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

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 rightangled Artin group is superrigid 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.

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

“…Theorem 1 covers a much larger class of right-angled Artin groups than our previous work [HH20b], including many examples with infinite outer automorphism group. For example, it applies to all right-angled Artin groups whose defining graph is a tree of diameter at least 3, which are usually less rigid from other viewpoints (for instance, they are all quasi-isometric to each other [BN08], and the problem of their measure equivalence classification is open).…”

“…In [HH20b], we started to investigate the class of right-angled Artin groups from the viewpoint of measured group theory. These groups are of basic importance (see e.g.…”

“…On the rigidity side, we proved in [HH20b] that if two right-angled Artin groups G Γ , G Λ with finite outer automorphism groups admit free, ergodic, measure-preserving actions on standard probability spaces which are orbit equivalent, or merely stably orbit equivalent (equivalently if the groups are measure equivalent), then G Γ and G Λ are isomorphic. However, rigidity fails beyond this context: given any right-angled Artin group G Γ , and any group H which is a graph product of countably infinite amenable groups over the same graph Γ, we can build free, ergodic, probability measure-preserving actions of G Γ and H which are orbit equivalent [HH20b, Proposition 4.2].…”

“…However, rigidity fails beyond this context: given any right-angled Artin group G Γ , and any group H which is a graph product of countably infinite amenable groups over the same graph Γ, we can build free, ergodic, probability measure-preserving actions of G Γ and H which are orbit equivalent [HH20b, Proposition 4.2]. In fact, our proof of [HH20b,Proposition 4.2] shows that starting from any action G Γ Z as above, we can find a blown-up action G Γ Ẑ (i.e. coming with a G Γ -equivariant map Ẑ → Z) which fails to be superrigid for orbit equivalence.…”

“…coming with a G Γ -equivariant map Ẑ → Z) which fails to be superrigid for orbit equivalence. We can also build two actions of G Γ which are orbit equivalent but not conjugate [HH20b,Remark 4.4].…”