a1 |U | ) ⩾ 1 2n0(b1 + 2) log |U |. If log |U | < 2 log a1, we obtain ω(U) ⩾ 1 n0(b1 + 2) log 2 ⩾ log 2 2n0(b1 + 2) log a1 log |U |. 104 2.4. Shortening and shortening-free words 2.4 Shortening and shortening-free words In the context of classical small cancellation theory, Greendlinger’s Lemma states that if a word over the free generating set of a free group represents the identity element in a small cancellation quotient, then it should contain a subword corresponding to a large portion of a relator. This section is structured as follows. First, we are going to formalise the notion of “large portion of a relator” with the definition of shortening word in the context of actions by isometries on hyperbolic spaces. Then, we are going to find a lower bound for the number of shortening-free words of free subgroups generated by reduced subsets of low energy. Finally, we will see that these shortening-free words embedd in geometric small cancellation quotients of appropriate parameters after using a suitable version of Greendlinger’s Lemma (Lemma 2.1.33). Global parameters and hypothesis for this section. Let δ0 and ∆0 be the constants of the Small Cancellation Theorem (Lemma 2.1.27). We fix once for all during this section L0 > 0, and τ0 = 106(δ0 + L0 + ∆0). Let 0 < δ ⩽ δ0, α ⩾ 200δ0, and τ ⩾ τ0. Let G be a group acting by isometries on a δ-hyperbolic space X. Let U ⊂ G be an α-reduced subset at p ∈ X (Definition 2.2.1). Let Q be a loxodromic moving family (Definition 2.1.24). We assume that 0 < L(U, p) ⩽ L0, and ∆(Q, X) ⩽ ∆0. 2.4.1 Shortening words Here we study shortening words. Part of this subsection is based on [36, Section 3.1]. Definition 2.4.1 (Shortening word). — Let w ≡ u1 · · ·un be an element of F(U). Let (H, Y ) ∈ Q. We say that w is a τ -shortening word over (H, Y ) if it satisfies the following. Consider the points x0 = p and xn = wp. Let y0 and yn be respective projections of x0 and xn on Y . Then, (S1) |y0 − yn| > τ . (S2) |x0 − y0| < 1 2 |u1p− p| − 100δ, and |xn − yn| < 1 2 |unp− p| − 100δ. 105 Chapter 2 – Uniform uniform exponential growth in small cancellation groups A minimal τ -shortening word over (H, Y ) is a τ -shortening word over (H, Y ) none of whose proper prefixes are τ -shortening words over (H,Y ). Remark 2.4.2. — Applying the triangle inequality, we observe that the choice τ ⩾ τ0 implies that τ -shortening words over (H,Y ) are distinct form the identity: |x0 − xn| ⩾ |y0 − yn| − |x0 − y0| − |xn − yn| > 0. Proposition 2.4.3.— Let w ≡ u1 · · ·un be a τ -shortening word over (H,Y ) ∈ Q. Consider the sequence of n+ 1 points x0 = p, x1 = u1p, x2 = u1u2p, · · · , xn = u1 · · ·unp. Let yi be a projection of xi on Y , for every i ∈ J0, nK. Then, |xi − yi| < 1 2 min{|uip− p|, |ui+1p− p|} − 100δ, for every i ∈ J1, n− 1K. Proof. — Let i ∈ J1, n− 1K. Let zi be a projection of xi on [y0, yn]. Since Y is 10δ-quasi- convex (Lemma 2.1.14), there exist z′ i ∈ Y such that |zi − z′ i| ⩽ 11δ. By definition, |xi − yi| ⩽ d(xi, Y ) + δ ⩽ |xi − z′ i| + δ. By the triangle inequality, |xi − z′ i| ⩽ |xi − zi| + |zi − z′ i|. By definition, |xi − zi| ⩽ d(xi, [y0, yn]) + δ. According to Lemma 2.1.3, d(xi, [y0, yn]) ⩽ (y0, yn)xi + 4δ. We claim that (y0, yn)xi ⩽ (x0, xn)xi + 2δ. It follows from the four point inequality that min{(x0, y0)xi , (y0, yn)xi , (yn, xn)xi } ⩽ (x0, xn)xi + 2δ. One can argue using the Broken Geodesic Lemma (Lemma 2.2.3) and the fact that w is a τ -shortening to prove that the minimum must be attained by (y0, yn)xi . Now applying the 106 2.4. Shortening and shortening-free words Broken Geodesic Lemma (Lemma 2.2.3 (ii)), (x0, xn)xi ⩽ (xi−1, xi+1)xi + 2δ. Moreover, (xi−1, xi+1)xi = (u−1 i p, ui+1p)p. Since the subset U is α-reduced and α ⩾ 200δ, (u−1 i p, ui+1p)p < 1 2 min {|uip− p|, |ui+1p− p|} − 118δ. Combining all the estimations, we obtain |xi − yi| < 1 2 min{|uip− p|, |ui+1p− p|} − 100δ. Proposition 2.4.4. — Let w ≡ u1 · · ·un be a τ -shortening word over (H, Y ) ∈ Q. The following holds. (i) We have |w|U ⩾ τ − 50δ L(U, p) . (ii) If w is a minimal τ -shortening word over (H,Y ), then |w|U ⩽ τ α + 2. Proof. — Consider the sequence of n+ 1 points x0 = p, x1 = u1p, x2 = u1u2p, · · · , xn = u1 · · ·unp. Let yi be a projection of xi on Y , for every i ∈ J0, nK. (i) Since L(U, p) > 0 and w is distinct from the identity (Remark 2.4.2), it follows from the triangle inequality that, |w|U ⩾ |x0 − xn| L(U, p) . According to (S1), we have |y0 −yn| > τ . Since Y is 10δ-quasi-convex (Lemma 2.1.14) and τ ⩾ 23δ, the strong contraction property of Y (Lemma 2.1.6) implies |x0 − xn| ⩾ |x0 − y0| + |y0 − yn| + |yn − xn| − 46δ. 107 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Consequently, |x0 − xn| > τ − 50δ. Therefore, |w|U ⩾ τ − 50δ L(U, p) . (ii) Assume that w is a minimal τ -shortening word over (H,Y ). Let w′ ≡ u1 · · ·un−1. By definition, |w|U = |w′|U + 1. In view of Proposition 2.2.4 (ii), we deduce |w′p− p| ⩾ 1 2 |u1p− p| + 1 2 |un−1p− p| + α(|w′|U − 1). By the triangle inequality, |w′p− p| ⩽ |xn−1 − yn−1| + |yn−1 − y0| + |y0 − x0|. Since w is a τ -shortening word over (H, Y ), the property (S2) implies |x0 − y0| < 1 2 |u1p− p| − 100δ. According to Proposition 2.4.3, |xn−1 − yn−1| < 1 2 |un−1p− p| − 100δ. Therefore, since w′ is not a τ -shortening over (H, Y ), we have |yn−1 − y0| ⩽ τ . Consequently, |w′|U ⩽ τ α + 1. Thus, |w|U ⩽ τ α + 2. Proposition 2.4.5. — Let (H1, Y1), (H2, Y2) ∈ Q. Let w ∈ F(U). If w is a τ -shortening word over both (H1, Y1) and (H2, Y2), then (H1, Y1) = (H2, Y2). Proof. — Assume that w is a τ -shortening word over (H1, Y1) and (H2, Y2). In order to prove that (H1, Y1) = (H2, Y2), it is enough to show that diam(Y +20δ 1 ∩Y +20δ 2 ) > ∆(Q, X). Since the subsets Y1 and Y2 are 10δ-quasi-convex (Lemma 2.1.14), it follows from Lemma 2.1.8 that diam(Y +20δ 1 ∩ Y +20δ 2 ) ⩾ diam(Y +13δ 1 ∩ Y +13δ 2 ) ⩾ diam(Y +2L0 1 ∩ Y +2L0 2 ) − 4L0 − 4δ0. 108 2.4. Shortening and shortening-free words Let i ∈ J1, 2K. Let xi and zi be respective projections of p and wp on Yi. We claim that x1, z1 ∈ Y +2L0 1 ∩ Y +2L0 2 . Since w is a shortening word over (Hi, Yi), it follows from (S2) that max{|p− xi|, |wp− zi|} ⩽ L0. According to the triangle inequality, |x1 − x2| ⩽ |x1 − p| + |p− x2|, |z1 − z2| ⩽ |z1 − p| + |p− z2|. Consequently, max{|x1 − x2|, |z1 − z2|} ⩽ 2L0. Therefore, x1, z1 ∈ Y +2L0 2 . This proves the claim. Thus, diam(Y +2L0 1 ∩ Y +2L0 2 ) ⩾ |x1 − z1|. Since w is a shortening over (H1, Y1), it follows from (S1) that |x1 − z1| > τ . Finally, since τ ⩾ τ0, we obtain that diam(Y +20δ 1 ∩ Y +20δ 2 ) > ∆(Q, X). Figure 2.3 – Scheme for the proof of Proposition 2.4.5. Proposition 2.4.6.