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

Amandine Escalier, Camille Horbez

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Amandine Escalier, Camille Horbez. Graph products and measure equivalence: classification, rigidity, and quantitative aspects. 2024. hal-04389869

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

Amandine Escalier and Camille Horbez

January 9, 2024

## 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.
# Contents

<table><tr><td>1</td><td>Introduction</td><td>5</td></tr><tr><td>1.1</td><td>Measure equivalence classification of graph products</td><td>6</td></tr><tr><td>1.2</td><td>Quantitative point of view</td><td>11</td></tr><tr><td>1.3</td><td>Rigidity</td><td>14</td></tr><tr><td>1.4</td><td>Structure of the paper</td><td>17</td></tr></table>

## I Graph products and measured groupoids 18

<table><tr><td>2</td><td>Graph products and their geometry</td><td>18</td></tr><tr><td>2.1</td><td>Normal form for graph products</td><td>18</td></tr><tr><td>2.2</td><td>Parabolic subgroups</td><td>20</td></tr><tr><td>2.3</td><td>Extension graphs, right-angled buildings and their automorphisms</td><td>23</td></tr><tr><td>2.4</td><td>Amenable actions on boundaries of trees</td><td>27</td></tr></table>

<table><tr><td>3</td><td>Measure and orbit equivalence</td><td>30</td></tr><tr><td>3.1</td><td>Background and definitions</td><td>30</td></tr><tr><td>3.2</td><td>Link with commensurability</td><td>33</td></tr><tr><td>3.3</td><td>Restricting couplings to subgroups</td><td>35</td></tr></table>

<table><tr><td>4</td><td>Measured groupoids</td><td>37</td></tr><tr><td>4.1</td><td>Measured groupoids</td><td>38</td></tr><tr><td>4.2</td><td>Normalization and amenable groupoids</td><td>41</td></tr><tr><td>4.3</td><td>Action-like cocycles</td><td>42</td></tr><tr><td>4.4</td><td>Kida's blueprint</td><td>44</td></tr></table>

## II Measure equivalence classification of graph products 50

<table><tr><td>5</td><td>From vertex recognition to rigidity</td><td>53</td></tr><tr><td>5.1</td><td>The Vertex Recognition Property</td><td>53</td></tr><tr><td>5.2</td><td>Factoring the self-coupling</td><td>54</td></tr><tr><td>5.3</td><td>Proof of the measure equivalence classification theorem</td><td>56</td></tr><tr><td>5.4</td><td>Applications: Commensurability and von Neumann algebras</td><td>59</td></tr></table>

<table><tr><td>6</td><td>Combinatorial and groupoid-theoretic tools</td><td>61</td></tr><tr><td>6.1</td><td>Combinatorial tools</td><td>61</td></tr><tr><td>6.2</td><td>Groupoid-theoretic tools</td><td>67</td></tr></table>

<table><tr><td>7</td><td>The irreducible case</td><td>73</td></tr><tr><td>7.1</td><td>Overview of the proof</td><td>74</td></tr><tr><td>7.2</td><td>Recognizing maximal products</td><td>76</td></tr><tr><td>7.3</td><td>Recognizing factors in products</td><td>79</td></tr><tr><td>7.4</td><td>Proof of the Vertex Recognition Property</td><td>86</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>8</td>
      <td>The reducible case and proof of Proposition 5.3</td>
      <td>90</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Goal of the section</td>
      <td>90</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Proof of the Vertex Recognition Property</td>
      <td>90</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Amenable vertex groups and untransvable vertices</td>
      <td>94</td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Main result</td>
      <td>94</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Detecting clique subgroups.</td>
      <td>96</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Proof of Proposition 9.1</td>
      <td>97</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Examples</td>
      <td>99</td>
    </tr>
    <tr>
      <td>III</td>
      <td>Quantitative aspects and the inverse problem for RAAGs</td>
      <td>102</td>
    </tr>
    <tr>
      <td>11</td>
      <td>From the factors to the graph product</td>
      <td>102</td>
    </tr>
    <tr>
      <td>12</td>
      <td>From the graph product to the factors</td>
      <td>104</td>
    </tr>
    <tr>
      <td>12.1</td>
      <td>The general statement and applications</td>
      <td>104</td>
    </tr>
    <tr>
      <td>12.2</td>
      <td>Integrability at the vertex level</td>
      <td>106</td>
    </tr>
    <tr>
      <td>12.3</td>
      <td>Proof of the main theorem.</td>
      <td>108</td>
    </tr>
    <tr>
      <td>13</td>
      <td>The inverse problem for right-angled Artin groups</td>
      <td>109</td>
    </tr>
    <tr>
      <td>13.1</td>
      <td>Main theorem.</td>
      <td>109</td>
    </tr>
    <tr>
      <td>13.2</td>
      <td>Proof of Theorem 1.13</td>
      <td>110</td>
    </tr>
    <tr>
      <td>IV</td>
      <td>Rigidity under ergodicity assumptions</td>
      <td>112</td>
    </tr>
    <tr>
      <td>14</td>
      <td>Rigidity among graph products</td>
      <td>112</td>
    </tr>
    <tr>
      <td>14.1</td>
      <td>Some terminology.</td>
      <td>112</td>
    </tr>
    <tr>
      <td>14.2</td>
      <td>Rigidity of actions within the class of graph products</td>
      <td>113</td>
    </tr>
    <tr>
      <td>14.3</td>
      <td>An application to von Neumann algebras.</td>
      <td>126</td>
    </tr>
    <tr>
      <td>15</td>
      <td>Superrigidity</td>
      <td>127</td>
    </tr>
    <tr>
      <td>V</td>
      <td>Measure equivalence rigidity of some graph products</td>
      <td>129</td>
    </tr>
    <tr>
      <td>16</td>
      <td>An ME-rigidity criterion for graph products</td>
      <td>130</td>
    </tr>
    <tr>
      <td>16.1</td>
      <td>Self-couplings of clique subgroups</td>
      <td>131</td>
    </tr>
    <tr>
      <td>16.2</td>
      <td>Factoring the coupling through the right-angled building</td>
      <td>132</td>
    </tr>
    <tr>
      <td>16.3</td>
      <td>Properties of the H-action on D<sub>G</sub>.</td>
      <td>134</td>
    </tr>
    <tr>
      <td>16.4</td>
      <td>Exploiting the structure of simple complex of groups</td>
      <td>138</td>
    </tr>
    <tr>
      <td>17</td>
      <td>An ME-rigid group which is not QI-rigid</td>
      <td>139</td>
    </tr>
    <tr>
      <td>17.1</td>
      <td>Homomorphisms between Higman groups</td>
      <td>140</td>
    </tr>
    <tr>
      <td>17.2</td>
      <td>Measure equivalence rigidity of the graph product G</td>
      <td>141</td>
    </tr>
    <tr>
      <td>17.3</td>
      <td>Flexibility in quasi-isometry</td>
      <td>142</td>
    </tr>
  </tbody>
</table>
<table><tr><td>Appendix</td><td>143</td></tr><tr><td>A On the isomorphism problem for graph products</td><td>143</td></tr><tr><td>A.1 Proof of Theorem A.1</td><td>144</td></tr><tr><td>A.2 Acylindrical hyperbolicity for automorphisms of graph products</td><td>155</td></tr><tr><td>Index</td><td>159</td></tr><tr><td>Notations</td><td>160</td></tr><tr><td>Bibliography</td><td>166</td></tr></table>
# 1 Introduction

Graph products of groups, introduced by Green in [Gre90], form a natural class of groups that interpolates between direct products and free products. If $\Gamma$ is a finite simple graph and $(G_v)_{v \in V_\Gamma}$ is a family of groups, the graph product over $\Gamma$ with vertex groups $(G_v)_{v \in V_\Gamma}$ is the group

$$G := \left( \underset{v \in V_\Gamma}{*} G_v \right) / \langle\langle [g, h] | g \in G_u, h \in G_v, (u, v) \in E_\Gamma \rangle\rangle.$$

The class of graph products encompasses several particularly important families of groups, among which right-angled Artin groups (when all vertex groups are isomorphic to $\mathbb{Z}$) and right-angled Coxeter groups (when all vertex groups are isomorphic to $\mathbb{Z}/2\mathbb{Z}$).

Basic questions about graph products remain open. For instance, their classification up to isomorphism, or up to commensurability, is still open in general. Moving to the framework of geometric group theory, the problem of their quasi-isometry classification is also still wide open. In a very different direction, the notion of a graph product can be defined for other structures than groups, e.g. von Neumann algebras [CF17]; and from this viewpoint the $W^*$-classification of graph products and their measure-preserving actions is also an important open question that has recently attracted a lot of attention [CKE21, CDD22, CDD23].

