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Measure equivalence rigidity among the Higman groups

Camille Horbez, Jingyin Huang

TL;DR

This work proves measure equivalence rigidity for generalized Higman groups Hig_σ with at least five generators, showing that any countable group ME to Hig_σ is virtually isomorphic to Hig_σ. The authors develop a two-pronged strategy: (i) a general ME rigidity principle for groups acting acylindrically on CAT(-1) complexes, and (ii) a combinatorial rigidity that recovers Hig_σ from the intersection pattern of its Baumslag–Solitar subgroups via an intersection graph Θ. A key technical achievement is proving Aut(X_σ) and Aut(Θ_σ) are isomorphic to the extended group Ŝ Hig_σ, enabling a robust transfer of ME data to the Higman structure. These results yield orbit equivalence and W*-rigidity consequences for actions of Higman groups, including unique Cartan subalgebras and W*-superrigidity for all free ergodic actions. The work further clarifies the role of line- and intersection-graph combinatorics in capturing the global rigidity of these negatively curved, polygonal-group complexes, and outlines open problems for k=4 and other parameter regimes.

Abstract

We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by $a_ia_{i+1}^{m_i}a_i^{-1}=a_i^{n_i}$, with $i$ varying in $\mathbb{Z}/k\mathbb{Z}$ and nonzero integers $|m_i|\neq |n_i|$ for each $i$. We prove that every countable group which is measure equivalent to $H$, is in fact virtually isomorphic to $H$. A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on $\mathrm{CAT}(-1)$ polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and $W^*$-superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.

Measure equivalence rigidity among the Higman groups

TL;DR

This work proves measure equivalence rigidity for generalized Higman groups Hig_σ with at least five generators, showing that any countable group ME to Hig_σ is virtually isomorphic to Hig_σ. The authors develop a two-pronged strategy: (i) a general ME rigidity principle for groups acting acylindrically on CAT(-1) complexes, and (ii) a combinatorial rigidity that recovers Hig_σ from the intersection pattern of its Baumslag–Solitar subgroups via an intersection graph Θ. A key technical achievement is proving Aut(X_σ) and Aut(Θ_σ) are isomorphic to the extended group Ŝ Hig_σ, enabling a robust transfer of ME data to the Higman structure. These results yield orbit equivalence and W*-rigidity consequences for actions of Higman groups, including unique Cartan subalgebras and W*-superrigidity for all free ergodic actions. The work further clarifies the role of line- and intersection-graph combinatorics in capturing the global rigidity of these negatively curved, polygonal-group complexes, and outlines open problems for k=4 and other parameter regimes.

Abstract

We prove that all (generalized) Higman groups on at least generators are superrigid for measure equivalence. More precisely, let , and let be a group with generators , and Baumslag-Solitar relations given by , with varying in and nonzero integers for each . We prove that every countable group which is measure equivalent to , is in fact virtually isomorphic to . A key ingredient in the proof is a general statement providing measured group theoretic invariants for groups acting acylindrically on polyhedral complexes with control on vertex and edge stabilizers. Among consequences of our work, we obtain rigidity theorems for generalized Higman groups with respect to lattice embeddings and automorphisms of their Cayley graphs. We also derive an orbit equivalence and -superrigidity theorem for all free, ergodic, probability measure-preserving actions of generalized Higman groups.
Paper Structure (29 sections, 46 theorems, 11 equations, 3 figures)

This paper contains 29 sections, 46 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General context and statement of the main theorem
  3. Several open questions.
  4. Some consequences
  5. Orbit equivalence rigidity and $W^*$-rigidity.
  6. Proof ingredients
  7. On the proof of Theorem \ref{['theo:combination']}
  8. A combinatorial/geometric rigidity theorem
  9. Background on Higman groups
  10. Generalized Higman groups and associated complexes
  11. Metric on $X$ and some consequences
  12. Weak acylindricity of the action
  13. A finite extension of $\mathrm{Hig}_\sigma$
  14. Line-preserving bijections of Baumslag--Solitar groups
  15. Combinatorial rigidity of the intersection graph
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $k\ge 5$, and let $\sigma=((m_1,n_1),\dots,(m_k,n_k))$ be a $k$-tuple of pairs of non-zero integers, with $|m_i|\neq |n_i|$ for every $i\in\{1,\dots,k\}$. Then the generalized Higman group $\mathrm{Hig}_\sigma$ is superrigid for measure equivalence: if a countable group $G$ is measure equivalent

Figures (3)

  • Figure 1: Illustration of the conditions in the proof of Lemma \ref{['lemma:equally-spaced']}.
  • Figure 2: Illustration of the configuration in the proof of Proposition \ref{['prop:combinatorial rigid BS']}.
  • Figure 3: The vertices with a cross indicate a contradiction with Lemma \ref{['lemma:intersection of vertex stabilizer']} (2).

Theorems & Definitions (98)

  • Theorem 1.1
  • Corollary 1.3: see Corollary \ref{['cor:consequences']}
  • Corollary 1.4: see Corollaries \ref{['cor:oe']} and \ref{['cor:w']}
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: see Theorem \ref{['theo:crigidity0']}
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 88 more