— For every (H, Y ) ∈ Q, there exist at most two minimal τ -shortening words over (H,Y ). 109 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Proof. — Let (H, Y ) ∈ Q. Let η− and η+ be the points of ∂X fixed by H and γ : R → X be an 103δ-local (1, δ)-quasi-geodesic joining η− to η+. Let q be a projection of p on γ. Without loss of generality, we may assume that q = γ(0). Let S(H,Y ) denote the set of elements in F(U) that are τ -shortening words over (H,Y ). Assume that S(H,Y ) is non-empty, otherwise the statement is true. We decompose S(H,Y ) in two sets as follows: an element w ∈ S(H,Y ) belongs to S + (H,Y ) (respectively, S − (H,Y )) if there is a projection γ(t) of wp on γ with t ⩾ 0 (respectively, t ⩽ 0). Observe that a priori the sets S − (H,Y ) and S + (H,Y ) are not disjoint, but that will not be an issue for the rest of the proof. Let w1, w2 ∈ S + (H,Y ). Let q1 = γ(t1) and q2 = γ(t2) be the respective projections of w1p and w2p on γ. Without loss of generality, we may assume that 0 ⩽ t1 ⩽ t2. Claim 2.4.7. — The word w1 is a prefix of w2. Proof. — We are going to apply the Geodesic Extension Property (Proposition 2.2.6). By the triangle inequality, (p, w2p)w1p ⩽ |w1p− q1| + (w2p, p)q1 . (2.4.1) Assume that w1 ≡ u1 · · ·um. (a) Let’s estimate |w1p− q1|. By definition, the H-invariant cylinder Y is contained in the 20δ-neighbourhood of γ. Consequently, |w1p− q1| = d(w1p, γ) ⩽ d(w1p, Y ) + 20δ. Since w1 is a τ -shortening word over (H,Y ), the property (S2) implies d(w1p, Y ) < 1 2 |ump− p| − 100δ. Therefore, |w1p− q1| < 1 2 |ump− p| − 80δ. (2.4.2) (b) Let’s estimate (w2p, p)q1 . By definition, (w2p, p)q1 = 1 2(|w2p− q1| + |p− q1| − |w2p− p|). 110 2.4. Shortening and shortening-free words Since w2 is a τ -shortening word over (H, Y ), the property (S1) implies |q2 − q| > τ. Since Y is 10δ-quasi-convex (Lemma 2.1.14) and τ ⩾ 23δ, the strong contraction property of Y (Lemma 2.1.6) implies |w2p− p| ⩾ |w2p− q2| + |q2 − q| + |q − p| − 46δ. Again by definition, |q2 − q| = |q2 − q1| + |q1 − q| − 2(q2, q)q1 . According to Lemma 2.1.15 (i), (q2, q)q1 ⩽ 6δ. Note that here we have used the assumption 0 ⩽ t1 ⩽ t2. By the triangle inequality, |w2p− q1| ⩽ |w2p− q2| + |q2 − q1|. Therefore, |w2p− p| ⩾ |w2p− q1| + |q1 − p| − 58δ. Consequently, (w2p, p)q1 ⩽ 29δ. (2.4.3) Finally, combining Equation 2.4.1, Equation 2.4.2 and Equation 2.4.3, we obtain (p, w2p)w1p ⩽ 1 2 |ump− p| − δ. Therefore, the Geodesic Extension Property (Proposition 2.2.6) implies that w1 is a prefix of w2. This proves our claim. If w1 is not a proper prefix of w2, then the claim above implies that w1 = w2. Therefore S + (H,Y ) has at most one element satisfying the statement of the proposition. By symmetry, S − (H,Y ) has at most one element satisfying the statement. Therefore S(H,Y ) has at most two elements satisfying the statement. 111 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Figure 2.4 – Scheme for the proof of Proposition 2.4.6. 2.4.2 The growth of shortening-free words Here we count shortening-free words. The counting is based on [36, Section 3.22]. Definition 2.4.8 (Shortening-free word). — Let w ≡ u1 · · ·un be an element of F(U). Let (H, Y ) ∈ Q. We say that w contains a τ -shortening word over (H,Y ) if w splits as w ≡ w0w1w2, where w1 is a τ -shortening word over (H,Y ). We say that w is a τ -shortening- free word if for every (H, Y ) ∈ Q, the word w does not contain any τ -shortening word over (H,Y ). We denote by F (τ) ⊂ F(U) the subset of τ -shortening-free words. Recall that the natural homomorphism F(U) → G is injective (Proposition 2.2.5). Hence, we can safely identify the elements of F(U) with their images in G. The ball BU (n) ⊂ F(U) of radius n is the set of reduced words over the alphabet U ⊔U−1 of length |w|U ⩽ n, for every n ⩾ 0. Note that BU(n) = (U ⊔ U−1 ⊔ {1})n when n ⩾ 1. Recall that we have fixed global hypothesis at the beginning of this section. The goal of this subsection is to obtain the following estimation. Proposition 2.4.9. — For every θ ∈ (0, 1/2), there exist τ1 ⩾ τ0 depending on θ, δ0, L0 and ∆0 with the following property. If |U | ⩾ 2 and τ ⩾ τ1, then for every n ⩾ 0, we have |F (τ) ∩BU(n+ 1)| ⩾ (1 − θ)(2|U | − 1)|F (τ) ∩BU(n)|. 112 2.4. Shortening and shortening-free words In particular, for every n ⩾ 0 |F (τ) ∩BU(n)| ⩾ (1 − θ)n(2|U | − 1)n. We are going to divide the proof of Proposition 2.4.9 into a few lemmas. First we fix some notations. We let Z = {w ∈ F(U) : w ≡ w0u,w0 ∈ F (τ), u ∈ U ⊔ U−1 }. For every (H, Y ) ∈ Q, we denote by Z(H,Y ) ⊂ Z the set of elements w ∈ Z that split as w ≡ w1w2, where w1 ∈ F (τ) and w2 is a τ -shortening word over (H, Y ). Lemma 2.4.10. — The set Z is contained in the disjoint union of F (τ) and ⋃(H,Y )∈Q Z(H,Y ). Proof. — The sets F (τ) and ⋃ (H,Y )∈Q Z(H,Y ) are disjoint as a direct consequence of the definitions. Let w ∈ Z − F (τ). Since w ∈ Z, there exist w0 ∈ F (τ) and u ∈ U ⊔ U−1 such that w ≡ w0u. Since w /∈ F (τ), there exist (H, Y ) ∈ Q and a subword w2 of w that is a τ -shortening word over (H, Y ). It follows from the definition of F (τ) that every subword of w0 must also be in F (τ). In particular, the word w2 cannot be a subword of w0. Hence, the only possibility is that w2 is a suffix of w. Therefore, w ∈ Z(H,Y ). Our Lemma 2.4.10 implies that for every n ⩾ 0, |F (τ) ∩BU(n)| ⩾ |Z ∩BU(n)| − ∑ (H,Y )∈Q |Z(H,Y ) ∩BU(n)|. (2.4.4) The next step is to estimate each term in the right side of the above inequality. The following lemma is a direct consequence of the definition of Z. Lemma 2.4.11. — For every n ⩾ 0, |Z ∩BU(n+ 1)| = (2|U | − 1)|F (τ) ∩BU(n)|. Lemma 2.4.12. — Let a = 2, b = ⌈ τ0 200δ0 + 2 ⌉ + 1, M = ⌊ τ − 50δ0 L0 ⌋ . 