Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups

Abstract

We prove that for any free ergodic probability measure preserving action $Γ\actson (X,μ)$ of a non-elementary hyperbolic group, or a lattice in a rank one simple Lie group, the associated group measure space II_1 factor $L^{\infty }(X)⋊\Gamma$ has L^\infty(X) as its unique Cartan subalgebra, up to unitary conjugacy.

v3: Considerably simplified proof of the main result. v2: Improved exposition; new result proving uniqueness of Cartan for crossed products by irreducible lattices in products of rank one Lie groups

References in corpus (7)

• Examples of groups which are not weakly amenable
• 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
• Examples of group actions which are virtually W*-superrigid
• On co-amenability for groups and von Neumann algebras
• One-cohomology and the uniqueness of the group measure space decomposition of a II_1 factor

Cited by in corpus (6)

• W*-superrigidity for group von Neumann algebras of left-right wreath products
• A Unique Prime Decomposition Result for Wreath Product Factors
• Type II_1 factors satisfying the spatial isomorphism conjecture
• Examples of factors which have no Cartan subalgebras
• Type II${}_1$ factors with arbitrary countable endomorphism group
• On bi-exactness of discrete quantum groups