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

For any $n⩾4$ let ${\tilde{B}}_n=B_n\mathrm{/}Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,\mu)$isW$^*$-superrigidinthefollowingsense:if$L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimes\Lambda$, for an arbitrary free ergodic pmp action $\mathrm{\Lambda }\curvearrowright (Y,\nu )$, then the actions ${\tilde{B}}_n\curvearrowright X,\mathrm{\Lambda }\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$isreplacedwithafiniteindexsubgroupofthedirectproduct$\tilde B_{n_1}\times\cdots\times\tilde B_{n_k}$,forsome$n_1,\ldots,n_k\geqslant 4$. 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}.