113 Chapter 2 – Uniform uniform exponential growth in small cancellation groups If |U | ⩾ 2, then for every n ⩾ 0, ∑ (H,Y )∈Q |Z(H,Y ) ∩BU(n)| ⩽ a(2|U | − 1)b|F (τ) ∩BU(n−M)|. Proof. — Assume that |U | ⩾ 2. Let n ⩾ 0. Note that for every (H, Y ) ∈ Q, the set Z(H,Y ) is empty whenever there is no τ -shortening word over (H,Y ). We denote by Q0 the set of (H,Y ) ∈ Q for which there exist a τ -shortening word over (H, Y ). We have, ∑ H∈Q |Z(H,Y ) ∩BU(n)| = ∑ (H,Y )∈Q0 |Z(H,Y ) ∩BU(n)|. The desired estimation is obtained from the two estimations of the claims below: Claim 2.4.13. — |Z(H,Y ) ∩BU(n)| ⩽ a|F (τ) ∩BU(n−M)|, for every (H,Y ) ∈ Q0. Proof. — Let (H,Y ) ∈ Q0. Let w ∈ Z(H,Y ) ∩ BU(n). Since w ∈ Z(H,Y ), there exist w1 ∈ F (τ) and a τ -shortening word w2 over (H,Y ) such that w ≡ w1w2. We are going to describe the possible choices of w1 and w2. Since w is a reduced word over U ⊔ U−1, |w1|U = |w|U − |w2|U . According to Proposition 2.4.4 (i), |w2|U ⩾ τ − 50δ0 L0 ⩾M ⩾ 0. Therefore, w1 ∈ F (τ) ∩BU(n−M). Since w ∈ Z, the prefix consisting of all but the last letter is a τ -shortening free word. Thus, no proper prefix of w2 is a τ -shortening word. It follows from Proposition 2.4.6 that there are most a = 2 possible choices for w2. Therefore, there are at most a|F (τ) ∩BU(n−M)| choices for w. This proves our claim. Claim 2.4.14. — |Q0| ⩽ (2|U | − 1)b Proof. — Let d = ⌈ τ0 200δ0 + 2 ⌉ . Since the free group F(U) has rank |U | ⩾ 2, we have |BU(d)| = |U |(2|U | − 1)d − 1 |U | − 1 ⩽ (2|U | − 1)d+1 = (2|U | − 1)b. Consequently, it suffices to show that there exists an injective map χ : Q0 → BU(d). Let (H,Y ) ∈ Q0. By definition, there exist a τ -shortening word w over (H, Y ). Note that since 114 2.4. Shortening and shortening-free words τ ⩾ τ0, we have that w is a τ0-shortening word over (H,Y ). Let w′ be the shortest prefix of w that is a τ0-shortening word over (H, Y ). In particular, w′ is a minimal τ0-shortening word over (H,Y ). We define χ(H, Y ) = w′. Since α ⩾ 200δ0, according to Proposition 2.4.4 (ii), |w′|U ⩽ d. According to Proposition 2.4.5, there exist at most one (H,Y ) ∈ Q such that w′ is a τ0-shortening word over (H,Y ). Hence χ is well-defined and injective. This proves our claim. Lemma 2.4.15. — For every θ ∈ (0, 1/2) and a, b ⩾ 1, there exist M0 ⩾ 0 with the following property. Let µ = (1 − θ)(2|U | − 1), ξ = a(2|U | − 1)b, and σ = θ 2(1 − θ)ξ . If |U | ⩾ 2, then for every M ⩾M0, we have 1 µM ⩽ σ. Proof. — Let θ ∈ (0, 1/2) and a, b ⩾ 1. Let M0 = max { b, d1 d2 } , where d1, d2 are constants depending only on θ, a, b whose exact value will be precised below. Let µ, ξ, σ as above. Assume that |U | ⩾ 2. Let M ⩾M0. In order to prove that 1 µM ⩽ σ, it is enough to show that log ( 1 σµM ) ⩽ 0. A first computation yields log ( 1 σµM ) = − log σ − log ( µM ) , log(σ) = log ( θ 2(1 − θ)a ) − b log(2|U | − 1), log ( µM ) = M log(1 − θ) +M log(2|U | − 1). Consequently, log ( 1 σµM ) ⩽ (b−M) log(2|U | − 1) −M log(1 − θ) − log ( θ 2(1 − θ)a ) . Since M ⩾ b and |U | ⩾ 2, we have (b−M) log(2|U | − 1) ⩽ (b−M) log 3. 115 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Therefore, log ( 1 σµM ) ⩽ −M [log 3 + log(1 − θ)] + b log 3 − log ( θ 2(1 − θ)a ) . We put d1 = b log 3 + log(2a) − log ( θ 1 − θ ) , d2 = log 3 + log(1 − θ). Since a ⩾ 1, b ⩾ 1 and θ ∈ (0, 1/2), we have min{d1, d2} > 0. Finally, since M ⩾ d1 d2 , we obtain, log ( 1 σµM ) ⩽ 0. We are ready to prove the proposition. Proof of Proposition 2.4.9. — Let θ ∈ (0, 1/2). We are going to define the constant τ1. Let a = 2, b = ⌈ τ0 200δ0 + 2 ⌉ + 1. Let M0 ⩾ 0 be the constant of Lemma 2.4.15 depending on θ, a, b. We put τ1 = max{τ0, L0(M0 + 1) + 50δ0}. Assume that |U | ⩾ 2 and τ ⩾ τ1. We define the auxiliary parameters µ = (1 − θ)(2|U | − 1), ξ = a(2|U | − 1)b, σ = θ 2ξ(1 − θ) , and M = ⌊ τ − 50δ0 L0 ⌋ . In particular, M ⩾M0. For every n ⩾ 0, we let c(n) = |F (τ) ∩BU(n)|. We must prove that for every n ⩾ 1, c(n) ⩾ µc(n− 1). The proof goes by induction on n: Base step. We claim that c(1) ⩾ µ. Note that BU(1) = U ⊔ U−1 ⊔ {1}. Therefore, it is enough to show that U ⊔ U−1 ⊔ {1} is contained in F (τ). Let w ∈ U ⊔ U−1 ⊔ {1}. In particular, |w|U = 1. Therefore, w ∈ F (τ) if and only if for every (H,Y ) ∈ Q, the element w is not a τ -shortening word over (H, Y ). According to Proposition 2.4.4 (i), for every 116 2.4. Shortening and shortening-free words (H,Y ) ∈ Q and for every τ -shortening word v over (H,Y ), we have |v|U ⩾ τ−50δ0 L0 . Since τ ⩾ τ0, we have 1 < τ−50δ0 L0 . Consequently, w ∈ F (τ). This proves our claim. Inductive step. Let n ⩾ 1. Assume that c(m) ⩾ µc(m − 1), for every m ∈ J1, nK. We claim that c(n+ 1) ⩾ µc(n). According to Equation 2.4.4, c(n+ 1) ⩾ |Z ∩BU(n+ 1)| − ∑ (H,Y )∈Q |Z(H,Y ) ∩BU(n+ 1)|. It follows from Lemma 2.4.11 and Lemma 2.4.12 that c(n+ 1) ⩾ (2|U | − 1)c(n) − ξc(n+ 1 −M). The induction hypothesis implies that for every k ⩾ 0, we have c(n− k) ⩽ µ−kc(n). Note that M − 1 ⩾ 0. Therefore, specifying the choice k = M − 1, we obtain c(n+ 1) ⩾ ( 1 − ξµ 2|U | − 1 1 µM ) (2|U | − 1)c(n). Recall that we defined µ = (1 − θ)(2|U | − 1). Hence, in order to prove our claim, it is enough to show that ξµ 2|U | − 1 1 µM ⩽ θ. Since M ⩾M0, it follows from Lemma 2.4.15 that 1 µM ⩽ σ. Finally, note that ξµ 2|U | − 1σ = ξ(1 − θ)(2|U | − 1) 2|U | − 1 θ 2ξ(1 − θ) = θ 2 ⩽ θ. This proves our claim. 2.4.3 The injection of shortening-free words Let ρ0 be the constant of the Small Cancellation Theorem (Lemma 2.1.27). Let τ1 ⩾ τ0 be the constant of Proposition 2.4.9 depending on θ = 1/3, δ0, L0 and ∆0. Let ρ ⩾ max{ρ0, log(2[4τ1 + 23δ0] + 1)}. 117 Chapter 2 – Uniform uniform exponential growth in small cancellation groups In addition to the global hypothesis for this section, we assume that T(Q, X) ⩾ 100π sinh ρ. Denote K = ⟨⟨H | (H,Y ) ∈ Q⟩⟩ and Ḡ = G/K. The goal of this subsection is to prove: Proposition 2.4.16. — There exists τ2 ⩾ τ1 depending on δ0, L0 and ∆0 with the following property. The restriction of the natural homomorphism F(U) → G to the subset of τ2-shortening-free words is an injection. Lemma 2.4.17. — Let w ≡ u1 · · ·un be an element of F(U). Let (H,Y ) ∈ Q. Let y0 and yn be respective projections of p and wp on Y . If |y0 − yn| > 2τ , then w contains a (2τ − τ0)-shortening word over a conjugate of (H,Y ). Proof. — Consider the sequence of n+ 1 points x0 = p, x1 = u1p, x2 = u1u2p, · · · , xn = u1 · · ·unp. Let yi be a projection of xi on Y , for every i ∈ J0, nK. Assume that |y0 − yn| > 2τ . Since Y is 10δ-quasi-convex (Lemma 2.1.14) and τ ⩾ 23δ, the strong contraction property of Y (Lemma 2.1.6) implies that there exist y′ 0, y ′ n ∈ [p, wp] such that max{|y0 − y′ 0|, |yn − y′ n|} ⩽ 23δ ⩽ 23δ0. Consider the broken geodesic γw = n⋃ i=1 (u1 · · ·ui−1)[p, uip]. Let y′′ 0 and y′′ n be respective projections of y′ 0 and y′ n on γw. Up to permuting y′ 0 and y′ n we may assume that p, y′′ 0 , y′′ n and wp are ordered in this way along γw. In particular, there are i ⩽ n − 1 and j ⩽ n − 1 such that y′′ 0 ∈ (u1 · · ·ui)[p, ui+1p] and y′′ n ∈ (u1 · · ·uj)[p, uj+1p]. Since y′′ 0 comes before y′′ n on γw, we have i ⩽ j. Let w0 ≡ u1 · · ·ui+1 and take the word w1 such that w0w1 ≡ u1 · · ·uj. We are going to prove that w1 is a (2τ − τ0)-shortening word over (w−1 0 Hw0, w −1 0 Y ). The property (S2) follows from the fact that U is 200δ0- reduced at p and from the Broken Geodesic Lemma (Lemma 2.2.3). Let’s prove (S1), i.e. |yi+1 − yj| > 2τ − τ0. By the triangle inequality, |yi+1 − yj| ⩾ |y0 − yn| − |y0 − yi+1| − |yn − yj|, 118 2.4. Shortening and shortening-free words |y0 − yi+1| ⩽ |y0 − y′ 0| + |y′ 0 − y′′ 0 | + |y′′ 0 − xi+1| + |xi+1 − yi+1|, |yn − yj| ⩽ |yn − y′ n| + |y′ n − y′′ n| + |y′′ n − xj| + |xj − yj|. Since [x0, xn] is contained in the 5δ-neighbouhood of γw (Lemma 2.2.3 (iii)), max{|y′ 0 − y′′ 0 |, |y′ n − y′′ n|} ⩽ 5δ ⩽ 5δ0. Since y′′ 0 ∈ (u1 · · ·ui)[p, ui+1p] and y′′ n ∈ (u1 · · ·uj)[p, uj+1p], max{|y′′ 0 − xi+1|, |y′′ n − xj|} ⩽ L(U, p) ⩽ L0. It follows from (S2) that, max{|xi+1 − yi+1|, |xj − yj|} ⩽ L(U, p) ⩽ L0. Combining the previous estimations, we obtain |yi+1 −yj| > 2τ−τ0. Note that 2τ−τ0 ⩾ τ0. Proof of Proposition 2.4.16. — We put τ2 = 2τ1 − τ0. Let w1, w2 ∈ F(U) be two τ2- shortening-free words such that w1w2 ∈ K. Assume for a contradiction that w1w2 is not the identity as an element of G. According to Greendlinger’s Lemma (Lemma 2.1.33), there exist (H,Y ) ∈ Q such that if y0 and y2 are respective projections of p and w1w2p on Y , then |y0 − y2| > T (H,X) − 2π sinh ρ− 23δ. By definition, T (H,X) ⩾ T (Q, X). By hypothesis T (Q, X) ⩾ 100π sinh ρ, and δ ⩽ δ0. Therefore, |y0 − y2| > eρ − 1 2 − 23δ0. The choice of ρ now implies that |y0 − y2| > 4τ1 Let y1 be a projection of w1p on Y . Note that w−1 1 y1 and w−1 1 y2 are respective projections of p and w2p on w−1 1 Y . Also, (w−1 1 Hw1, w −1 1 Y ) ∈ Q. Since w1 and w2 are τ2-shortening-free 119 Chapter 2 – Uniform uniform exponential growth in small cancellation groups words, it follows from Lemma 2.4.17 that max {|y0 − y1|, |y1 − y2|} < 2τ1. By the triangle inequality, |y0 − y2| ⩽ |y0 − y1| + |y1 − y2| ⩽ 4τ1. Contradiction. Hence w1w2 = 1. 2.5 Growth in small cancellation groups The goal of this section is to prove Theorem 0.6.2. We start with the following lemma. Lemma 2.5.1. — Let a ⩾ 0, b ⩾ a. Let G be a group acting acylindrically on a δ-hyperbolic space X. Let U ⊂ G be a finite symmetric subset containing the identity such that L(U) ⩽ b. Let Γ = ⟨U⟩. One of the following holds. (i) Γ is elliptic. (ii) There exist n ⩾ 1 depending on U such that a < L(Un) ⩽ 2b. Proof. — Assume that Γ is not elliptic. Since the action of G on X is acylindrical, there exists a loxodromic element g ∈ Γ (Lemma 2.1.22). Claim 2.5.2. — There exists M0 ⩾ 1 depending on U such that for every M ⩾M0, L(UM) > a. Proof. — According to Lemma 2.1.13, the global injectivity radius T(G,X) is distinct from zero. Let m ⩾ a+δ T(G,X) . Since g ∈ Γ and U is a symmetric generating set, there exists M0 ⩾ 1 depending on U such that gm ∈ UM0 . Let M ⩾M0. Let p ∈ X almost-minimizing the ℓ∞-energy L(UM0). We have, L(UM , p) ⩾ L(UM0 , p) ⩾ |gmp− p| ⩾ ∥gm∥∞ = m∥g∥∞ > a+ δ. Hence L(UM) > a. This proves our claim. 120 2.5. Growth in small cancellation groups It follows from the claim above that there exist a smallest number n ⩾ 1 depending on U such that L(Un) > a. If n = 1, then we have L(U) ⩽ b by hypothesis. Therefore, L(U) ⩽ 2b. If n ⩾ 2, then n ⩽ 2(n− 1). Since U contains the identity, Un ⊂ U2(n−1). By the triangle inequality, L(Un) ⩽ L(U2(n−1)) ⩽ 2 L(Un−1) ⩽ 2a ⩽ 2b. Hypothesis for the remainder of this section. Recall that the constants of the Small Cancellation Theorem (Lemma 2.1.27) are δ0, δ̄, ∆0, ρ0. We can choose δ0 arbitrarily small (Remark 2.1.28). For convenience, we will assume δ0 ⩽ π sinh 104δ̄ 104 · 200 . We define the first geometric small cancellation parameter: λ ⩽ ∆0 100π sinh ρ0 . Let N > 0. Let c > 1 be the constant of Theorem 2.3.8 depending only on the acylindricity parameters (δ0, N). We fix an auxiliar parameter that will be used to bound the ℓ∞-energy: L0 = c · (2π sinh 104δ̄ + δ0). Let τ1 and τ2 be the constants of Proposition 2.4.16 depending on δ0, L0 and ∆0. Let ρ ⩾ max { ρ0, log(2[4τ1 + 23δ0] + 1), 5 · 104δ̄ } . Let δ > 0 and κ ⩾ δ. We define the second geometric small cancellation parameter: ε ⩾ 100π sinh ρ δ0 · κ δ . Let G be a group acting (κ,N)-acylindrically on a δ-hyperbolic space X. Let Q be a loxodromic moving family satisfying the geometric C ′′(λ, ε)-small cancellation condition 121 Chapter 2 – Uniform uniform exponential growth in small cancellation groups for the action of G on X. We define a rescaling parameter σ = min { δ0 κ , ∆0 ∆(Q, X) } . Remark 2.5.3. — Instead of working with the action of G on X, we will work with the action of G on the rescaled space X . The space X is σδ-hyperbolic and the action of G on X is (σκ,N)-acylindrical. Note that σδ ⩽ σκ ⩽ δ0, where the first inequality comes from the hypothesis κ ⩾ δ. In particular, the action of G on X is (δ0, N)-acylindrical for the hyperbolicity constant σδ. Besides, we have ∆(Q,X ) ⩽ σ∆(Q, X) ⩽ ∆0, T(Q,X ) ⩾ σT(Q, X) ⩾ σmax { εδ, ∆(Q, X) λ } ⩾ 100π sinh ρ. Note that the second equation is deduced after using the geometric C ′′(λ, ε)-small cancel- lation condition. Therefore G, X and Q satisfy the hypothesis of the Small Cancellation Theorem (Lemma 2.1.27). We denote K = ⟨⟨H | (H,Y ) ∈ Q⟩⟩ and Ḡ = G/K. We denote by Ā the image of any set A ⊂ G under the natural projection π : G↠ Ḡ. The following lemma is the core of the proof of our main theorem. It brings together Theorem 2.3.8, Proposition 2.4.9 and Proposition 2.4.16. Lemma 2.5.4. — There exist β ∈ (0, 1) depending only on N with the following property. Let U ⊂ G be a finite symmetric subset containing the identity such that L(U) ⩽ π sinh 104δ̄. Let Γ = ⟨U⟩. If Γ is non-elementary for the action on X , then ω(Ū) ⩾ βω(U) Proof. — We put β = sup θ∈(0,1) inf { θ · log 3 2 log (2c) , 1 − θ } · 1 c . Let U ⊂ G be a finite symmetric subset containing the identity such that L(U) ⩽ π sinh 104δ̄. Let Γ = ⟨U⟩ and assume that Γ is non-elementary for the action on X . We are going to choose a power of U and apply Theorem 2.3.8 to that power for the 122 2.5. Growth in small cancellation groups (δ0, N)-acylindrical action of G on the σδ-hyperbolic space X . By assumption, we have 104 · 200δ0 ⩽ π sinh 104δ̄. Since Γ is non-elementary, it follows from Lemma 2.5.1 that there exists n ⩾ 1 depending on U such that 104 · 200δ0 < L(Un) ⩽ 2π sinh 104δ̄. (2.5.1) Let Γ′ = ⟨Un⟩. Since U is symmetric and contains the identity, U ⊂ Un. Therefore Γ = Γ′. The fact that Γ is non-elementary now implies that Γ′ is non-elementary. Let p ∈ X be a point almost-minimizing the ℓ∞-energy L(Un). It follows from Theorem 2.3.8 that there exist a subset S ⊂ G such that (i) S ⊂ U cn, (ii) |S| ⩾ 1 c |Un|, (iii) S is 200δ0-reduced at p. We are going to estimate ω(Ū). Let r ⩾ 1. Since U is symmetric and contains the identity, (i) implies BS(r) ⊂ U cnr. Let F (τ2) be the set of τ2-shortening-free words associated to U and Q. We have |Ū cnr| ⩾ |B̄S(r)| ⩾ |F̄ (τ0) ∩ B̄S(r)|. Further, L(S, p) ⩽ L(U cn, p) ⩽ cL(Un, p) ⩽ L0, where the first inequality is (i) and the second one is the triangle inequality. The third one is due to the upper bound of Equation 2.5.1, together with the fact that the point p is almost-minimizing the ℓ∞-energy L(U). Hence we can apply Proposition 2.4.9 and Proposition 2.4.16 to obtain, respectively |F̄ (τ2) ∩ B̄S(r)| = |F (τ2) ∩BS(r)|, and |F (τ2) ∩BS(r)| ⩾ [1 2(2|S| − 1) ]r . Applying Fekete’s Subadditive Lemma, |Un| ⩾ enω(U). 