RT Journal Article
T1 Parabolic subgroups acting on the additional length graph
A1 Antolín Pichel, Yago
A1 Cumplido, María
AB Let 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.
PB Mathematical Sciences Publishers (MSP)
SN 1472-2747
YR 2021
FD 2021
LK https://hdl.handle.net/20.500.14352/8624
UL https://hdl.handle.net/20.500.14352/8624
LA eng
NO [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 Zbl
NO Ministerio de Ciencia e Innovación (MICINN)
NO Ministerio de Economía y Competitividad (MINECO)
NO Junta de Andalucía
DS Docta Complutense
RD 28 abr 2024