In the present work, we adopt the point of view of measured group theory, and obtain classification and rigidity theorems for graph products in measure equivalence. Interestingly, using this viewpoint, we are able to also obtain several consequences and draw connections with the algebraic, geometric and functional analytic approaches mentioned in the above paragraph.

Besides being a particularly natural class of groups, and an important object of study on their own, graph products will also enable us to exhibit new phenomena in measured group theory, like examples of groups that are superrigid in measure equivalence (among torsion-free groups) but not in quasi-isometry (Part V). They will also be a good framework to develop the theory of quantitative measure equivalence (in the sense of [DKLMT22]) beyond the world of amenable groups (Part III).

**Measure equivalence.** Introduced by Gromov in [Gro93], measure equivalence is a measurable counterpart to the notion of quasi-isometry between finitely generated groups. Two countable groups $G$ and $H$ are *measure equivalent* if there exists a standard measure space $(\Omega, m)$ equipped with commuting, free, measure-preserving actions of $G$ and $H$ by Borel automorphisms, having finite measure Borel fundamental domains $X_G, X_H$ respectively. The space $(\Omega, m)$ is called a *measure equivalence coupling* between $G$ and $H$. As an important example, any two lattices in the same locally compact second countable group $G$ are measure equivalent, by letting $(\Omega, m) = (G, \text{Haar})$, equipped with the actions of the two lattices by left/right multiplication.

Measure equivalence is intimately connected to the ergodic-theoretic notion of *orbit equivalence*, which finds its roots in the work of Dye [Dye59, Dye63]. Two countable groups $G$ and $H$ are *orbit equivalent* if there exist free probability measure-preserving actions $G \to X$ and $H \to Y$ on standard probability spaces, and a measure space
isomorphism $f: X \to Y$ sending orbits to orbits, i.e. such that for almost every $x \in X$, one has $f(G \cdot x) = H \cdot f(x)$. It turns out that orbit equivalence is a stronger notion than measure equivalence: two groups are orbit equivalent precisely when one can find a measure equivalence coupling $(\Omega, m)$ as above, with $X_G = X_H$ (see e.g. **PME5** in [Gab05]).

The classification of countable groups up to measure equivalence is fundamentally different in the amenable and in the non-amenable case. A celebrated theorem of Ornstein and Weiss [OW80], building on the aforementioned work of Dye, asserts that all countably infinite amenable groups are orbit equivalent. At the other extreme, certain non-amenable groups satisfy a very strong form of rigidity. Furman proved in [Fur99a] that every countable group which is measure equivalent to a lattice in a higher-rank simple Lie group **G** with trivial center, is virtually isomorphic to a lattice in **G**. Some (non-amenable) groups **G** are even more rigid, in that every countable group **H** which is measure equivalent to **G**, must be virtually isomorphic to **G**: this is the case of most mapping class groups of finite-type surfaces (Kida [Kid10]), or of $\text{Out}(F_N)$ for $N \ge 3$ (Guirardel-Horbez [GH21b]).

# 1.1 Measure equivalence classification of graph products

We now present our first main theorem (Theorem 1.2 below, proved in Part II), giving a full measure equivalence classification of graph products with countably infinite vertex groups defined over transvection-free graphs with no partial conjugation. Let us mention that Monod-Shalom [MS06] previously obtained measure equivalence classification results for a wide class of direct products. In the opposite direction, Alvarez-Gaboriau [AG12] classified a large family of free products. As a motivating example for our work, we first review the case of right-angled Artin groups treated in [HH22a].

## 1.1.1 The case of right-angled Artin groups: review

In [HH22a], Huang and the second-named author proved the following measure equivalence classification theorem for right-angled Artin groups, which turns out to match the quasi-isometry classification for the groups under consideration [Hua17, Theorem 1.1].

**Theorem 1.1 ([HH22a, Theorem 1])**. Let $A_1$ and $A_2$ be two right-angled Artin groups such that $\text{Out}(A_i)$ is finite for every $i \in \{1, 2\}$. Then $A_1$ and $A_2$ are measure equivalent if and only if they are isomorphic.

The condition on the finiteness of the outer automorphism groups is important, and can be read off from the defining graphs as follows. Recall that if $\Gamma$ is a finite simple graph, and $v \in V\Gamma$, then

* the *link* of $v$ is the subgraph $\text{lk}(v)$ of $\Gamma$ induced by all vertices adjacent to $v$;

* the *star* of $v$ is the subgraph $\text{star}(v)$ of $\Gamma$ induced by $v$ itself and its neighbors.

A theorem of Laurence [Lau95], confirming a conjecture of Servatius [Ser89], ensures that if $A_\Gamma$ is the right-angled Artin group over $\Gamma$, then $|\text{Out}(A_\Gamma)| < +\infty$ if and only if

* $\Gamma$ is *transvection-free*, that is, for any two distinct vertices $v, w \in V\Gamma$, one has $\text{lk}(v) \notin \text{star}(w)$, and
• $\Gamma$ has no partial conjugation, that is, for every vertex $v \in V\Gamma$, the subgraph induced by $V\Gamma\setminus V\text{star}(v)$ is connected.

### 1.1.2 Main theorem and discussion on the hypotheses

Our first main theorem is proved in Part II. Its statement vastly generalizes Theorem 1.1 to all graph products of countably infinite groups, and its proof relies on different tools (see the discussion at the beginning of Part II).

**Theorem 1.2.** Let $\Gamma_G, \Gamma_H$ be two finite simple graphs, not reduced to one vertex, with no transvection and no partial conjugation. Let $G, H$ be graph products of countably infinite groups over $\Gamma_G, \Gamma_H$, respectively. Then the following assertions are equivalent.

1. The groups $G$ and $H$ are measure equivalent.

2. The groups $G$ and $H$ are orbit equivalent.

3. There exists a graph isomorphism $\sigma: \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, the groups $G_v$ and $H_{\sigma(v)}$ are orbit equivalent.

The implication $2 \Rightarrow 1$ is obvious, and $3 \Rightarrow 2$ was proved in [HH22a, Proposition 4.2] by adapting an argument of Gaboriau [Gab05] from free products to graph products. Our contribution in the present paper is the (hardest) implication $1 \Rightarrow 3$.

Examples illustrating the above theorem (and some variations obtained in Part II) are provided in Section 10.

**Necessity of all assumptions.** The assumptions in Theorem 1.2 are necessary.

First, the theorem does not hold in general if the vertex groups are finite. For instance, for $n \ge 5$, the right-angled Coxeter group $W_n$ with defining graph an $n$-gon acts properly and cocompactly on the real hyperbolic plane $\mathbb{H}_R^2$, as the group generated by the reflections over a right-angled hyperbolic $n$-gon. Therefore all groups $W_n$ are cocompact Fuchsian, so they are all measure equivalent (in fact commensurable). The classification of right-angled Coxeter groups up to measure equivalence is a widely open problem. We refer to the survey [Dan20, Sections 5 and 6] and the references therein for an overview of the current knowledge regarding their commensurability and quasi-isometry classification.

Assuming that both $\Gamma_G$ and $\Gamma_H$ have no partial conjugation is also necessary, even in the case of right-angled Artin groups. Indeed, every non-cyclic right-angled Artin group $G$ with $|\text{Out}(G)| < +\infty$ has proper finite-index subgroups that are themselves right-angled Artin groups, defined over graphs which are transvection-free but have partial conjugations. An example is given by the two leftmost graphs in Figure 9, page 57, with all vertex groups isomorphic to $\mathbb{Z}$. More generally, we refer to [Hua17, Theorem 1.5 and Section 6] for a full description of all finite-index right-angled Artin subgroups of $G$.

Furthermore, assuming that $\Gamma_G, \Gamma_H$ have no transvection is also necessary. For instance, when $\Gamma_G = \Gamma_H =: \Gamma$ is either a complete graph or a square, the same group $G$ can be defined as a graph product over $\Gamma$ in two different ways, and therefore the third conclusion of the theorem cannot hold in general. One can obtain counterexamples by considering a graph product over a square with appropriate vertex groups $A, B, C * D, E$ (in cyclic order), which is isomorphic to the graph product over a square with vertex
groups $A * C, B, D, E$. On the other hand, we mention that it is not possible to de-
compose the same group $G$ in two different ways over a finite simple graph (not
reduced to a point) with no transvection and no partial conjugation (see Corollary A.2,
page 143).