123 Chapter 2 – Uniform uniform exponential growth in small cancellation groups Together with (ii), this implies 2|S| − 1 ⩾ |S| ⩾ 1 c enω(U). Combining our estimations, we deduce |Ū cnr| ⩾ max {[1 2(2|S| − 1) ]r , [ 1 2ce nω(U) ]r} . (2.5.2) We have, ω(Ū) = lim sup r→∞ 1 cnr log |Ū cnr|. Let θ ∈ (0, 1). Consider the positive number γ = log 2c θω(U) . ▶ If n ⩽ γ, we use the first bound of Equation 2.5.2 to obtain ω(Ū) ⩾ 1 cn · log [1 2(2|S| − 1) ] . Since n ⩽ γ, we have 1 n ⩾ 1 γ . Further, |S| ⩾ 2. Consequently, ω(Ū) ⩾ θ · log 3 2 log 2c · 1 c · ω(U). ▶ If n ⩾ γ, we use the second bound of Equation 2.5.2 to obtain ω(Ū) ⩾ 1 c ( ω(U) − 1 n log 2c ) . Since n ⩾ γ, we have 1 n ⩽ 1 γ . Consequently, 1 n log 2c ⩽ θω(U). Therefore, ω(Ū) ⩾ (1 − θ) · 1 c · ω(U). 124 2.5. Growth in small cancellation groups Finally, combining the cases n ⩽ γ and n ⩾ γ, we obtain: ω(Ū) ⩾ βω(U). Theorem 2.5.5 (Theorem 0.6.2 (i)). — Let ξ > 0. If G has ξ-uniform uniform exponential growth, then every geometric C ′′(λ, ε)-small cancellation quotient of G has ξ′-uniform uniform exponential growth. The constant ξ′ depends only on ξ and N . Proof. — Let ξ > 0. Assume that G has ξ-uniform uniform exponential growth. Let Ū ⊂ Ḡ be a finite symmetric subset containing the identity and denote Γ̄ = ⟨Ū⟩. Recall that V stands by the set of apices of the cone-off space Ẋρ(Q, X). There are two cases: Case 1. There exist v̄ ∈ V̄ such that Ū is contained in Stab(v̄). Let v ∈ V be a preimage of v̄. Let (H,Y ) ∈ Q such that v is the apex of the cone Z(Y ). The natural projection π : G↠ Ḡ induces an isomorphism Stab(Y )/H ∼−→ Stab(v̄) (Lemma 2.1.27 (iii)). Since the moving family Q is loxodromic, H has finite index in Stab(Y ). Hence Γ̄ is finite, in particular virtually nilpotent. Case 2. The set Ū is not contained in Stab(v̄), for every v̄ ∈ V̄ . The quotient space X̄ρ is δ̄-hyperbolic (Lemma 2.1.27 (i)) and the action of Γ̄ on X̄ρ is acylindrical (Lemma 2.1.35). Then Γ̄ falls exactly in one of the following three situations (Lemma 2.1.22): (a) Γ̄ is elliptic, or equivalently one (hence any) orbit of Γ̄ is bounded. Since the set Ū is not contained in Stab(v̄), for every v̄ ∈ V̄ , there exists an elliptic subgroup E ⊂ G for the action of G on X such that the natural projection π : G↠ Ḡ induces an isomorphism E ∼−→ Γ̄ (Lemma 2.1.31). Since G has ξ-uniform uniform exponential growth, the subgroup E is either virtually nilpotent or has ξ-uniform exponential growth. In combination with the isomorphism F ∼−→ Γ̄, we deduce that Γ̄ either is virtually nilpotent or has ξ-uniform exponential growth. (b) Γ̄ is loxodromic, or equivalently Γ̄ is virtually cyclic and contains a loxodromic element. Then Γ̄ is virtually nilpotent. 125 Chapter 2 – Uniform uniform exponential growth in small cancellation groups (c) Γ̄ is non-elementary, or equivalently Γ̄ contains a free group F2 of rank 2 and one (hence any) orbit of F2 is unbounded. There are two subcases: (E1) Large energy: L(Ū) > 104δ̄. Then ω(Ū) ⩾ 1 103 log 2 (Lemma 2.1.22 and Lemma 2.1.23). Note that here we do not require any control over the parameters of the acylindrical action of Γ̄ on X̄ρ. (E2) Small energy: L(Ū) ⩽ 104δ̄. Since Ū is not contained in Stab(v̄), for every v̄ ∈ V̄ , and 104δ̄ ⩽ ρ/5, there exists a pre-image U ⊂ G of Ū of energy L(U) ⩽ π sinh 104δ̄ (Lemma 2.1.32). Without loss of generality, we may assume that U is symmetric and contains the identity. Since Γ̄ is non-elementary for the action on X̄ρ, the subgroup Γ is non-elementary for the action on X (Lemma 2.1.29). According to Lemma 2.5.4, there exists β ∈ (0, 1) depending on N such that ω(Ū) ⩾ βω(U). Since G has ξ-uniform uniform exponential growth and Γ is non-elementary, we have ω(U) ⩾ ξ. Therefore, ω(Ū) ⩾ βξ. This completes the proof of our theorem. Theorem 2.5.6 (Theorem 0.6.2 (ii)). — Let ξ > 0. If there exists a geometric C ′′(λ, ε)- small cancellation quotient of G that has ξ-uniform uniform exponential growth, then G has ξ′-uniform uniform exponential growth. The constant ξ′ depends only on ξ. Proof. — Let ξ > 0. Assume that Ḡ has ξ-uniform uniform exponential growth. Let U ⊂ G be a finite symmetric subset containing the identity and denote Γ = ⟨U⟩. Then Γ falls exactly in one of the following three situations (Lemma 2.1.22): (a) Γ is elliptic, or equivalently one (hence any) orbit of Γ is bounded. The projection π : G ↠ Ḡ induces an isomorphism Γ ∼−→ Γ̄ (Lemma 2.1.30). Since Ḡ has ξ-uniform uniform exponential growth, the subgroup Γ̄ is either virtually nilpotent or has ξ-uniform exponential growth. In combination with the isomorphism Γ ∼−→ Γ̄, we deduce that Γ is either virtually nilpotent or has ξ-uniform exponential growth. (b) Γ is loxodromic, or equivalently Γ is virtually cyclic and contains a loxodromic element. Then Γ is virtually nilpotent. 126 2.5. Growth in small cancellation groups (c) Γ is non-elementary, or equivalently Γ contains a free group F2 of rank 2 and one (hence any) orbit of F2 is unbounded. There are two subcases: (E1) Large energy: L(U) > 104δ0. Then ω(U) ⩾ 1 103 log 2 (Lemma 2.1.22 and Lemma 2.1.23). Note that here we do not require any control over the parameters of the acylindrical action of Γ on X . (E2) Small energy: L(U) ⩽ 104δ0. By definition, ω(U) ⩾ ω(Ū). Since Γ is non-elementary for its action on X , we have ω(U) > 0. Since 104δ0 ⩽ π sinh 104δ̄, it follows from Lemma 2.5.4 that ω(Ū) > 0. In particular Γ̄ is not virtually nilpotent. Since Ḡ has ξ-uniform uniform exponential growth, we deduce that ω(Ū) ⩾ ξ. Therefore, ω(U) ⩾ ξ. 127 Appendix A PROPERTIES OF CONSTRICTING SUBSETS Contents A.1 Standard properties . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.2 Behrstock inequality . . . . . . . . . . . . . . . . . . . . . . . . 