On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs

TL;DR

This paper investigates the action of right-angled Artin groups on extension graphs, demonstrating a finiteness property and analyzing the rationality and denominators of asymptotic translation lengths.

Contribution

It introduces a finiteness property for these actions, proves rationality of translation lengths, and constructs examples with arbitrary denominators.

Findings

Asymptotic translation lengths are always rational.

If the defining graph has girth ≥ 6, all translation lengths share a common denominator.

For trees or small syllable length elements, translation lengths are integers.

Abstract

We study the right-angled Artin group action on the extension graph. We show that this action satisfies a certain finiteness property, which is a variation of a condition introduced by Delzant and Bowditch. As an application we show that the asymptotic translation lengths of elements of a given right-angled Artin group are always rational and once the defining graph has girth at least 6, they have a common denominator. We construct explicit examples which show the denominator of the asymptotic translation length of such an action can be arbitrary. We also observe that if either an element has a small syllable length or the defining graph for the right-angled Artin group is a tree then the asymptotic translation lengths are integers.