**Remark 1.3** (New instances of the measure equivalence classification of RAAGs). Even among right-angled Artin groups, our main theorem yields new classification results that were not covered by [HH22a], and were not accessible to our knowledge by previously available techniques. For example, let $\Gamma$ be a pentagon and let $G$ and $H$ be the graph products over $\Gamma$ with respective vertex groups $(F_2)_{v \in V_\Gamma}$ and $(F_3)_{v \in V_\Gamma}$. By our main result $G$ and $H$ are not measure equivalent, because the vertex groups $F_2$ and $F_3$ are not orbit equivalent [Gab00]. But this was not possible to tell using hitherto known techniques. For instance $G$ and $H$ both have cost 1 because they have chain-commuting generating sets, see [Gab00, Critères VI.24(1)]. They have proportional $\ell^2$-Betti numbers: these all vanish except in dimension 2, as follows from the work of Davis and Leary which relates the $\ell^2$-Betti numbers of a right-angled Artin group, to the classical homology of the flag completion of its defining graph [DL03, Corollary 2]. And they have infinite outer automorphism groups (which contain transvections), so [HH22a] does not apply – one could also prove that they have isomorphic extension graphs in the sense of Kim-Koberda [KK13], so the situation is very different from [HH22a]. We finally remark that $G$ and $H$ are quasi-isometric (see also our next remark on this point).

Sticking to the problem of the measure equivalence classification of right-angled Artin
groups, we also refer to Section 9, where we establish the *untrans vectorable extension graph*
as a new invariant, thereby improving the results from [HH22a].

**Remark 1.4** (Measure equivalence versus quasi-isometry). No analogue of Theorem 1.2 for quasi-isometry is known. An easy observation (see e.g. Lemma 17.2, page 142) shows that if $G$ and $H$ are graph products over the same finite simple graph $\Gamma$, and if the vertex groups $G_v$ and $H_v$ are bi-Lipschitz equivalent for every $v \in V_\Gamma$ (i.e. there exists a bi-Lipschitz bijection from $G_v$ to $H_v$), then $G$ and $H$ are bi-Lipschitz equivalent. But no converse of this statement is known, except in some specific situations, see e.g. [HKS23, Corollary A.3] and the conjecture [HKS23, Remark A.4].

Notice that, in contrast to the situation from Theorem 1.1, the measure equivalence and quasi-isometric classifications do not match for graph products in general. This is because the difference between quasi-isometry and bi-Lipschitz equivalence is significantly looser than the difference between measure equivalence and orbit equivalence. Indeed, a theorem of Whyte [Why99, Theorem 2] ensures that two non-amenable groups which are quasi-isometric are in fact bi-Lipschitz equivalent. On the other hand, measure equivalent groups need not be orbit equivalent: for example, a group $G$ with a positive $\ell^2$-Betti number is never orbit equivalent to any proper finite-index subgroup [Gab02]. As a consequence, if $G$ is a graph product of non-amenable groups, then changing one vertex group $G_v$ to $G_v \times \mathbb{Z}/2\mathbb{Z}$ yields a group which is always quasi-isometric to $G$, but not measure equivalent to $G$ in general.

See also Theorem 1.16 for an example of a group which is rigid in measure equivalence but not in quasi-isometry among torsion-free groups.
**Remark 1.5** (Fundamental groups and von Neumann algebras). When proving Theorem 1.2, we actually show (see Proposition 5.9, page 56) that all measure equivalence couplings between $G$ and $H$ are actually orbit equivalence couplings, i.e. there is a common Borel fundamental domain for $G$ and $H$.

A consequence is the following. Given a group $G$ as in Theorem 1.2, and a free, ergodic, measure-preserving $G$-action on a standard probability space $X$, the orbit equivalence relation $\mathcal{R}$ of the action has trivial fundamental group, i.e. it is not isomorphic to $\mathcal{R}_{|U}$ for any positive measure Borel subset $U \subseteq X$.

Now, to every action $G \not\to X$ as above, one can associate a von Neumann algebra (a $\Pi_1$ factor), via Murray and von Neumann's group measure space construction [MvN36]. By combining the above with a Cartan rigidity theorem established for graph products by Chifan and Kunnawalkam Elayavalli [CKE21] in the framework of Popa's deformation/rigidity theory, we also deduce that the von Neumann algebra $L(G \not\to X)$ has trivial fundamental group. This is proved in Section 5.4.2, page 60. See also Section 14.3, page 126, for another application of our work to a $W^*$-rigidity theorem.

Finally, let us mention that we also obtain variations of Theorem 1.2, for instance Corollary 5.7, page 55. We refer to Section 10 for examples of applications.

### 1.1.3 On the commensurability classification

Recall that two groups $G$ and $H$ are *commensurable* if they share isomorphic finite-index subgroups. Following [JS01], we say that they are *strongly commensurable* if there exist isomorphic finite-index subgroups $G^0 \subseteq G$ and $H^0 \subseteq H$ such that $[G : G^0] = [H : H^0]$. Januszkiewicz and *Świątkowski* proved in [JS01, Theorem 1] that if $G, H$ are two graph products over finite simple graphs $\Gamma_G$ and $\Gamma_H$ respectively, and if there exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, the vertex groups $G_v$ and $H_{\sigma(v)}$ are strongly commensurable, then $G$ and $H$ are (strongly) commensurable. In [JS01, Section 5], they discuss the necessity of strong commensurability of the vertex groups (as opposed to mere commensurability) in various examples.

It turns out that commensurability (resp. strong commensurability) is exactly the property of having a measure equivalence coupling (resp. orbit equivalence coupling) on a countable set $\Omega$, see Lemma 3.2, page 33. By specifying Theorem 1.2 to the setting of discrete couplings, we attain the following theorem which provides a converse to the work of Januszkiewicz and *Świątkowski* when the defining graphs have no transvection and no partial conjugation.

**Theorem 1.6.** Let $\Gamma_G, \Gamma_H$ be two finite simple graphs, not reduced to a point, with no transvection and no partial conjugation. Let $G, H$ be graph products of countably infinite groups over $\Gamma_G, \Gamma_H$, respectively. Then the following assertions are equivalent.

1. The groups $G$ and $H$ are commensurable.

2. The groups $G$ and $H$ are strongly commensurable.

3. There exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, the groups $G_v$ and $H_{\sigma(v)}$ are strongly commensurable.

**Remark 1.7** (On the classification of graph products up to isomorphism). The same theorem is also true if “commensurable” and “strongly commensurable” are replaced by
"isomorphic". In this case one can altogether remove the assumption that the vertex
groups are countably infinite. See Corollaries A.2 and A.26, pages 143 and 155. This
classification theorem up to isomorphism was essentially proved by Genevois [Gen18,
Theorem 8.2], with the extra assumption that vertex groups do not further split as graph
products. Our approach, which was inspired from the setting of measure equivalence, is
different, and enables us to remove this assumption.

1.1.4 A word on proof techniques

A complete outline of the proof is available in the introduction of Part II, whose reading
can be completed with the introduction of Section 7. We therefore refer the reader to
these sections, and here we only describe our main tools.

A measure equivalence coupling $\Omega$ between $G$ and $H$ yields two actions $G \to H \setminus \Omega$ and $H \to G \setminus \Omega$, whose orbits coincide on a common positive measure subset $U$ of the two spaces [Fur99b]. This can be reformulated in the language of measured groupoids (see Section 4), by saying that we have a measured groupoid $\mathcal{G}$ over $U$, equipped with two cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$. Our proof strategy can be decomposed in several steps.

* We first show that subgroupoids of the form $\rho_G^{-1}(G_v)$ for some $v \in V\Gamma_G$, are also of the form $\rho_H^{-1}(hH_wh^{-1})$ for some $w \in V\Gamma_H$ and $h \in H$, up to a countable Borel partition of $U$.

* Following a strategy of Kida [Kid10] (reviewed in Section 4.4, page 44), we use the above to find a Borel $(G \times H)$-equivariant map $\Omega \to \text{Isom}(\Gamma_G^e, \Gamma_H^e)$, where $\Gamma_G^e, \Gamma_H^e$ are the extension graphs of $G, H$ in the sense of Kim-Koberda [KK13] – they play for us the same role as the curve graph in Kida’s proof of the measure equivalence rigidity of surface mapping class groups.