134 A.3 Morseness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 In this section, we fix constants µ ⩾ 1, ν ⩾ 0 and a (µ, ν)-path system space (X,P). We will be looking closely at the geometric features of the constricting subsets of X. A.1 Standard properties The goal of this subsection is to bring together the essential properties of constricting maps that can be deduced from the definition. Proposition A.1.1. — For every δ ⩾ 0, there exist a constant θ ⩾ 0 and a pair of maps, σ : R⩾1 × R⩾0 → R⩾0 and ζ : R⩾0 → R⩾0, such that any δ-constricting map πA : X → A satisfies the following properties: (1) Coarse nearest-point projection. For every x ∈ X, we have |x− πA(x)| ⩽ µd(x,A) + θ. (2) Coarse gate map. Let x ∈ X and a ∈ A. Let γ ∈ P joining x to a. If a′ is the entrance point of γ on A+δ, then |a′ − πA(x)| ⩽ θ. (3) Coarse equivariance. Let G be a group acting by isometries on A such that P is G-invariant. Then for every g ∈ G and for every x ∈ X, we have |πA(gx) − gπA(x)| ⩽ θ. 128 A.1. Standard properties (4) Coarse Lipschitz map. For every x, y ∈ X, we have |x− y|A ⩽ µ|x− y| + θ. (5) Intersection–Image. For every γ ∈ P, we have | diam(A+δ ∩ γ) − diamA(γ)| ⩽ θ. (6) Behrstock inequality. Let πB : X → B be a δ-constricting map. Then for every x ∈ X, we have min {dA(x,B), dB(x,A)} ⩽ θ. (7) Tight contraction. Let x ∈ X and r = 1 µ d(x,A). Then diamA(BX(x, r)) ⩽ θ. (8) Morseness. Let κ ⩾ 1, l ⩾ 0. Let α be a (κ, l)-quasi-geodesic of X with endpoints in A. Then α ⊂ A+σ(κ,l). (9) Coarse invariance. Let ε ⩾ 0. Let B ⊂ X be a subset such that dHaus(A,B) ⩽ ε. Then B is ζ(ε)- constricting. (10) Coarse uniqueness. Let ε ⩾ 0. Let πB : X → B be a δ-constricting map such that dHaus(A,B) ⩽ ε. Then for every x ∈ X, we have |πA(x) − πB(x)| ⩽ ζ(ε). Proof. — Our proves are based on the sketches of the following references. For (1), (4), (5) and (7), see [74, Lemma 2.4]. For (2), see [74, Lemma 5.2 (3)], [74, Lemma 5.2 (3)]. For (6), see [74, Lemma 2.5]. For (8), see [74, Lemma 2.8 (1)]. Let δ ⩾ 0. We put θ = max 1⩽i⩽7 θi and for every ε ⩾ 0, we put ζ(ε) = max 9⩽i⩽10 ζi(ε) ⩾ 0, where θi ⩾ 0 and ζi(ε) ⩾ 0 are constants whose exact values will be precised below, with the exception of θ6, which is the constant of Proposition A.2.1. Let πA : X → A be a δ-constricting map. (1) Let x ∈ X. Let a ∈ A such that |x− a| ⩽ d(x,A) + 1. Assume first that |x− a| ⩽ δ. By the triangle inequality, |x− πA(x)| ⩽ |x− a| + |a− πA(a)| + |a− x|A. By (CS1), we have |a− πA(a)| ⩽ δ. Thus, |x− πA(x)| ⩽ d(x,A) + 2δ + 1. 129 Chapter A – Appendix to Chapter 1 Assume now that |x− a|A > δ. Let γ ∈ P joining x to a. By (CS2), there exists a point p in γ such that |p− πA(x)| ⩽ δ. By the triangle inequality, |x− πA(x)| ⩽ |x− p| + |p− πA(x)|. Since γ is a (µ, ν)-quasi-geodesic, |x− p| ⩽ µ|x− a| + ν. In conclusion, |x− πA(x)| ⩽ µd(x,A) + µ+ ν + δ. Finally, we put θ1 = max {2δ + 1, µ+ ν + δ}. (2) Let x ∈ X and a ∈ A. Let γ ∈ P joining x to a. Let a′ be the entrance point of γ on A+δ. Assume first that |x− a′|A ⩽ δ. By the triangle inequality, |πA(x) − a′| ⩽ |x− a′|A + |πA(a′) − a′|. It follows from (1) that |πA(a′) − a′| ⩽ µδ + θ1. Consequently, |πA(x) − a′| ⩽ δ + µδ + θ1. Assume now that |x− a′|A > δ. Since [x, a′]γ ∈ P, it follows from (CS2) that there exists a point p in [x, a′]γ such that |πA(x) − p| ⩽ δ. By definition of a′, we have p = a′ and hence |πA(x) − a′| ⩽ δ. Finally, we put θ2 = max {δ + µδ + θ1, δ}. (3) Let G be a group acting by isometries on A such that P is G-invariant. Let g ∈ G and x ∈ X. Let γ ∈ P joining x to πA(x). Let a′ be the entrance point of γ on A+δ. By the triangle inequality, |πA(gx) − gπA(x)| ⩽ |πA(gx) − ga′| + |ga′ − gπA(x)|. Since A is G-invariant, the element ga′ is the entrance point of gγ on A+δ. Since P is G-invariant, the path gγ belongs to P. It follows from (2) that max {|πA(gx) − ga′|, |ga′ − gπA(x)|} ⩽ θ2. 130 A.1. Standard properties Consequently, |πA(gx) − gπA(x)| ⩽ 2θ2. Finally, we put θ3 = 2θ2. (4) Let x, y ∈ X. It suffices to assume that |x− y|A > δ. Let γ ∈ P joining x to y. By (CS2), there exist p, q ∈ γ such that max {|πA(x) − p|, |πA(y) − q|} ⩽ δ. By the triangle inequality, |x− y|A ⩽ |πA(x) − p| + |p− q| + |q − πA(y)|. Since γ is a (µ, ν)-quasi-geodesic, |p− q| ⩽ µ|x− y| + ν. Consequently, |x− y|A ⩽ µ|x− y| + 2δ + ν. Finally, we put θ4 = 2δ + ν. (5) Let γ ∈ P. First we prove that diamA(γ) ⩽ diam(A+δ ∩ γ) + θ5. Let x, y ∈ γ. It suffices to assume that |x− y|A > δ. Since [x, y]γ ∈ P , there exist p, q ∈ [x, y]γ such that max {|πA(x) − p|, |πA(y) − q|} ⩽ δ. By the triangle inequality, |x− y|A ⩽ |πA(x) − p| + |p− q| + |q − πA(y)|. Since p, q ∈ A+δ ∩ γ, we have |p− q| ⩽ diam(A+δ ∩ γ). Hence, |x− y|A ⩽ diam(A+δ ∩ γ) + 2δ. Now we prove that diam(A+δ ∩ γ) ⩽ diamA(γ) + θ5. Let x, y ∈ A+δ ∩ γ. By the 131 Chapter A – Appendix to Chapter 1 triangle inequality, |x− y| ⩽ |x− πA(x)| + |x− y|A + |πA(y) − y|. By (1), max {|πA(x) − x|, |πA(y) − y|} ⩽ µδ + θ1. Since πA(x), πA(y) ∈ πA(γ), we have |x− y|A ⩽ diamA(γ). Hence, |x− y| ⩽ diamA(γ) + 2µδ + 2θ1. Finally, we put θ5 = max{2δ, 2µδ + 2θ1}. (6) We refer to Proposition A.2.1 for this proof. (7) Let x ∈ X and r = 1 µ d(x,A). It suffices to prove that for every y ∈ X, if |x− y| ⩽ r, then |x − y|A ⩽ 3δ + ν. We argue by contraposition. Let y ∈ X such that |x − y|A > 3δ + ν. Let γ : [0, L] → X be a path of P joining x to y. Since γ is a (µ, ν)-quasi-geodesic, |x− y| ⩾ 1 µ L− 1 µ ν. By (CS2), there exist p, q ∈ γ such that max {|πA(x) − p|, |πA(y) − q|} ⩽ δ. Let s, t ∈ [0, L] such that p = γ(s) and q = γ(t). We note that L ⩾ max {s, t} ⩾ min {s, t} + |s− t|. Thus, |x− y| ⩾ 1 µ min {s, t} + 1 µ |s− t| − 1 µ ν. By the triangle inequality, s ⩾ |x− p| ⩾ |x− πA(x)| − |πA(x) − p|, t ⩾ |x− q| ⩾ |x− πA(y)| − |πA(y) − q|, |s− t| ⩾ |p− q| ⩾ |x− y|A − |p− πA(x)| − |q − πA(y)|. 