Acylindricity of the action of right-angled Artin groups on extension graphs

TL;DR

This paper proves that the $r$-quasi-stabilizer in the action of right-angled Artin groups on extension graphs is linearly bounded and explores lattice properties to improve understanding of group actions.

Contribution

It shows the $r$-quasi-stabilizer is contained in a cyclic group with linear bounds, refining previous exponential bounds and advancing the understanding of group actions.

Findings

The $r$-quasi-stabilizer is a subset of a cyclic group.

The cardinality of the $r$-quasi-stabilizer is linearly bounded by $r$.

Improved lower bounds for minimal asymptotic translation length.

Abstract

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called $r$-quasi-stabilizer of a pair of distant points is bounded above by a function of $r$. The known upper bound of the cardinality is an exponential function of $r$. In this paper we show that the $r$-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of $r$. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right angled Artin group on its extension graph.