* Another combinatorial object associated to a graph product $G$ is its right-angled building $\mathcal{D}_G$, introduced by Davis [Dav98]. In fact, by comparison of the automorphism groups of $\Gamma_G^e$ and $\mathcal{D}_G$, the above yields a $(G \times H)$-equivariant Borel map from $\Omega$ to $\text{Isom}(\mathcal{D}_G, \mathcal{D}_H)$. Using that $G$ and $H$ have a common fundamental domain on $\mathcal{D}_G \approx \mathcal{D}_H$, and therefore on $\text{Isom}(\mathcal{D}_G, \mathcal{D}_H)$, we deduce by pulling back a common fundamental domain on $\Omega$. Therefore $G$ and $H$ are orbit equivalent. And we also use the right-angled building to show that $\Omega$ induces an orbit equivalence coupling at the level of the vertex groups.

**Remark 1.8.** The first step is the translation to measured groupoids of the following group-theoretic statement: every isomorphism $f: G \to H$ sends vertex groups to conjugates of vertex groups. Its proof requires new combinatorial arguments, which we believe to be of independent interest for the study of graph products. We have written a proof of the above group-theoretic statement, with assumptions that are slightly more general than those from Theorem 1.2, in the appendix of the paper (see Theorem A.1, page 143). This is used to obtain some new results on graph products: this is the key for the isomorphism classification as in Remark 1.7, and also allows us to slightly improve a theorem of Genevois regarding the acylindrical hyperbolicity of their automorphism groups [Gen18].
## 1.2 Quantitative point of view

As previously mentioned, some classes of groups turn out to be very rigid from the measure equivalence point of view while amenable groups or lattices in $SL_2(\mathbb{R})$, on the other hand, are quite flexible: the class of groups that are measure equivalent to one of these groups is actually very large.

This motivates the study of refinements of measure (or orbit) equivalence, that are more sensitive to the geometry of the groups inside these flexible classes. To that end, an integrable version of these notions has been defined and allowed Bader, Furman and Sauer [BFS13] to obtain rigidity phenomena for lattices in $SO(n, 1)$ when $n \ge 2$.

This refined version has recently been generalized by Delabie, Koivisto, Le Maître and Tessera [DKLMT22] to what they called quantitative measure (and orbit) equivalence (see Definition 1.9 below). In particular, geometric notions such as the growth (see Bowen's appendix to Austin's article [Aus16], see also [DKLMT22, Theorem 3.1]) or the isoperimetric profile [DKLMT22, Theorem 1.1] are captured by this quantitative version of measure equivalence, and can thus be used to distinguish amenable groups.

But quantitative measure equivalence has not been extensively studied beyond the amenable world (where growth or isoperimetric profiles no longer provide meaningful invariants), apart from some strong rigidity theorems in integrable measure equivalence for some specific families of groups [BFS13, Bow17, HH23].

In Part III of the present work, we extend our main classification theorem for graph products to the quantitative framework (Theorems 1.10 and 1.11), thereby providing the first systematic study of quantitative measure equivalence in a non-amenable setting. As an application, we answer the inverse problem of the quantification for a very large family of right-angled Artin groups (Theorem 1.13). But let us first recall some notations and terminology introduced in [DKLMT22].

### 1.2.1 Quantification of couplings

Let $G$ and $H$ be two measure equivalent groups over some measured space $(\Omega, m)$ with respective fundamental domains $X_G$ and $X_H$. In this framework one can define two measure equivalence cocycles denoted $c: G \times X_H \to H$ and $c': H \times X_G \to G$ by letting $c(g, x)$ be the unique element of $H$ that brings $g \ast x$ back to the fundamental domain $X_H$ (see Figure 1) and $c'$ is defined analogously by reverting the roles of $G$ and $H$. In this section we will need to keep track of the fundamental domains $X_G$ and $X_H$, therefore we will denote by $(\Omega, m, X_G, X_H)$ a measure equivalence coupling from $G$ to $H$.

If $S_G$ denotes a finite generating set of $G$, we denote by $|g|_{S_G}$ the word length of $g$ with respect to $S_G$.

**Definition 1.9.** Let $G = \langle S_G \rangle$ and $H = \langle S_H \rangle$ be two finitely generated groups. Given a measure equivalence coupling $(\Omega, m, X_G, X_H)$ and two non-decreasing functions $\varphi$ and $\psi$, one says that the coupling is $(\varphi, \psi)$-integrable (from $G$ to $H$) if for all $g \in G$ and all $h \in H$ we have

$$ \int_{X_H} \varphi(|c(g, x)|_{S_H}) dm(x) < +\infty \quad \text{and} \quad \int_{X_G} \psi(|c'(h, x)|_{S_G}) dm(x) < +\infty. $$
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Figure 1: Definition of $c(g, x)$</figcaption></figure>

We will say that the coupling is $(\varphi, L^0)$-integrable when there are no conditions on the integrability of the cocycle $c'$.

In the particular case where $X_G = X_H =: X$, the coupling $(\Omega, m, X, X)$ will be called a $(\varphi, \psi)$-orbit equivalence coupling.

### 1.2.2 Main quantification results for graph products

We first show in Section 11 that the integrability between vertex groups passes to the graph products. The following theorem gives a quantitative version of the implication “3 ⇒ 2” of Theorem 1.2, valid with no condition on the defining graph.

**Theorem 1.10.** Let $G$ and $H$ be two graph products over a finite simple graph $\Gamma$, with finitely generated vertex groups $(G_v)_{v \in \Gamma}$ and $(H_v)_{v \in \Gamma}$, respectively.

Let $\varphi, \psi: [1, +\infty[ \to [1, +\infty[$ be two non-decreasing functions. Assume that for every $v \in \Gamma$, there exists a $(\varphi, \psi)$-integrable orbit equivalence coupling from $G_v$ to $H_v$.

Then there exists a $(\varphi, \psi)$-integrable orbit equivalence coupling from $G$ to $H$.

In fact the coupling was already constructed in [HH22a], using an argument of Gaboriau [Gab05], and here we only need to check its integrability.

Remark that by Theorem 1.2, orbit equivalence can not be replaced by measure equivalence in the above theorem. For a basic example, consider $G := \{1\} * \mathbb{Z}/2\mathbb{Z}$ and $H = \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$. Even though $\{1\}$ and $\mathbb{Z}/2\mathbb{Z}$ are measure equivalent, the two graph products $G$ and $H$ are not: $G$ is finite and $H$ infinite amenable (see e.g. **P**<sub>ME1</sub> and **P**<sub>ME10</sub> in [Gab05]).

Our main theorem in Part III is a converse to Theorem 1.10, which provides a quantitative version of the implication “1 ⇒ 3” in Theorem 1.2.

**Theorem 1.11** (see Theorem 12.1, page 104). Let $G, H$ be graph products with infinite finitely generated vertex groups over finite simple graphs $\Gamma_G, \Gamma_H$ with no transvection and no partial conjugation. Let $\varphi, \psi: [1, +\infty[ \to [1, +\infty[$ be two non-decreasing functions.
If there exists a $(\varphi, \psi)$-integrable measure equivalence coupling from $G$ to $H$, then there exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$, such that for every $v \in V\Gamma_G$, there exists a $(\varphi, \psi)$-integrable measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$.

We refer to Theorem 12.1 for an even more general statement, with variations on the assumptions made on the defining graphs $\Gamma_G, \Gamma_H$.

Notice that Theorem 1.2 already gives us a measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$. We crucially use the convexity of the vertex groups inside the graph product (exploited in the form of Green's normal form theorem for graph products [Gre90]) to obtain the desired integrability.

**Remark 1.12** (Discussion on the asymmetry orbit/measure equivalence). We do not know whether, as could be suggested by Theorem 1.2, the conclusion of Theorem 1.11 can be improved to a $(\varphi, \psi)$-integrable orbit equivalence coupling from $G_v$ to $H_{\sigma(v)}$. The difficulty is that having a $(\varphi, \psi)$-integrable measure equivalence coupling $(\Omega, m, X_G, X_H)$, and knowing that there also exists a common fundamental domain $Y$ for the actions of $G$ and $H$, is not enough to deduce that $(\Omega, m, Y, Y)$ is $(\varphi, \psi)$-integrable. Whence the slight asymmetry in our statements of Theorems 1.10 and 1.11, one with orbit equivalence, and the second with measure equivalence.

### 1.2.3 An application to the inverse problem for quantifications

In Section 13 we construct couplings with right-angled Artin groups with prescribed integrability. Formally, we will say that a $(\Phi, L^0)$-integrable coupling from $G$ to $H$ is *optimal* if any $(\varphi, L^0)$-integrable coupling from $G$ to $H$ verifies $\varphi \le \Phi$ (that is to say if $\varphi(x) = \mathcal{O}(\Phi(Cx))$ for some $C > 0$, as $x$ goes to $+\infty$). In Section 13 we investigate the following question — also called “the inverse problem”: given a group $H$ and an increasing function $\varphi$, can one find a group $G$ and a $(\varphi, L^0)$-integrable coupling from $G$ to $H$ such that the integrability is optimal?