132 A.1. Standard properties In particular, min {s, t} ⩾ d(x,A) − δ, |s− t| > δ + ν. Therefore, |x− y| > 1 µ d(x,A). Finally, we put θ7 = 2(3δ + ν). (8) We refer to Proposition A.3.4 for this proof. (9) For every ε ⩾ 0, we put ζ9(ε) = δ + 2(ε+ 2). Let ε ⩾ 0. Let B ⊂ X be a subset such that dHaus(A,B) ⩽ ε. We define a map πB : X → B as follows. Since A ⊂ B+ε+1, for every x ∈ X, there exists b ∈ B such that |b − πA(x)| ⩽ ε + 2. We put πB(x) = b. We prove that the map πB : X → B is ζ9-constricting. Let x ∈ B. By the triangle inequality, |πB(x) − x| ⩽ |πB(x) − πA(x)| + |πA(x) − x|. By (CS1), we have |πA(x) − x| ⩽ δ. Therefore, we obtain |πB(x) − x| ⩽ ζ9. This establishes (CS1). Let y, z ∈ X such that |y − z|B > ζ9. Let γ ∈ P joining y to z. By the triangle inequality, |y − z|A ⩾ |y − z|B − |πB(y) − πA(y)| − |πB(z) − πA(z)|. Consequently, we have |y− z|A > δ. Therefore, it follows from (CS2) that there exist p, q ∈ γ such that max {|πA(y) − p|, |πA(z) − q|} ⩽ δ. By the triangle inequality, |πB(y) − p| ⩽ |πB(y) − πA(y)| + |πA(y) − p|. Therefore, we have |πB(y) − p| ⩽ ζ9. By symmetry, we obtain |πB(z) − q| ⩽ ζ9. This establishes (CS2). (10) Let ε ⩾ 0. Let πB : X → B be a δ-constricting map such that dHaus(A,B) ⩽ ε. Let x ∈ X. We bound |πA(x) − πB(x)|. Let γ ∈ P joining x to πA(x). Let a′ be the entrance point of γ on A+ε+1+δ. By the triangle inequality, |πA(x) − πB(x)| ⩽ |x− a′|A + |πA(a′) − a′| + |a′ − πB(a′)| + |a′ − x|B. 133 Chapter A – Appendix to Chapter 1 Since a′ ∈ A+ε+1+δ and A+ε+1+δ ⊂ B+2ε+2+δ, it follows from (1) that max {|πA(a′) − a′|, |a′ − πB(a′)|} ⩽ θ1(2ε+ 2 + δ) + θ1. Applying now (5) we obtain, max {|x− a′|A, |a′ − x|B} ⩽ max {diam(A+δ ∩ [x, a′]γ), diam(B+δ ∩ [x, a′]γ)} + θ5. Since A+δ, B+δ ⊂ A+ε+1+δ and since [x, a′]γ ∩ A+ε+1+δ = {a′}, max {|x− a′|A, |a′ − x|B} ⩽ θ5. Therefore, we have |πA(x) − πB(x)| ⩽ 2θ5 + 2θ1(2ε+ 2 + δ) + 2θ1. Finally, we put ζ10(ε) = 2θ5 + 2θ1(2ε+ 2 + δ) + 2θ1. A.2 Behrstock inequality The goal of this subsection is to introduce a variant of Behrstock inequality, [11], in the context of Masur-Minsky subsurface projections. Proposition A.2.1 ([74, Lemma 2.5]). — For every δ ⩾ 0, there exists θ ⩾ 0 satisfying the following. Let πA : X → A and πB : X → B be δ-constricting maps. Then for every x ∈ X, min {dA(x,B), dB(x,A)} ⩽ θ. Remark A.2.2. — The idea is that if dA(x,B) is large then A is “between” x and B. Proof. — Let δ ⩾ 0. Let θ0 = θ0(δ) ⩾ 0 be the constant of Proposition A.1.1. Let θ > θ0 +1. Its exact value will be precised below. Let πA : X → A and πB : X → B be δ-constricting maps. Let x ∈ X. By symmetry, it suffices to show that if dA(x,B) > θ, then dB(x,A) ⩽ θ. Assume that dA(x,B) > θ. Let b ∈ B and consider a path γ ∈ P joining x to b. Claim A.2.3. — A+δ ∩ γ ̸= ∅. 134 A.2. Behrstock inequality By Proposition A.1.1 (5) Intersection–Image, diam(A+δ ∩γ) ⩾ diamA(γ)−θ0. Moreover, diamA(γ) ⩾ |x− b|A ⩾ dA(x,B). Since dA(x,B) > θ0 + 1, we obtain diam(A+δ ∩ γ) > 0. This proves the claim. Since A+δ ∩ γ ̸= ∅ we can consider the entrance point a′ of γ on A+δ. Claim A.2.4. — B+δ ∩ [x, a′]γ = ∅. To argue by contradiction, assume that there exists y ∈ B+δ ∩ [x, a′]γ. In particular, there exists b′ ∈ B such that |y − b′| ⩽ δ + 1. By the triangle inequality, dA(x,B) ⩽ |x− b′|A ⩽ |x− y|A + |y − b′|A. Since [x, a′]γ ∈ P and A+δ ∩ [x, a′]γ = {a′}, it follows from Proposition A.1.1 (5) Intersection–Image that |x− y|A ⩽ diamA([x, a′]γ) ⩽ diam(A+δ ∩ [x, a′]γ) + θ0 ⩽ θ0. By Proposition A.1.1 (4) Coarse Lipschitz map, |y − b′|A ⩽ µ(δ + 1) + θ0. Hence dA(x,B) ⩽ θ. Contradiction. Therefore B+δ ∩ [x, a′]γ = ∅. This proves the claim. Finally, we estimate dB(x,A). Let a ∈ A. By the triangle inequality, dB(x,A) ⩽ |x− πA(x)|B ⩽ |x− a′|B + |a′ − πA(x)|B. Since B+δ ∩ [x, a′]γ = ∅, it follows from Proposition A.1.1 (5) Intersection–Image that |x− a′|B ⩽ diamB([x, a′]γ) ⩽ diam(B+δ ∩ [x, a′]γ) + θ0 ⩽ θ0. Applying together Proposition A.1.1 (2) Coarse gate map and (4) Coarse Lipschitz map, we have |a′ − πA(x)|B ⩽ µθ0 + θ0. Consequently, we obtain dB(x,A) ⩽ θ for θ = max {θ0 + 1, µ(δ + 1) + 2θ0, 2θ0 + µθ0}. 135 Chapter A – Appendix to Chapter 1 A.3 Morseness There is a large number of different notions of convexity that coincide with quasi- convexity in hyperbolic spaces but differ in more general metric spaces. One of them is the notion of Morseness. DefinitionA.3.1 (Morseness). — Let σ : R⩾1 ×R⩾0 → R⩾0. A subset Y ⊂ X is σ-Morse if for every κ ⩾ 1, l ⩾ 0, any (κ, l)-quasi-geodesic of X with endpoints in Y is contained in the σ(κ, l)-neighbourhood of Y . Example A.3.2. — A geodesic metric space X is hyperbolic if and only if there exists σ : R⩾1 × R⩾0 → R⩾0 such that any geodesic segment of X is σ-Morse [61, Lemma 7.2]. The goal of this section is to show that constricting subsets of X are Morse. We would like to emphasize that it is possible to give a proof that does not involve the path system, following the argument of [74, Lemma 2.8]. For this reason, we introduce the following notion of convexity. Definition A.3.3 (Weak contraction). — Let M ⩾ 1, ∆ ⩾ 0. A map πA : X → A from X to a subset A ⊂ X is (M,∆)-weakly contracting if it verifies the following properties. (WC1) Coarse nearest-point projection. For every x ∈ X, we have |x− πA(x)| ⩽Md(x,A) + ∆. (WC2) Contraction. Let x ∈ X and r = 1 M d(x,A) − ∆. Then diamA(BX(x, r)) ⩽ ∆. A subset A ⊂ X is (M,∆)-weakly contracting if there exists a (M,∆)-contracting map πA : X → A. It follows from Proposition A.1.1 (1) Coarse nearest point projection and Propo- sition A.1.1 (7) Tight contraction that δ-constricting subsets of X are always weakly contracting with constants depending on µ, ν, δ. PropositionA.3.4 ([74, Lemma 2.8]). — For every M ⩾ 1, ∆ ⩾ 0, there exists σ : R⩾1 × R⩾0 → R⩾0 such that any (M,∆)-weakly contracting subset A ⊂ X is σ-Morse. In particular, constricting subsets are Morse. Remark A.3.5. — There exist a metric space X containing a Morse subset that is not weakly contracting, [69, Example 3.8]. In particular this Morse subset is not constricting with respect to any path system of X. 136 A.3. Morseness From now on in this subsection, we fix M ⩾ 1, ∆ ⩾ 0. Let πA : X → A be a (M,∆)-weakly contracting map. The idea of the proof consists on finding a large enough neighbourhood of A so that given a quasi-geodesic with endpoints in A, the subpaths that intersect this neighbourhood only at both endpoints have uniformly bounded length. The purpose of the following lemmas is to estimate these lengths. The neighbourhood will depend only on the rescaling constant of the quasi-geodesic. Lemma A.3.6. — For every κ ⩾ 1, l ⩾ 0, η ⩾ 0, there exists θ ⩾ 0 with the following property. Let α be a (κ, l)-quasi-geodesic of X such that α ∩ A+η = {α−, α+}. Then |α− − α+|A ⩾ 1 κ ℓ(α) − θ. Proof. — Let κ ⩾ 1, l ⩾ 0, η ⩾ 0. Let θ ⩾ 0. Its exact value will be precised below. Let α be a (κ, l)-quasi-geodesic of X such that α ∩ A+η = {α−, α+}. By the triangle inequality, |α− − α+|A ⩾ |α− − α+| − |α− − πA(α−)| − |α+ − πA(α+)|. Since α is a (κ, l)-quasi-geodesic, |α− − α+| ⩾ 1 κ ℓ(α) − 1 κ l. Since α−, α+ ∈ A+η, it follows from (WC1) that max {|α− − πA(α−)|, |α+ − πA(α+)|} ⩽Mη + ∆. Finally, we put θ = 1 κ l + 2Mη + 2∆ Lemma A.3.7. — For