Euclidean Artin–Tits groups are acylindrically hyperbolic

Abstract

In this paper, we prove that all Euclidean Artin–Tits groups are acylindrically hyperbolic. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the additional length graph and they used it to show that central quotients of Artin–Tits groups of spherical type are acylindrically hyperbolic. In general, a Euclidean Artin–Tits group is not a priori a Garside group but McCammond and Sulway have shown that it embeds into an infinite-type Garside group which they call a crystallographic Garside group. We associate a hyperbolic additional length graph to this crystallographic Garside group and we exhibit elements of the Euclidean Artin–Tits group which act loxodromically and weakly properly discontinuously on this hyperbolic graph.