Orbit equivalence rigidity of irreducible actions of right-angled 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 the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes 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 nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.