Measure equivalence rigidity of the handlebody groups

TL;DR

This paper proves that the handlebody group of genus at least 3 exhibits superrigidity under measure equivalence, meaning any group measure equivalent to it is virtually isomorphic, with applications in lattice embeddings and orbit equivalence.

Contribution

It establishes measure equivalence superrigidity for handlebody groups, a significant advancement in understanding their rigidity properties.

Findings

Handlebody group is superrigid for measure equivalence.

Any group measure equivalent to the handlebody group is virtually isomorphic.

Applications include rigidity results for lattice embeddings and orbit equivalence.

Abstract

Let $V$ be a connected $3$-dimensional handlebody of finite genus at least $3$. We prove that the handlebody group $\mathrm{Mod}(V)$ is superrigid for measure equivalence, i.e. every countable group which is measure equivalent to $\mathrm{Mod}(V)$ is in fact virtually isomorphic to $\mathrm{Mod}(V)$. Applications include a rigidity theorem for lattice embeddings of $\mathrm{Mod}(V)$, an orbit equivalence rigidity theorem for free ergodic measure-preserving actions of $\mathrm{Mod}(V)$ on standard probability spaces, and a $W^*$-rigidity theorem among weakly compact group actions.