Euclidean Artin-Tits groups are acylindrically hyperbolic

TL;DR

This paper proves that Euclidean Artin-Tits groups are acylindrically hyperbolic by embedding them into crystallographic Garside groups and analyzing their actions on associated hyperbolic graphs.

Contribution

It extends acylindrical hyperbolicity results to Euclidean Artin-Tits groups via embedding into crystallographic Garside groups and hyperbolic graph actions.

Findings

Euclidean Artin-Tits groups act loxodromically on hyperbolic graphs.

They exhibit WPD elements indicating acylindrical hyperbolicity.

Embedding into Garside groups enables analysis of their hyperbolic actions.

Abstract

In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the \emph{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 \emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an \emph{infinite-type} Garside group which they call a \emph{crystallographic Garside group}. We associate a \emph{hyperbolic} additional length graph to this crystallographic Garside group and we exhibit elements of the euclidean Artin-Tits group which act loxodromically and WPD on this hyperbolic graph.