Dilatation of outer automorphisms of Right-angled Artin Groups

TL;DR

This paper investigates the growth rates of words under automorphisms of right-angled Artin groups, establishing conditions under which dilatation is realized on specific subgroups or strata.

Contribution

It introduces a measure of dilatation for automorphisms of right-angled Artin groups and characterizes where this dilatation is realized under certain conditions.

Findings

Dilatation is realized on free abelian subgroups or strata.

Positive dilatation implies existence of specific invariant subgroups.

Provides a framework for understanding automorphism dynamics in RAAGs.

Abstract

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(\Gamma) = \langle V | E \rangle$ and an automorphism $\phi \in Out(A(\Gamma))$ there is a natural measure of how fast the length of a word of $A(\Gamma)$ grows after $n$ iterations of $\phi$ as a function of $n$, which we call the dilatation of $w$ under $\phi$. We define the dilatation of $\phi$ as the supremum over dilatations of all $w \in A(\Gamma)$. Assuming that $\phi$ is a pure and square map, we show that if the dilatation of $\phi$ is positive, then either there exists a free abelian special subgroup on which that dilatation is realized; or there exists a strata of either free or free abelian groups on which the dilatation is realized.