The construction of prescribed couplings with optimal (or close to be) integrability has been solved by the first named author in [Esc22] for various classes of amenable groups (including $\mathbb{Z}$ or the usual lamplighter group). We offer here an extension of these results for the same class of functions as in the latter reference but outside of the amenable framework. This is done by constructing couplings with prescribed integrability with a right-angled Artin group $H$. To do so we rely on Theorem 1.10 to obtain a coupling from a graph product $G$ with amenable vertex groups, to $H$, with the wanted integrability. Finally, relying on the geometry of the vertex groups, we prove that these couplings are optimal (up to some logarithmic error). This is formalized in Theorem 1.13 below.

A vertex $v \in V\Gamma$ is *untrans vectorable* if there does not exist any $w \in V\Gamma\setminus\{v\}$ such that $\text{lk}(v) \subseteq \text{star}(w)$.

**Theorem 1.13.** Let $H$ be a right-angled Artin group over a finite simple graph $\Gamma$, and assume that $\Gamma$ contains an untrans vectorable vertex. Let $\rho : [1, +\infty) \to [1, +\infty)$ be a non-decreasing map such that $x \mapsto x/\rho(x)$ is non-decreasing.

Then there exist a group $G$ and an orbit equivalence coupling that is $(\varphi_\varepsilon, \exp \circ \rho)$-integrable from $G$ to $H$ for all $\varepsilon > 0$, where $\varphi_\varepsilon(x) := \rho \circ \log(x)/(\log \circ \rho \circ \log(x))^{1+\varepsilon}$.

Moreover, this coupling is optimal up to a log factor in the following sense: if there exists a $(\varphi, L^0)$-integrable measure equivalence coupling from $G$ to $H$, then $\varphi \le \rho \circ \log$.
Theorem 1.13 applies, for instance, to every right-angled Artin group with finite outer automorphism group (see the above Section 1.1.1). But Theorem 1.13 also applies, for instance, to all right-angled Artin groups whose defining graph is a tree of diameter at least 3, a class which is not covered by the results in [HH22a].

## 1.3 Rigidity

All the results presented so far are classification theorems within the class of graph products. We will now consider measure equivalence couplings between a graph product $G$ and an arbitrary countable group $H$, and aim for rigidity theorems, i.e. give conditions ensuring that $H$ has to be virtually isomorphic to $G$. There are two ways in which rigidity can be achieved, which occupy Part IV and Part V of the present work, respectively.

* In Part IV, we impose extra ergodicity conditions on the coupling between $G$ and $H$. This is in the spirit of the work of Monod-Shalom on rigidity for irreducible actions of direct products [MS06], and [HHI23] for right-angled Artin groups – in fact we do follow the strategy of [HHI23] very closely.

* In Part V, we impose some strong form of measure equivalence rigidity for the vertex groups (but no extra conditions on the coupling), and use graph products to obtain examples of groups that are superrigid in measure equivalence, but not in quasi-isometry.

### 1.3.1 Orbit equivalence rigidity under ergodicity assumptions

In Part IV, we consider probability measure-preserving actions of graph products that satisfy extra ergodicity conditions, and obtain strong rigidity results. Here we will state a simple version of our theorems where we assume all actions to be mixing. In Part IV, we investigate various sets of ergodicity assumptions that ensure rigidity. We refer in particular to Theorem 14.2, page 114 and Theorem 15.1, page 127 for the full statements.

Recall that a free, measure-preserving action of a countable group $G$ on a standard probability space $(X, \mu)$ is *mixing* if for any Borel subsets $U, V \subseteq X$, and any sequence $(g_n)_{n \in \mathbb{N}} \in G^{\mathbb{N}}$ consisting of pairwise distinct elements, one has

$$ \lim_{n \to +\infty} \mu(g_n \cdot U \cap V) = \mu(U)\mu(V). $$

Recall also that two probability measure-preserving actions $G \rightsquigarrow X$ and $H \rightsquigarrow Y$ are *conjugate* if there exists a group isomorphism $\alpha : G \to H$ such that for all $g \in G$ and a.e. $x \in X$ one has $f(g \cdot x) = \alpha(g) \cdot f(x)$.

**Theorem 1.14.** Let $G, H$ be two torsion-free countable groups. Let $G \rightsquigarrow X$ and $H \rightsquigarrow Y$ be two mixing measure-preserving actions on standard probability spaces. Assume that $G$ splits as a graph product over a connected finite simple graph $\Gamma$, with no vertex joined to every other vertex.

If the two actions $G \rightsquigarrow X$ and $H \rightsquigarrow Y$ are orbit equivalent, then they are conjugate.

Part IV is organized as follows.

* In Section 14.2, we consider the case where both $G$ and $H$ are graph products, over connected graphs with no vertex joined to every other vertex. We prove rigidity
results that are internal to the class of graph products, under softer ergodicity conditions. For instance, assuming that the actions $G \to X$ and $H \to Y$ are both vertexwise ergodic (i.e. every vertex group acts ergodically), and that $G$ and $H$ have the unique root property (if $g_1^k = g_2^k$ with $k > 0$, then $g_1 = g_2$), we prove that (stably) orbit equivalent actions $G \to X$ and $H \to Y$ are automatically conjugate. Several variations over this statement are given.

* In Section 15, we consider the superrigidity problem, where the group $H$ is arbitrary. As in the work of Monod–Shalom, a mild mixing condition on the action $H \to Y$ enables to transfer the rigidity within the class of graph products, to a superrigidity theorem, through composition of couplings.

**Remark 1.15.** As in Remark 1.5, combining the above theorem with theorems regarding the uniqueness of the Cartan subalgebra also yields $W^*$-rigidity theorems for actions of graph products satisfying extra ergodicity conditions. See Section 14.3.

### Comparison with previously known results.

* In [MS06], Monod and Shalom proved rigidity theorems for actions of products of groups having some negative curvature property (class $C_{\text{reg}}$ – see Example 16.2, page 130 for the definition), assuming that each factor acts ergodically. Our proof deeply relies on their work, using the fact that graph products contain many product subgroups. Interestingly, we do not need to assume that the vertex groups belong to the class $C_{\text{reg}}$ in our theorems: the graph product structure alone is enough to ensure the rigidity.

* In [HHI23], Huang, Ioana and the second-named author proved the above theorem (and the variations presented in Part IV) in the case of right-angled Artin groups. We follow their strategy very closely, and crucially use our work from Part II to replace several arguments from [HHI23].

* Finally, for Bernoulli actions, a particular case of *malleable* actions (see [Pop07, Definition 4.3]), very strong rigidity theorems were proved by Popa in the framework of his deformation/rigidity theory [Pop07, Pop08]. His theorems cover the Bernoulli actions of numerous graph products, and in fact many more classes (including, in a different direction, all Bernoulli actions of Property (T) groups).

## 1.3.2 Measure equivalence rigidity for graph products

The final chapter of this work (Part V) addresses the question of the rigidity of graph products whose vertex groups are rigid in measure equivalence. We say that a countable group $G$ is *rigid in measure equivalence* if every countable group $H$ which is measure equivalent to $G$, is virtually isomorphic to $G$. Our main results are the following.

* In Section 16, we give a sufficient criterion to ensure that a graph product $G$ is rigid in measure equivalence. This criterion requires to know that every vertex group $G_v$ is rigid in a very strong sense, that is, every self measure equivalence coupling of $G_v$ factors through the tautological one (by left-right multiplication on $G_v$ itself, or at least on a finite index extension of $G_v$). We also need extra assumptions on the vertex groups, including having some control on possible homomorphisms between them. See Theorem 16.1, page 130.
• In Section 17, using the above criterion with Higman groups as vertex groups, we construct a graph product which, to our knowledge, is the first example of a group which is rigid (among torsion-free groups) in measure equivalence, but not in quasi-isometry.

As mentioned, our example involves families of Higman groups [Hig51], whose rigidity in measure equivalence was proven by Huang and the second-named author [HH22b]. Given $k \ge 4$, let

$$ \mathrm{Hig}_k = \langle a_1, \dots, a_k \mid a_i a_{i+1} a_i^{-1} = a_{i+1}^2 \quad \forall i \pmod k \rangle. $$

We attain the following theorem.

