Measure equivalence rigidity via s-malleable deformations

Abstract

We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $Γ_1,\dots,Γ_n\in {\mathscr{M}}$ and $Γ_1\times\dots\timesΓ_n$ is measure equivalent to a product $Λ_1\times\dots\timesΛ_m$ of infinite icc groups, then $n \ge m$, and if $n = m$ then, after permutation of the indices, $Γ_i$ is measure equivalent to $Λ_i$, for all $1\leq i\leq n$. This provides an analogue of Monod and Shalom's theorem \cite{MS02} for groups that belong to ${\mathscr{M}}$. Class ${\mathscr{M}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $Γ$ for which either (i) $Γ$ is an arbitrary wreath product group with amenable base or (ii) $Γ$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\mathscr{M}}$. Finally, for groups $Γ$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(Γ)$ are ``rigid". In particular, we provide an alternative solution to a question of Popa that was recently answered in \cite{DKEP22}.