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Measure equivalence rigidity of the handlebody groups

Sebastian Hensel, Camille Horbez

TL;DR

The paper proves that the handlebody group $\mathrm{Mod}(V)$ for a connected 3-manifold handlebody of genus $g\ge3$ is ME-superrigid: any countable group ME-equivalent to $\mathrm{Mod}(V)$ is virtually isomorphic to it. The authors reduce ME-rigidity to cocycle rigidity for actions of $\mathrm{Mod}(V)$ by developing a measurement-groupoid framework, introducing strongly Schottky substructures, and classifying subgroupoids by meridian type via canonical reductions to the disk graph. A key step is constructing a cocycle-conjugacy to $\mathrm{Mod}^{\pm}(V)$ by mapping stabilizers of meridians to vertices of the disk graph and invoking the automorphism theorem of Korkmaz–Schleimer, which yields cocycle-independence. They then derive consequences for lattice embeddings, orbit equivalence, and von Neumann algebras, obtaining OE-superrigidity and $W^{*}$-rigidity results for handlebody actions. Altogether, the work extends ME and OE-rigidity phenomena from surface mapping class groups and $\mathrm{Out}(F_N)$ to handlebody groups, with substantial implications for 3-manifold topology and measured group theory.

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.

Measure equivalence rigidity of the handlebody groups

TL;DR

The paper proves that the handlebody group for a connected 3-manifold handlebody of genus is ME-superrigid: any countable group ME-equivalent to is virtually isomorphic to it. The authors reduce ME-rigidity to cocycle rigidity for actions of by developing a measurement-groupoid framework, introducing strongly Schottky substructures, and classifying subgroupoids by meridian type via canonical reductions to the disk graph. A key step is constructing a cocycle-conjugacy to by mapping stabilizers of meridians to vertices of the disk graph and invoking the automorphism theorem of Korkmaz–Schleimer, which yields cocycle-independence. They then derive consequences for lattice embeddings, orbit equivalence, and von Neumann algebras, obtaining OE-superrigidity and -rigidity results for handlebody actions. Altogether, the work extends ME and OE-rigidity phenomena from surface mapping class groups and to handlebody groups, with substantial implications for 3-manifold topology and measured group theory.

Abstract

Let be a connected -dimensional handlebody of finite genus at least . We prove that the handlebody group is superrigid for measure equivalence, i.e. every countable group which is measure equivalent to is in fact virtually isomorphic to . Applications include a rigidity theorem for lattice embeddings of , an orbit equivalence rigidity theorem for free ergodic measure-preserving actions of on standard probability spaces, and a -rigidity theorem among weakly compact group actions.
Paper Structure (23 sections, 54 theorems, 15 equations, 2 figures)

This paper contains 23 sections, 54 theorems, 15 equations, 2 figures.

Table of Contents

  1. Handlebody and mapping class group facts
  2. Mapping Class Group background
  3. Handlebody background
  4. Improved rotationless mapping classes
  5. Infinite conjugacy classes
  6. Background on measured groupoids
  7. Measure equivalence rigidity of the handlebody group
  8. Groupoids with cocycles to a free group, after Adams, Kida
  9. Canonical reduction sets, after Kida
  10. Exploiting amenable normalized subgroupoids, after Kida
  11. Uniqueness statements
  12. Property $(\mathrm{P}_{\mathrm{nsep}})$ and subgroupoids of non-separating meridian type
  13. Stabilizers of separating meridians do not satisfy Property $(\mathrm{P}_{\mathrm{nsep}})$
  14. Admissible decorated multicurves and their active subgroups
  15. Compatibility
  16. ...and 8 more sections

Key Result

Theorem 1

Let $V$ be a connected $3$-dimensional handlebody of finite genus at least $3$. Then $\mathrm{Mod}(V)$ is ME-superrigid.

Figures (2)

  • Figure 1: On the left: a meridian$d$, i.e. an essential curve bounding a disk in the handlebody. On the right: two curves $\alpha_1, \alpha_2$ which individually do not bound disks in the handlebody, and which are not homotopic on the boundary surface, but bound a properly embedded annulus in the handlebody.
  • Figure 2: The setup in the proof of Lemma \ref{['lem:cover-homology-inclusion']}. The subsurface $X_S$ is decomposed into a surface $Y$ with a single boundary, and a bordered sphere. To control the homology classes defined by the boundary curves of that sphere, we construct auxiliary curves $\beta_i$.

Theorems & Definitions (108)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • ...and 98 more