Cocycle and orbit superrigidity for lattices in SL(n,R) acting on homogeneous spaces

Abstract

We prove cocycle and orbit equivalence superrigidity for lattices in SL(n,R) acting linearly on R^n, as well as acting projectively on certain flag manifolds, including the real projective space. The proof combines operator algebraic techniques with the property (T) in the sense of Zimmer for the action of SL(n,Z) on R^n, n \geq 4. We also show that the restriction of the associated orbit equivalence relation to a subset of finite Lebesgue measure, provides a II_1 equivalence relation with property (T) and yet fundamental group equal to R_+.

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• A remark on $\mathbb{T}$-valued cohomology groups of algebraic group actions
• A classification of all finite index subfactors for a class of group-measure space II_1 factors
• Cocycle superrigidity for coinduced actions
• The Palm groupoid of a point process and factor graphs on amenable and Property (T) groups