Acylindrical hyperbolicity and Artin-Tits groups of spherical type

TL;DR

This paper proves that irreducible Artin-Tits groups of spherical type, when quotiented by their centers, are acylindrically hyperbolic, using the study of associated length graphs and generic element actions.

Contribution

It establishes acylindrical hyperbolicity for these groups by analyzing their length graphs and generic element actions, a novel approach in this context.

Findings

The quotient of an irreducible Artin-Tits group of spherical type by its center is acylindrically hyperbolic.

Constructs a specific element with loxodromic and WPD actions on the length graph.

Shows that generic elements act loxodromically, either via random walks or in large Cayley graph balls.

Abstract

We prove that, for any irreducible Artin-Tits group of spherical type $G$, the quotient of $G$ by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on $G$, and constructing a specific element $x_G$ of $G/Z(G)$ whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that "generic" elements of $G$ act loxodromically, where the word "generic" can be understood in either of the two common usages: as a result of a long random walk or as a random element in a large ball in the Cayley graph.