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

TL;DR

This paper establishes unique decomposition results for certain group measure space II_1 factors, leading to W*-superrigidity for actions of specific groups, including free products and Bernoulli actions.

Contribution

It proves a unique crossed product decomposition for a broad class of group measure space II_1 factors, demonstrating W*-superrigidity for actions of groups like PSL(n,Z) and their free products.

Findings

Unique decomposition results for group measure space II_1 factors.

W*-superrigidity of free, mixing actions of certain amalgamated free products.

Bernoulli actions of many groups in the class are W*-superrigid.

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 \Gamma in 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 \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes \Lambda, comes from a conjugacy of the actions. We also prove that for many groups \Gamma in the family G, the Bernoulli actions of \Gamma are W*-superrigid.