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Parabolic subgroups acting on the additional length graph

Yago Antolín and María Cumplido

Algebraic & Geometric Topology 21 (2021) 1791–1816

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.

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