Acylindrical hyperbolicity and Artin-Tits groups of spherical type

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 xG 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.