Quasi-isometric classification of right-angled Artin groups I: the finite out case

TL;DR

This paper proves that under certain conditions, right-angled Artin groups are quasi-isometric if and only if they are isomorphic, and provides an algorithm to determine quasi-isometry based on their defining graphs.

Contribution

It establishes a quasi-isometric classification for RAAGs with finite outer automorphism groups and introduces an algorithm for checking quasi-isometry from defining graphs.

Findings

RAAGs with finite Out are quasi-isometric iff they are isomorphic.

If only Out(G) is finite, G' is quasi-isometric to G iff G' is a finite index subgroup.

An algorithm is provided to determine quasi-isometry from defining graphs.

Abstract

Let $G$ and $G'$ be two right-angled Artin groups (RAAG). We show they are quasi-isometric iff they are isomorphic, under the assumption that $Out(G)$ and $Out(G')$ are finite. If only $Out(G)$ is finite, then $G'$ is quasi-isometric $G$ iff $G'$ is isomorphic to a finite index subgroup of $G$. In this case, we give an algorithm to determine whether $G$ and $G'$ are quasi-isometric by looking at their defining graphs.