$W^*$-Superrigidity for arbitrary actions of central quotients of braid 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 virtually $\hbox {W}^*$-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 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 Popa and Vaes (212, 141–198, 2014) in combination with the OE superrigidity theorem for actions of mapping class groups from Kida (131, 99–109, 2008). Similar techniques allow us to prove that if a group $\Gamma$ is hyperbolic relative to a finite family of proper, finitely generated, residually finite, infinite subgroups, then the $\hbox {II}_1$ factor $L^{\infty }(X)\rtimes \Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic pmp action $\Gamma \curvearrowright (X,\mu )$.