**Theorem 1.16.** Let $\Gamma$ be a finite simple graph with no transvection and no partial conjugation, such that $\mathrm{Aut}(\Gamma) = \{1\}$. Let $(k_v)_{v \in V\Gamma}$ be a family of integers, and for every $v \in V\Gamma$, let $G_v = \mathrm{Hig}_{\sigma_v}$. Assume that for every $v \ne w$, the integer $k_v$ does not divide $k_w$. Let $G$ be the graph product over $\Gamma$ with vertex groups $(G_v)_{v \in V\Gamma}$. Then

1. Every torsion-free countable group $H$ which is measure equivalent to $G$, is isomorphic to $G$.

2. There exists an infinite family of finitely generated torsion-free groups that are all quasi-isometric to $G$, and pairwise not commensurable.

To build a group that is quasi-isometric to $G$ but not commensurable, one only needs to replace one vertex groups $G_v$ by $G_v \times \mathbb{Z}/n\mathbb{Z}$. See Remark 1.4. We now say a word about our proof of the measure equivalence rigidity of $G$, a more detailed outline is given at the beginning of Section 16.

**A word on the proof.** Using the work from Part II, we know that for every self measure equivalence coupling $\Omega$ of $G$, there is a $(G \times G)$-equivariant measurable map $\Omega \to \mathrm{Aut}(\mathbb{D}_G)$, where $\mathbb{D}_G$ is the right-angled building of the graph product. We use a standard technique that dates back to the work of Furman [Fur99a], to transfer rigidity from self-couplings to arbitrary couplings. We deduce a homomorphism $H \to \mathrm{Aut}(\mathbb{D}_G)$, i.e. an action of $H$ on $\mathbb{D}_G$. And we prove that the cell stabilizers for the $H$-action on $\mathbb{D}_G$ are measure equivalent to their respective $G$-stabilizers. Using the measure equivalence rigidity of the Higman groups, in fact $G$ and $H$ act on $\mathbb{D}_G$ with isomorphic cell stabilizers.

The crucial point is to use a volume (coupling index) argument to show that $G$ and $H$ act on $\mathbb{D}_G$ with a common strict fundamental domain $Y$. For this, it is crucial to know that $\mathrm{Hig}_\sigma$ has no proper finite-index subgroup [Hig51].

The existence of a strict fundamental domain for the $H$-action ensures that $H$ has the structure of simple complex of groups over $Y$, in the sense of Bridson-Haefliger [BH99, Chapter II.12]. The inclusion homomorphisms are inclusions between direct products of Higman groups, and these are very constrained: they must be factor inclusions. Consequently, the complex of group structure of $H$ is nothing but a graph product structure, in other words $H$ is isomorphic to $G$.
## 1.4 Structure of the paper

The present article is decomposed in five parts and one appendix. The first part contains the necessary material about graph products, measure and orbit equivalence, and measured groupoids and – depending on the reader’s background – can be skipped on first reading.

Part II revolves around our measure equivalence classification of graph products, namely the proof of Theorem 1.2. In particular we develop in Section 7 most of the groupoid theoretic machinery needed to prove the latter result. Let us mention that a translation of this argument in the language of groups can be found in Appendix A, where it is used to show the isomorphic classification theorem analogous to Theorem 1.2. Part II is also where we prove our corollaries about commensurability (Theorem 1.6) and fundamental groups of equivalence relations and von Neumann algebras (Corollary 5.15, page 61). Both of the latter proofs can be found in Section 5.4, page 59. The last section of the part is devoted to examples (Section 10, page 99).

In Part III we study the quantitative version of this measure equivalence classification and prove Theorems 1.10, 1.11 and 1.13.

The last two parts deal with rigidity behaviours. In Part IV we impose some ergodicity assumptions on the actions and obtain some rigidity statements that are in the same spirit as Monod-Shalom rigidity theorem for direct products (Theorem 1.14). Part V focuses on exhibiting an example of a group that is superrigid for measure equivalence but not for quasi-isometry (Theorem 1.16).

Finally, in Appendix A we obtain an isomorphic classification theorem (Corollary A.2, page 143) of graph products. We use it to slightly improve a theorem of Genevois and show the acylindrical hyperbolicity of Aut($G$) whenever $G$ does not split as a direct product, and is not an infinite dihedral group (Theorem A.27, page 156). For the reader less familiar with the language of groupoids, the proof of Corollary A.2 can also serve as a guide through the proofs of Section 7.

## Acknowledgements

We warmly thank Jingyin Huang for many stimulating discussions that inspired several ideas from the present work.

This project was initiated during the trimester *Groups acting on fractals, Hyperbolicity and Self-Similarity* held at the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université) during Spring 2022. Both authors thank the IHP for its hospitality and support, through LabEx CARMIN, ANR-10-LABX-59-01.

Both authors were supported by the European Union (ERC, Artin-Out-ME-OA, 101040507). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

The first-named author was also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure.
# Part I

# Graph products and measured groupoids

This first part contains background and preliminary facts regarding graph products and their geometry (Section 2), measure and orbit equivalence and its quantitative versions (Section 3), and measured groupoids and their use in measure equivalence classification problems (Section 4).

## 2 Graph products and their geometry

In the following $\Gamma$ will always denote a finite simple graph (namely a graph with no loop edge and no multiple edges between two vertices) and $(G_v)_{v \in V_\Gamma}$ a family of non-trivial vertex groups. We let $G$ be the graph product over $\Gamma$ with vertex groups $(G_v)_{v \in V_\Gamma}$. An *induced subgraph* $\Lambda \subseteq \Gamma$ is a subgraph such that two vertices of $\Lambda$ are joined by an edge in $\Lambda$ if and only if they are joined by an edge in $\Gamma$. Recall that the *link* of a vertex $v \in V_\Gamma$ is the induced subgraph with vertex set all vertices adjacent to $v$. We denote it $\mathrm{lk}(v)$. The *star* of $v \in V_\Gamma$ is the subgraph of $\Gamma$ induced by $v$ itself and its neighbors and is denoted by $\mathrm{star}(v)$.

Recall that a finite simple graph $\Gamma$ is *transvection-free* if there do not exist two distinct vertices $v, w \in V\Gamma$ such that $\mathrm{lk}(v) \subseteq \mathrm{star}(w)$. We say that $\Gamma$ *has no partial conjugation* if there does not exist any vertex $v \in V\Gamma$ such that $\mathrm{star}(v)$ disconnects $\Gamma$. This terminology comes from the setting of right-angled Artin groups, where these conditions prevent having transvections and partial conjugations in $\mathrm{Out}(G_\Gamma)$ – and in fact any finite simple graph $\Gamma$ which is transvection-free and has no partial conjugation determines a right-angled Artin group $G_\Gamma$ with $|\mathrm{Out}(G_\Gamma)| < +\infty$ by [Lau95, Ser89].

A finite simple graph $\Gamma$ is called *reducible* if it splits as a join of (at least) two non-empty subgraphs. We will call *irreducible* a graph which is not reducible.

**Remark 2.1.** Let $\Gamma$ be a finite simple graph, not reduced to one vertex, and let $\Gamma = \Gamma_1 \circ \cdots \circ \Gamma_k$ be its join decomposition into irreducible subgraphs (well-defined up to permutation of the factors). If $\Gamma$ is transvection-free, then no $\Gamma_i$ is reduced to one vertex, and every $\Gamma_i$ is transvection-free. And if $\Gamma$ has no partial conjugation, then for every $i \in \{1, \dots, k\}$, the graph $\Gamma_i$ has no partial conjugation.

### 2.1 Normal form for graph products

This section reviews work of Green [Gre90]. Given $g \in G$, a *word* representing $g$ is a tuple $\mathbf{w} = (g_1, \dots, g_k)$, where every $g_i$ is an element of some vertex group, such that $g = g_1 \dots g_k$. Following [Gre90, Definition 3.5], we say that a word $\mathbf{w}$ is *reduced* if, letting $v_i \in V\Gamma$ be such that $g_i \in G_{v_i}$, the following two conditions hold:

* for every $i \in \{1, \dots, k\}$, one has $g_i \neq 1$;

* for every $i < j$, if $v_i = v_j$, then there exists $i < k < j$ such that $v_k$ is neither equal nor adjacent to $v_i$.
The second condition implies in particular that $v_{i+1} \neq v_i$ for every $i \in \{1, \dots, k\}$. Notice that a subword of a reduced word is again reduced, and that if $(g_1, \dots, g_n)$ is reduced, so is $(g_n^{-1}, \dots, g_1^{-1})$.

