W*-superrigidity for arbitrary actions of central quotients of braid groups

TL;DR

This paper proves W*-superrigidity for actions of central quotients of braid groups and their finite index subgroups, showing that their von Neumann algebra invariants determine the actions up to conjugacy.

Contribution

It establishes W*-superrigidity for a broad class of actions of central quotients of braid groups and their finite index subgroups, extending previous rigidity results.

Findings

W*-superrigidity holds for actions of $ ilde B_n$ and certain subgroups.

Von Neumann algebra isomorphisms imply conjugacy of actions.

Results extend to finite index subgroups of product groups.

Abstract

For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,\mu)$ is W$^*$-superrigid in the following sense: if $L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimes\Lambda$, for an arbitrary free ergodic pmp action $\Lambda\curvearrowright (Y,\nu)$, then the actions $\tilde B_n\curvearrowright X,\Lambda\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with a finite index subgroup of the direct product $\tilde B_{n_1}\times\cdots\times\tilde B_{n_k}$, for some $n_1,\ldots,n_k\geqslant 4$. The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from \cite{PV11} in combination with the OE superrigidity theorem for actions of mapping class groups from \cite{Ki06}.