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

Abstract

For any $n⩾4$ 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${}^{\ast }$-superrigid in the following sense: if $L^{\infty }(X)⋊{\tilde{B}}_n\cong L^{\infty }(Y)⋊\Lambda$, 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 $n_1,\dots ,n_k⩾4$. 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}.

v2: added a new result (Thm C) which shows that groups that are hyperbolic relative to a finite family of finitely generated, residually finite subgroups, are Cartan-rigid; v3: improved exposition

References in corpus (5)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups
• On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra
• Examples of group actions which are virtually W*-superrigid
• Cocycle superrigidity for profinite actions of irreducible lattices

Cited by in corpus (12)

• Classification of type III Bernoulli crossed products
• OE and W* superrigidity results for actions by surface braid groups
• Primeness results for von Neumann algebras associated with surface braid groups
• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• Orbit equivalence rigidity for product actions
• Prime II${}_1$ factors arising from irreducible lattices in products of rank one simple Lie groups
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• Cocycle superrigidity for coinduced actions
• Classification of Tensor Decompositions of II${}_1$ Factors Associated With Poly-Hyperbolic Groups
• Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-superrigidity
• Classification of tensor decompositions for II${}_1$ factors