Superrigidity, measure equivalence, and weak Pinsker entropy

TL;DR

This paper proves that certain groups satisfying Popa's Cocycle Superrigidity are measure equivalence invariant, introduces Weak Pinsker entropy as an orbit-equivalence invariant, and applies these results to nonamenable lattices.

Contribution

It extends the invariance of Popa's class under measure equivalence to groupoids and introduces Weak Pinsker entropy as a new invariant for group actions.

Findings

Popa's class $\\mathscr{B}$ is measure equivalence invariant.

Nonamenable lattices in product groups belong to $\\mathscr{B}$.

Weak Pinsker entropy is an orbit-equivalence invariant for groups in $\\mathscr{B}$.

Abstract

We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G.