Group measure space decomposition of II_1 factors and W*-superrigidity

Abstract

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γin a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X) \rtimes Γand an arbitrary group measure space factor L^\infty(Y) \rtimes Λ, comes from a conjugacy of the actions. We also prove that for many groups Γin the family G, the Bernoulli actions of Γare W*-superrigid.

Final version. Some extra details have been added to improve the exposition

References in corpus (4)

• Bass-Serre rigidity results in von Neumann algebras
• Cocycle superrigidity for profinite actions of irreducible lattices
• Measure equivalence rigidity of the mapping class group
• Actions of F_\infty whose II_1 factors and orbit equivalence relations have prescribed fundamental group

Cited by in corpus (13)

• On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups
• Strong solidity of group factors from lattices in SO(n,1) and SU(n,1)
• Some unique group-measure space decomposition results
• W*-superrigidity for arbitrary actions of central quotients of braid groups
• OE and W* superrigidity results for actions by surface braid groups
• Primeness results for von Neumann algebras associated with surface braid groups
• W*-rigidity paradigms for embeddings of II${}_1$ factors
• Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups
• Measure equivalence classification of transvection-free right-angled Artin groups
• Orbit equivalence rigidity for product actions
• Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations
• Orbit equivalence superrigidity for type III${}_0$ actions
• W*-superrigidity for discrete quantum groups