Measure equivalence classification of transvection-free right-angled Artin groups

TL;DR

This paper classifies measure equivalence of transvection-free right-angled Artin groups, showing it aligns with isomorphism classes for groups with finite outer automorphism groups, but reveals limitations in measure superrigidity.

Contribution

It establishes measure equivalence classification for transvection-free right-angled Artin groups and compares it with quasi-isometry, highlighting new rigidity and flexibility results.

Findings

Measure equivalence implies isomorphic extension graphs for these groups.

Groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic.

No right-angled Artin group is superrigid for measure equivalence in the strongest sense.

Abstract

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group $G$ is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when $G$ is nonabelian, the automorphism group of the universal cover of the Salvetti complex of $G$ always contains infinitely generated (non-uniform) lattices.