Measure equivalence rigidity of the handlebody groups

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 $W^{\ast }$-rigidity theorem among weakly compact group actions.

arXiv commentv2: Revised version after a referee report

References in corpus (6)

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• A survey of Measured Group Theory
• Measure equivalence classification of transvection-free right-angled Artin groups
• Automorphisms of the disk complex
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• (Un)distorted stabilisers in the handlebody group