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Parabolic subgroups acting on the additional length graph

Yago Antolín and María Cumplido

Algebraic & Geometric Topology 21 (2021) 1791–1816
Abstract

Let AA1,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 AZ(A) is acylindrically hyperbolic. We use these results to find an element g A such that P,gP 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)}n of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.

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Keywords
braid groups, Artin groups, Garside groups, parabolic subgroups, acylindrically hyperbolic groups, growth of groups, relative growth
Mathematical Subject Classification 2010
Primary: 20F36, 20F65
References
Publication
Received: 23 September 2019
Revised: 10 July 2020
Accepted: 30 July 2020
Published: 18 August 2021
Authors
Yago Antolín
Departamento de Matemáticas
Universidad Autónoma de Madrid
Instituto de Ciencias Matemáticas
Madrid
Spain
Algebra Geometría y Topología
Universidad Complutense de Madrid
Madrid
Spain
María Cumplido
IMB, UMR 5584, CNRS
Université Bourgogne Franche-Comté
Dijon
France
Departamento de Álgebra
Universidad de Sevilla
Sevilla
Spain