Antolín Pichel, YagoCumplido, María2023-06-172023-06-172021[1] E Artin, Theory of braids, Ann. of Math. 48 (1947) 101–126 MR Zbl [2] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69–89 MR Zbl [3] E Brieskorn, K Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972) 245–271 MR Zbl [4] M Calvez, B Wiest, Acylindrical hyperbolicity and Artin–Tits groups of spherical type, Geom. Dedicata 191 (2017) 199–215 MR Zbl [5] M Calvez, B Wiest, Curve graphs and Garside groups, Geom. Dedicata 188 (2017) 195–213 MR Zbl [6] M Calvez, B Wiest, Hyperbolic structures for Artin–Tits groups of spherical type, preprint (2019) arXiv To appear in Contemp. Math. [7] H S M Coxeter, The complete enumeration of finite groups of the form r2i=(ri,rj)kij=1, J. Lond. Math. Soc. 10 (1935) 21–25 Zbl [8] M Cumplido, On the minimal positive standardizer of a parabolic subgroup of an Artin–Tits group, J. Algebraic Combin. 49 (2019) 337–359 MR Zbl [9] M Cumplido, V Gebhardt, J González-Meneses, B Wiest, On parabolic subgroups of Artin–Tits groups of spherical type, Adv. Math. 352 (2019) 572–610 MR Zbl [10] F Dahmani, V Guirardel, D Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 1156, Amer. Math. Soc., Providence, RI (2017) MR Zbl [11] P Dehornoy, L Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc. 79 (1999) 569–604 MR Zbl [12] P Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972) 273–302 MR Zbl [13] T Delzant, Sous-groupes à deux générateurs des groupes hyperboliques, from “Group theory from a geometrical viewpoint” (É Ghys, A Haefliger, A Verjovsky, editors), World Sci., River Edge, NJ (1991) 177–189 MR Zbl [14] R Flores, J González-Meneses, On the growth of Artin–Tits monoids and the partial theta function, preprint (2018) arXiv [15] E Godelle, Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d’Artin–Tits, PhD thesis, Université de Picardie Jules Verne (2001) [16] E Godelle, Normalisateur et groupe d’Artin de type sphérique, J. Algebra 269 (2003) 263–274 MR Zbl [17] E Godelle, Parabolic subgroups of Garside groups, J. Algebra 317 (2007) 1–16 MR Zbl [18] P de la Harpe, Topics in geometric group theory, Univ. Chicago Press (2000) MR Zbl [19] H van der Lek, The homotopy type of complex hyperplane complements, PhD thesis, Katholieke Universiteit Nijmegen (1983) [20] L Paris, Parabolic subgroups of Artin groups, J. Algebra 196 (1997) 369–399 MR Zbl [21] L Paris, Artin monoids inject in their groups, Comment. Math. Helv. 77 (2002) 609–637 MR Zbl1472-274710.2140/agt.2021.21.1791https://hdl.handle.net/20.500.14352/8624Let A ≠ A1;A2;I2m be an irreducible Artin–Tits group of spherical type. We show that the periodic elements of A and the elements preserving some parabolic subgroup of A act elliptically on the additional length graph CAL(A), a hyperbolic, infinite diameter graph associated to A constructed by Calvez and Wiest to show that A/Z(A) is acylindrically hyperbolic. We use these results to find an element g ∈ A such that <P,g> ≅ P * <g> for every proper standard parabolic subgroup P of A. The length of g is uniformly bounded with respect to the Garside generators, independently of A. This allows us to show that, in contrast with the Artin generators case, the sequence ω(An,S)(with n ∈ N) of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.engjournal articlehttps://doi.org/10.2140/agt.2021.21.1791https://msp.org/agt/2021/21-4/p06.xhtmlopen access512.54Braid groupsArtin groupsGeometric group theoryÁlgebraGrupos (Matemáticas)1201 Álgebra