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Graph products and measure equivalence: classification, rigidity, and quantitative aspects

Amandine Escalier, Camille Horbez

TL;DR

This work develops a comprehensive measured group theory framework for graph products, establishing a full measure equivalence classification (ME) for products over finite graphs with countably infinite vertex groups under transvection-free and no-partial-conjugation assumptions. Central to the approach is the Vertex Recognition Property, linking subgroupoids to vertex data via extension graphs and right-angled buildings, and enabling a vertex-by-vertex reconstruction that yields ME, OE, and isomorphism classifications, plus commensurability results and W*-rigidity consequences. The authors also cultivate a quantitative theory of ME, proving that vertex-level integrability transfers to the graph product and solving inverse problems for right-angled Artin groups with prescribed couplings, including optimality statements. A secondary thread provides rigidity results under ergodicity assumptions and constructs rigidity but non-quasi-isometry examples using Higman-type vertex groups, illustrating deep separations between ME/OE, quasi-isometry, and group-theoretic structure. Collectively, these results illuminate the interplay between graph-product combinatorics, measured groupoids, and operator-algebraic invariants, with broad implications for orbit equivalence, commensurability, and von Neumann algebras of group actions.

Abstract

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement are given with fewer assumptions on the defining graphs. We also provide a quantified version of our measure equivalence classification theorem, that keeps track of the integrability of associated cocycles. As an application, we solve an inverse problem in quantitative orbit equivalence for a large family of right-angled Artin groups. We then establish several rigidity theorems. First, in the spirit of work of Monod-Shalom, we achieve rigidity in orbit equivalence for probability measure-preserving actions of graph products, upon imposing extra ergodicity assumptions. Second, we establish a sufficient condition on the defining graph and on the vertex groups ensuring that a graph product G is rigid in measure equivalence among torsion-free groups (in the sense that every torsion-free countable group H which is measure equivalent to G, is in fact isomorphic to G). Using variations over the Higman groups as the vertex groups, we construct the first example of a group which is rigid in measure equivalence, but not in quasi-isometry, among torsion-free groups.

Graph products and measure equivalence: classification, rigidity, and quantitative aspects

TL;DR

This work develops a comprehensive measured group theory framework for graph products, establishing a full measure equivalence classification (ME) for products over finite graphs with countably infinite vertex groups under transvection-free and no-partial-conjugation assumptions. Central to the approach is the Vertex Recognition Property, linking subgroupoids to vertex data via extension graphs and right-angled buildings, and enabling a vertex-by-vertex reconstruction that yields ME, OE, and isomorphism classifications, plus commensurability results and W*-rigidity consequences. The authors also cultivate a quantitative theory of ME, proving that vertex-level integrability transfers to the graph product and solving inverse problems for right-angled Artin groups with prescribed couplings, including optimality statements. A secondary thread provides rigidity results under ergodicity assumptions and constructs rigidity but non-quasi-isometry examples using Higman-type vertex groups, illustrating deep separations between ME/OE, quasi-isometry, and group-theoretic structure. Collectively, these results illuminate the interplay between graph-product combinatorics, measured groupoids, and operator-algebraic invariants, with broad implications for orbit equivalence, commensurability, and von Neumann algebras of group actions.

Abstract

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement are given with fewer assumptions on the defining graphs. We also provide a quantified version of our measure equivalence classification theorem, that keeps track of the integrability of associated cocycles. As an application, we solve an inverse problem in quantitative orbit equivalence for a large family of right-angled Artin groups. We then establish several rigidity theorems. First, in the spirit of work of Monod-Shalom, we achieve rigidity in orbit equivalence for probability measure-preserving actions of graph products, upon imposing extra ergodicity assumptions. Second, we establish a sufficient condition on the defining graph and on the vertex groups ensuring that a graph product G is rigid in measure equivalence among torsion-free groups (in the sense that every torsion-free countable group H which is measure equivalent to G, is in fact isomorphic to G). Using variations over the Higman groups as the vertex groups, we construct the first example of a group which is rigid in measure equivalence, but not in quasi-isometry, among torsion-free groups.
Paper Structure (144 sections, 117 theorems, 96 equations, 23 figures)

This paper contains 144 sections, 117 theorems, 96 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Measure equivalence classification of graph products
  3. The case of right-angled Artin groups: review
  4. Main theorem and discussion on the hypotheses
  5. On the commensurability classification
  6. A word on proof techniques
  7. Quantitative point of view
  8. Quantification of couplings
  9. Main quantification results for graph products
  10. An application to the inverse problem for quantifications
  11. Rigidity
  12. Orbit equivalence rigidity under ergodicity assumptions
  13. Measure equivalence rigidity for graph products
  14. A word on the proof.
  15. Structure of the paper
  16. ...and 129 more sections

Key Result

Theorem 1.1

Let $A_1$ and $A_2$ be two right-angled Artin groups such that $\mathrm{Out}(A_i)$ is finite for every $i\in\{1,2\}$. Then $A_1$ and $A_2$ are measure equivalent if and only if they are isomorphic.

Figures (23)

  • Figure 1: Definition of $c(g,x)$
  • Figure 2: Induced subgraphs and orthogonal
  • Figure 3: Definition of $\beta_A$
  • Figure 4: Illustration of the proof of \ref{['lemma:decomposition-boundary']}
  • Figure 8: Illustration for the proof of the equivariance of $\theta$
  • ...and 18 more figures

Theorems & Definitions (295)

  • Theorem 1.1: HH21
  • Theorem 1.2
  • Remark 1.3: New instances of the measure equivalence classification of RAAGs
  • Remark 1.4: Measure equivalence versus quasi-isometry
  • Remark 1.5: Fundamental groups and von Neumann algebras
  • Theorem 1.6
  • Remark 1.7: On the classification of graph products up to isomorphism
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 285 more