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

Abstract

For any (Formula presented.) let (Formula presented.) be the quotient of the braid group (Formula presented.) through its center. We prove that any free ergodic probability measure preserving (pmp) action (Formula presented.) is virtually (Formula presented.)-superrigid in the following sense: if (Formula presented.), for an arbitrary free ergodic pmp action (Formula presented.), then the actions (Formula presented.) are virtually conjugate. Moreover, we prove that the same holds if (Formula presented.) is replaced with a finite index subgroup of the direct product (Formula presented.), for some (Formula presented.). 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 (Formula presented.) is hyperbolic relative to a finite family of proper, finitely generated, residually finite, infinite subgroups, then the (Formula presented.) factor (Formula presented.) has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic pmp action (Formula presented.).