<i>OE</i>and<i>W</i>* superrigidity results for actions by surface braid groups

Abstract

We show that several important normal subgroups Γ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure-preserving action Γ ↷ X is stably O E -superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from Chifan–Ioana–Kida [‘ W * -superrigidity for arbitrary actions of central quotients of braid groups’, Math. Ann. 361 (2015) 925–959], we deduce that any free, ergodic, probability measure-preserving action of almost any surface braid group is stably W * -superrigid, that is, it can be completely reconstructed from its von Neumann algebra.