every κ > 0, there exists η ⩾ 0 with the following property. Let α be a path of X such that d(α,A) ⩾ η. Then |α− − α+|A ⩽ 1 κ ℓ(α) + ∆ + 1. Proof. — Let κ > 0. Let η = M(∆ + 1)κ + M∆. Let α : [0, L] → X be a path such that d(α,A) ⩾ η. We estimate |α− − α+|A. Let ζ = (∆ + 1)κ. Since ζ > 0, we can define m = ⌊ L ζ ⌋ + 1. We fix a partition 0 = t0 ⩽ t1 ⩽ · · · ⩽ tm = L of [0, L] such that |tm−1 − tm| ⩽ ζ and such that if m ⩾ 2, then for every i ∈ J0,m−2K, we have |ti − ti+1| = ζ. 137 Denote xi = α(ti). By the triangle inequality, |α− − α+|A ⩽ m−1∑ i=0 |xi − xi+1|A. Let i ∈ J0,m − 1K. Recall that, by convention, all of our paths are parametrised by arc length. Hence, |xi − xi+1| ⩽ ζ. Moreover, ζ = 1 M η − ∆. Consequently, |xi − xi+1| ⩽ 1 M d(xi, A) − ∆. Denote ri = 1 M d(xi, A) − ∆. By (WC2), |xi − xi+1|A ⩽ diamA(BX(xi, ri)) ⩽ ∆. Hence, |x− y|A ⩽ m(∆ + 1). By construction of the partition, m ⩽ ζ−1L+ 1. Therefore, |x− y|A ⩽ 1 κ L+ ∆ + 1. We are ready to proof the proposition: Proof of Proposition A.3.4. — Let κ ⩾ 1, l ⩾ 0. Let α be a (κ, l)-quasi-geodesic of X with endpoints in A. Let κ0 > κ. It follows from Lemma A.3.6 and Lemma A.3.7 that there exist η = η(κ0) ⩾ 0 and θ = θ(κ, l, η) ⩾ 0 such that for every subpath β of α satisfying β ∩ A+η = {β−, β+}, we have ℓ(β) ⩽ (1 κ − 1 κ0 )−1 (∆ + 1 + θ). Moreover, we can decompose α as an union of subpaths that either intersect A+η only at both endpoints or are contained in A+η. This is enough to prove that there exist σ : R⩾1 × R⩾0 → R⩾0 such that A is σ-Morse. 138 BIBLIOGRAPHY [1] C. R. Abbott and F. Dahmani. Property P-naive for acylindrically hyper- bolic groups. 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IMRN, (23):7259–7323, 2019. doi: 10.1093/imrn/rny001. 145 https://doi.org/10.2140/agt.2021.21.2995 https://arxiv.org/abs/1809.09303 https://doi.org/10.48550/ARXIV.1809.09303 https://doi.org/10.48550/ARXIV.1809.09303 https://doi.org/10.1007/s00209-021-02811-w https://doi.org/10.1007/s00209-021-02811-w https://doi.org/10.1007/s002220050172 https://doi.org/10.4171/LEM/59-1-6 https://doi.org/10.1515/crelle-2015-0093 https://doi.org/10.1515/crelle-2015-0093 http://www.numdam.org/item?id=PMIHES_1979__50__171_0 https://doi.org/10.2307/1970688 https://doi.org/10.1093/imrn/rny001 https://doi.org/10.1093/imrn/rny001 Titre : Croissance dans les groupes à courbure négative ou nulle Mot clés : théorie géométrique des groupes, groupes hyperboliques et leurs généralisations, croissance exponentielle, théorie de la petite simplification Résumé : L’objectif de cette thèse est d’obte- nir une meilleure compréhension du compor- tement des taux de croissance exponentiels au sein de la classe des groupes qui agissent de manière acylindrique dans un espace hy- perbolique au sens de Gromov. Pour ce faire, nous aborderons deux problèmes de nature différente. Dans le premier problème, nous étudie- rons les taux de croissance exponentiels des sous-groupes quasi-convexes. Nous compare- rons ces taux avec celui du groupe ambiant et nous déterminerons quand il est possible d’ob- tenir une égalité/inégalité stricte. Pour ce faire, nous allons exploiter des actions propres sur des espaces métriques, a priori, non hyperbo- liques, mais dont les isométries se comportent comme les isométries loxodromiques d’un es- pace hyperbolique. Le deuxième problème tourne autour de la croissance exponentielle uniforme uniforme. Nous prouverons que cette propriété est pré- servée si nous prenons des quotients à petite simplification de groupes qui agissent de ma- nière acylindrique sur un espace hyperbolique. En corollaire, nous obtiendrons qu’il existe une borne inférieure universelle sur le taux de croissance exponentielle uniforme pour la fa- mille des quotients à petite simplification clas- sique. Cette borne ne dépend que d’un des deux paramètres d’acylindricité. Title: Growth in groups of non-positive curvature Keywords: geometric group theory, hyperbolic groups and their generalisations, exponential growth, small cancellation theory Abstract: The aim of this thesis is to obtain a better understanding of the behavior of expo- nential growth rates within the class of groups that act acylindrically in a hyperbolic space in the sense of Gromov. To do this, we will ad- dress two problems of a different nature. In the first problem we will study the ex- ponential growth rates of quasi-convex sub- groups. We will compare these rates with that of the ambient group and we will deter- mine when it is possible to obtain strict equal- ity/inequality. To do so, we will exploit proper actions on metric spaces that, a priori, are not hyperbolic, but that have isometries that be- have like the loxodromic isometries of a hyper- bolic space. The second problem revolves around uni- form uniform exponential growth. We will prove that this property is preserved if we take small cancellation quotients of groups that act acylindrically on a hyperbolic space. As a corollary, we will obtain that there is a uni- versal lower bound on the uniform exponential growth rate for the family of classical small can- cellation quotients. This bound depends only on one of the two acylindricity parameters. Tesis Xabier Legaspi Juanatey Tittle page Table of contents Resumen en español Résumé en français Résumé Croissance des sous-groupes quasi-convexes Croissance exponentielle uniforme uniforme Introduction Abstract Growth of quasi-convex subgroups Uniform uniform exponential growth Notation Growth of quasi-convex subgroups in groups with a constricting element Path system geometry Growth estimation criteria Buffering sequences Quasi-convexity in the Intersection–Image property Finding a quasi-convex element Constricting elements A –invariant family Finding a constricting element Elementary closures Forcing a geometric separation Growth of quasi-convex subgroups Uniform uniform exponential growth in small cancellation groups Hyperbolic geometry Hyperbolicity Quasi-convexity Isometries Group action on a -hyperbolic space Small cancellation theory Reduced subsets Broken geodesics Quasi-isometric embedding of a free group Geodesic extension property Growth in groups acting on a -hyperbolic space Growth of maximal loxodromic subgroups. Producing reduced subsets Growth trichotomy Shortening and shortening-free words Shortening words The growth of shortening-free words The injection of shortening-free words Growth in small cancellation groups Properties of constricting subsets Standard properties Behrstock inequality Morseness Bibliography