# On the loxodromic actions of Artin-Tits groups

## TL;DR

This paper investigates the behavior of elements in Artin-Tits groups acting on a hyperbolic complex, proposing a conjecture that most elements act loxodromically, and provides conditions under which this holds.

## Contribution

It introduces a condition ensuring a positive proportion of loxodromic elements in large balls of the Cayley graph of certain Artin-Tits subgroups.

## Key findings

- The condition applies to Artin-Tits groups of spherical type.
- It holds for pure subgroups and some commutator subgroups.
- Most elements in these groups act loxodromically.

## Abstract

Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that "most" elements of Artin-Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup $G$ of an Artin-Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin-Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.