OE and W* superrigidity results for actions by surface braid groups

TL;DR

This paper proves superrigidity and measure equivalence rigidity for actions of certain subgroups of surface braid groups, showing they can be reconstructed from their von Neumann algebras.

Contribution

It establishes OE-superrigidity and measure equivalence rigidity for normal subgroups of surface braid groups, including Torelli groups and Johnson kernels, and describes their lattice embeddings.

Findings

Normal subgroups of surface braid groups are OE-superrigid.

These groups satisfy measure equivalence rigidity.

Actions of these groups are W*-superrigid, reconstructible from von Neumann algebras.

Abstract

We show that several important normal subgroups $\Gamma$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $\Gamma \curvearrowright X$ is stably OE-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 [CIK13] we deduce that any free, ergodic, probability measure preserving action of almost any surface braid group is stably W*-superrigid, i.e., it can be completely reconstructed from its von Neumann algebra.