Parabolic subgroups acting on the additional length graph

Antolín, Yago; Cumplido, María

doi:10.2140/agt.2021.21.1791

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Parabolic subgroups acting on the additional length graph

Yago Antolín and María Cumplido

Algebraic & Geometric Topology 21 (2021) 1791–1816

DOI: 10.2140/agt.2021.21.1791

Abstract

Let $A \neq A_{1}, A_{2}, I_{2 m}$ 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 $𝒞_{AL} (A)$ , 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 $g \in A$ 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 ${ω (A_{n}, 𝒮)}_{n \in ℕ}$ 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

Volume 21, issue 4 (2021)