By Green's normal form theorem [Gre90, Theorem 3.9], every element of $G$ is represented by a reduced word. In addition, two reduced words $\underline{w} = (g_1, \dots, g_k)$ and $\underline{w}' = (g'_1, \dots, g'_l)$ represent the same element if and only if $k = l$, and $\underline{w}$ and $\underline{w}'$ are obtained from one another by successive applications of the following operation: if $g_i$ and $g_{i+1}$ belong to adjacent vertex groups, swap them. More generally, in her proof, Green also showed (see [Gre90, Definition 3.10 and Property (a) on p.30]) that, starting from any word $\underline{w}$ representing $g$, one can obtain a reduced word representing $g$ by applying a sequence of the following operations:

* remove some $g_i = 1$;

* if $g_i$ and $g_{i+1}$ belong to the same vertex group, replace them by the product $g_ig_{i+1}$;

* if $g_i$ and $g_{i+1}$ belong to adjacent vertex groups, swap them.

For every $v \in V\Gamma$, let $S_v$ be a finite generating set of $G_v$, and let $S = \cup_{v \in V\Gamma} S_v$, which is a finite generating set of $G$. Given a word $\underline{w} = (g_1, \dots, g_k)$, we let $|\underline{w}|_S = \sum_{i=1}^k |g_i|_{S_{v_i}}$, where $v_i \in V\Gamma$ is such that $g_i \in G_{v_i}$ – here $|g_i|_{S_{v_i}}$ denotes the word length of $g_i$ in the generating set $S_{v_i}$. Observe that if $\underline{w}$ is a reduced word representing $g$, then $|g|_S = |\underline{w}|_S$ (see e.g. [BR22, Corollary 2.5]).

Given $g \in G$, the head of $g$ is the set of all elements $h \in G$ such that there exists a reduced word $\underline{w} = (g_1, \dots, g_k)$ representing $g$, with $g_1 = h$. Likewise, the tail of $g$ is the set of all elements $h \in G$ such that there exists a reduced word $\underline{w} = (g_1, \dots, g_k)$ representing $g$, with $g_k = h$. We say that an element $z \in G$ is a syllable of $g$ if for some (equivalently, any) reduced word $\underline{w} = (g_1, \dots, g_k)$ representing $g$, there exists $i \in \{1, \dots, k\}$ such that $z = g_i$.

**Lemma 2.2.** Let $g, h \in G$, let $v \in V\Gamma$, and let $z \in G_v \setminus \{1\}$. Assume that the heads of $g$ and $h$ do not contain any element of $G_v$.

Then $z$ is a syllable of $g^{-1}zh$. In particular, $|g^{-1}zh|_S \geq |z|_S$.

*Proof.* Given a reduced word $\underline{w} = (g_1, \dots, g_k)$ representing $g$, we let

$$i_0(\underline{w}) = \max\{i : \forall j \leq i, g_j \notin G_v, \text{ and } g_j \text{ centralizes } G_v\}.$$

Namely $i_0$ is the largest $i$ such that for every $j \leq i$, the element $g_j$ belongs to a vertex group $G_{w_j}$ with $w_j$ adjacent to $v$. We now choose $\underline{w}_g$ so as to maximize the value of $i_0$. Let $\underline{w}_h = (h_1, \dots, h_l)$ be a reduced word representing $h$, chosen with the same constraint (we denote by $j_0$ the index defined by the same property). Then $(g_k^{-1}, \dots, g_{i_0}^{-1}, \dots, g_1^{-1}, z, h_1, \dots, h_{j_0}, \dots, h_l)$ is a word that represents $g^{-1}zh$, and so is

$$(g_k^{-1}, \dots, g_{i_0+1}^{-1}, z, g_{i_0}^{-1}, \dots, g_1^{-1}, h_1, \dots, h_{j_0}, \dots, h_l).$$

The subword $\underline{w}' = (g_{i_0}^{-1}, \dots, g_1^{-1}, h_1, \dots, h_{j_0})$ might not be reduced. Let $\underline{w}'' = (k_1, \dots, k_r)$ be a reduced word that represents the same element as $\underline{w}'$, and observe that all $k_p$ belong to vertex groups $G_{w_p}$ with $w_p$ adjacent to $v$ (in particular, distinct from $v$). Now $g^{-1}zh$ is represented by $(g_k^{-1}, \dots, g_{i_0+1}^{-1}, z, k_1, \dots, k_r, h_{j_0+1}, \dots, h_l)$. Notice that $g_{i_0+1}$ and $h_{j_0+1}$ do not belong to vertex groups adjacent to $G_v$ by definition of $i_0$ and $j_0$, and they do
not belong to $G_v$ either by our assumption on the heads of $g$ and $h$. Therefore the latter
word is reduced, which concludes the proof. □

## 2.2 Parabolic subgroups

### 2.2.1 Definition and first facts

Let $\Lambda \subseteq \Gamma$ be an induced subgraph, possibly empty. Let $G_\Lambda \subseteq G$ be the subgroup generated by all subgroups $G_v$ with $v \in V\Lambda$, note that when $\Lambda = \emptyset$ we have $G_\Lambda = \{e\}$. As a consequence of the normal form theorem for graph products reviewed in the previous section, the group $G_\Lambda$ is isomorphic to the graph product over $\Lambda$ with vertex groups $(G_v)_{v \in V\Lambda}$.

A parabolic subgroup of $G$ is a subgroup $P$ which is conjugate to $G_\Lambda$ for some induced subgraph $\Lambda \subseteq \Gamma$.<fnref n="1" /> Notice that $\Lambda$ is uniquely determined by $P$: if $gG_\Lambda g^{-1} = hG_{\Lambda'} h^{-1}$, then $\Lambda = \Lambda'$, as follows from [AM15, Corollary 3.8]. We say that $\Lambda$ is the type of $P$.

An intersection of parabolic subgroups of $G$ is again a parabolic subgroup: this is
[AM15, Corollary 3.6] for finite intersections, and follows from [AM15, Corollary 3.18]
for infinite intersections. Remark also that if $P \subseteq G$ is a parabolic subgroup of type $\Lambda$,
then $P$ is naturally a graph product over $\Lambda$; parabolic subgroups of $P$ for this graph
product structure are then naturally identified with the parabolic subgroups of $G$ that
are contained in $P$.

Given an induced subgraph $\Lambda \subseteq \Gamma$, we denote by $\Lambda^{\perp}$ the induced subgraph spanned by all vertices of $\Gamma \setminus \Lambda$ that are joined to all vertices of $\Lambda$ (see Figure 2 for two examples). Remark that in particular if $\Lambda$ is reduced to one vertex $v$ then $\Lambda^{\perp} = \text{lk}(v)$. Given a parabolic subgroup $P = gG_{\Lambda}g^{-1}$ (with $g \in G$, and $\Lambda \subseteq \Gamma$ an induced subgraph), we let $P^{\perp} = gG_{\Lambda^{\perp}}g^{-1}$. That this is well-defined (i.e. does not depend on the choice of $g$) follows from [AM15, Proposition 3.13]. The normalizer of $P$ in $G$ is equal to $P \times P^{\perp}$, and is therefore again a parabolic subgroup [AM15, Proposition 3.13].

<figure><img src="image_5.png" /><figcaption>Figure 2: Induced subgraphs and orthogonal</figcaption></figure>

Given an induced subgraph $\Lambda \subseteq \Gamma$, there is a natural retraction $r_\Lambda : G_\Gamma \to G_\Lambda$, sending every $G_v$ with $v \in V\Lambda$ to itself, and every $G_w$ with $w \notin V\Lambda$ to $\{1\}$. More

<footnote n="1">¹The notion of parabolic subgroup does not merely depend on $G$, but also of its decomposition as a graph product. Throughout the paper, we always assume that $G$ comes equipped with this decomposition.</footnote>
generally, let $P$ be a parabolic subgroup, and let $\Lambda \subseteq \Gamma$ be its type. Then there exists a unique element $g \in G$ such that $P = gG_{\Lambda}g^{-1}$ and the tail of $g$ does not contain any element in $G_{\Lambda} \times G_{\Lambda}^{\perp}$. For such a $g$ we then define the retraction $r_{P}: G \to P$ to be the map sending $gG_{v}g^{-1}$ to itself if $v \in V\Lambda$, and $gG_{w}g^{-1}$ to $\{1\}$ if $w \notin V\Lambda$.

### 2.2.2 Splittings of a graph product

Every vertex $v \in V\Gamma$ determines a splitting of $G$ as $G = G_{\text{star}(v)} *_{G_{\text{lk}(v)}} G_{\Gamma\setminus\{v\}}$. We let $T_v$ be the Bass–Serre tree of this splitting, see [Ser77]. It is equipped with an isometric $G$-action whose vertex stabilizers are exactly the conjugates of $G_{\text{star}(v)}$ and of $G_{\Gamma\setminus\{v\}}$.

