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.

v2: Revised version after a referee report

References in corpus (3)

• Measure equivalence classification of transvection-free right-angled Artin groups
• Automorphisms of the disk complex
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type