Orbit equivalence rigidity of irreducible actions of right-angled Artin groups. Issue 4 (4th April 2023)

• Record Type: Journal Article
• Authors: Horbez, Camille; Huang, Jingyin; Ioana, Adrian
• Is Part Of: Compositio mathematica. Volume 159:Issue 4(2023)
• Page Start: 860
• Page End: 887
• Publication Date: 2023-04-04
• Subjects: Bernoulli actions -- cross-product von Neumann algebras -- mildly mixing actions -- orbit equivalence rigidity -- right-angled Artin groups -- stable orbit equivalence -- stable W*-equivalence -- W*-superrigidity; 37A20 -- 20F65 -- 20F36 -- 46L36; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X23007054
• Languages: English
• ISSNs: 0010-437X

Abstract

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.