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

TL;DR

This paper proves that irreducible, free measure-preserving actions of right-angled Artin groups are orbit equivalent only if they are conjugate, establishing a superrigidity result and $W^*$-superrigidity for certain Bernoulli actions.

Contribution

It demonstrates orbit equivalence superrigidity for irreducible right-angled Artin group actions and establishes $W^*$-superrigidity for a broad class of Bernoulli actions.

Findings

Orbit equivalence implies conjugacy for irreducible right-angled Artin group actions.

Superrigidity results extend to actions of groups stably orbit equivalent to mildly mixing actions.

$W^*$-superrigidity is established for Bernoulli actions of specific ICC groups, including many Artin groups.

Abstract

Let $G_\Gamma\curvearrowright X$ and $G_\Lambda\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma\curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use works of Popa and Ioana-Popa-Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all ICC groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended non-abelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.