The above splitting is non-trivial (i.e. the $G$-action on $T_v$ has no global fixed point) unless $v$ is joined to all other vertices of $\Gamma$ by an edge.

We will need a few standard facts regarding group actions on trees. Given an isometric action of a group $G$ on a tree $T$, we say that a subgroup $H \subseteq G$ is *elliptic* if it fixes a point in $T$. Notice that this happens as soon as $H$ as a finite orbit in $T$: indeed $H$ then fixes the circumcenter of this finite orbit. In particular, if $H$ has a finite-index subgroup $H^0$ which is elliptic in $T$, then $H$ is also elliptic in $T$ (because if $x \in T$ is a point fixed by $H^0$, then the $H$-orbit of $x$ is finite).

### 2.2.3 More properties of parabolic subgroups

We now establish a few extra properties of parabolic subgroups that will be used throughout the paper. They are based on the following results of Antolín–Minasyan.

**Proposition 2.3** (Antolín–Minasyan [AM15, Proposition 3.4 and Lemma 3.9]). Let $G$ be a graph product over a finite simple graph $\Gamma$.

1. Let $\Lambda_1, \Lambda_2 \subseteq \Gamma$ be induced subgraphs, and let $g \in G$. Then there exist an induced subgraph $\Upsilon \subseteq \Lambda_1 \cap \Lambda_2$, and $h \in G_{\Lambda_2}$, such that $gG_{\Lambda_1}g^{-1} \cap G_{\Lambda_2} = hG_{\Upsilon}h^{-1}$.

2. Let $P \subseteq G$ be a parabolic subgroup, and $g \in G$. If $gPg^{-1} \subseteq P$, then $gPg^{-1} = P$.

**Lemma 2.4.** Let $G$ be a graph product over a finite simple graph $\Gamma$ such that no vertex $v$ of $\Gamma$ satisfies $\Gamma = \text{star}(v)$. Let $H \subseteq G$ be a subgroup. If $H$ has a finite-index subgroup $H^0$ contained in a proper parabolic subgroup $P$ of $G$, then $H$ itself is contained in a proper parabolic subgroup $Q$ of $G$.

*Proof*. The group $P$ is conjugate to $G_{\Lambda}$ for some proper induced subgraph $\Lambda \subsetneq \Gamma$. Let $v$ be in $V\Gamma \setminus V\Lambda$. Then $H^0$ is elliptic in the tree $T_v$ because it is conjugate into $G_{\Gamma\setminus\{v\}}$. Therefore $H$ is also elliptic in $T_v$, and the conclusion follows (see the discussion in Section 2.2.2). $\square$

**Lemma 2.5.** Let $G$ be a graph product over a finite simple graph $\Gamma$ with infinite vertex groups. Let $P, Q \subseteq G$ be two parabolic subgroups, and assume that some finite-index subgroup of $P$ is contained in $Q$.

Then $P \subseteq Q$.

*Proof*. Up to a global conjugation, we can (and shall) assume that there exist two induced subgraphs $\Upsilon, \Lambda \subseteq \Gamma$ and $g \in G$ such that $Q = G_{\Lambda}$ and $P = gG_{\Upsilon}g^{-1}$. By Proposition 2.3(1), the intersection $P \cap Q$ is of the form $hG_{\Theta}h^{-1}$, with $h \in Q$ and $\Theta \subseteq \Lambda \cap \Upsilon$. Then for every $v \in V\Upsilon \setminus V\Theta$, the subgroup $gG_vg^{-1}$ intersects $P \cap Q$ trivially, again by
Proposition 2.3(1). But $gG_vg^{-1}$ is contained in $P$, so by assumption $gG_vg^{-1}$ has a finite-index (whence infinite) subgroup contained in $Q$. This proves that $\Theta = \Upsilon$. We have $P \cap Q = hG_\Theta h^{-1}$ and $P = gG_\Theta g^{-1}$, so Proposition 2.3(2) implies that $P \cap Q = P$, i.e. $P \subseteq Q$. $\square$

In the case where all vertex groups are torsion-free (or equivalently that $G$ is torsion-free [Gre90, Corollary 3.28]), we can extend Lemma 2.5 to all subgroups $H$ of $G$.

**Lemma 2.6.** Let $G$ be a graph product over a finite simple graph $\Gamma$, with infinite torsion-free vertex groups. Let $P \subseteq G$ be a parabolic subgroup, and let $H \subseteq G$ be a subgroup.

If $H$ has a finite-index subgroup $H^0$ contained in $P$, then $H \subseteq P$.

*Proof.* Without loss of generality, we will assume that $P$ is the smallest parabolic subgroup that contains $H^0$, namely $P$ is the intersection of all parabolic subgroups that contain $H^0$. Up to conjugation, we will assume that $P = G_\Lambda$ for some induced subgraph $\Lambda \subseteq \Gamma$. Let $v \in V\Gamma \setminus V\Lambda$. Then $H^0$ is elliptic in $T_v$, so $H$ is elliptic in $T_v$. So either $H$ is conjugate into $G_{\Gamma\setminus\{v\}}$, or else $H$ is conjugate into $G_{\text{star}(v)}$.

Let us first assume that $H$ is conjugate into $G_{\text{star}(v)}$. In particular, there exists $g \in G$ such that $H^0 \subseteq G_\Lambda \cap gG_{\text{star}(v)}g^{-1}$, namely $g^{-1}H^0g \subseteq g^{-1}G_\Lambda g \cap G_{\text{star}(v)}$. By Proposition 2.3(1), it follows that $H^0 \subseteq gG_{\text{lk}(v)}g^{-1}$. Since $G_{\text{star}(v)} = G_v \times G_{\text{lk}(v)}$ and since $G_v$ is torsion-free, it follows that $H \subseteq gG_{\text{lk}(v)}g^{-1}$: indeed otherwise, some element $h \in H$ has a nontrivial projection to $gG_vg^{-1}$, but then all powers of $h$ have a nontrivial projection to $gG_vg^{-1}$, contradicting that some power is contained in the finite-index subgroup $H^0$.

So in all cases, we can assume that $H$ is conjugate into $G_{\Gamma\setminus\{v\}}$. As this is true for every $v \in V\Gamma \setminus V\Lambda$, we can apply Proposition 2.3(1) again and deduce that $H$ is contained in a conjugate $hPh^{-1}$. In particular $H^0$ is contained in $P \cap hPh^{-1}$, and by minimality of $P$ this implies that $hPh^{-1} = P$. Therefore $H \subseteq P$, as desired. $\square$

Using the first point of Proposition 2.3 we can show that if $P = P_1 \times P_2$ is a product of two parabolic subgroups $P_1, P_2$ then its type $\Lambda$ can be written as the join $\Lambda = \Lambda_1 \circ \Lambda_2$, where $\Lambda_i$ is the type of $P_i$ for every $i \in \{1, 2\}$. Moreover if $P = gG_\Lambda g^{-1}$ for some $g \in G$, then $P_i = gG_{\Lambda_i}g^{-1}$. We can thus deduce the following two lemmas – details are left to the reader.

**Lemma 2.7.** Let $G$ be a graph product over a finite simple graph $\Gamma$. Let $P = P_1 \times P_2$ be a parabolic subgroup that splits as a direct product of two parabolic subgroups $P_1, P_2$. Let $Q \subseteq P$ be a parabolic subgroup.

Then $Q = (Q \cap P_1) \times (Q \cap P_2)$. $\square$

**Lemma 2.8.** Let $G$ be a graph product over a finite simple graph $\Gamma$. Let $P, Q$ be two distinct parabolic subgroups which are conjugate to vertex groups.

Then $P$ and $Q$ commute if and only if there exist $g \in G$ and two adjacent vertices $v, w \in V\Gamma$ such that $P = gG_vg^{-1}$ and $Q = gG_wg^{-1}$. $\square$

A vertex $v \in V\Gamma$ is *untrans vectorable* if there does not exist any vertex $w \neq v$ such that $\text{lk}(v) \subseteq \text{star}(w)$.
**Lemma 2.9.** Let $\Gamma$ be a finite simple graph which is not reduced to one vertex, let $G$ be a graph product over $\Gamma$. Let $C \subseteq G$ be a parabolic subgroup which is conjugate to $G_{\Upsilon}$ for some complete subgraph $\Upsilon \subseteq \Gamma$. Let $v \in V\Gamma$ be an untransvable vertex.

If $G_v \times G_v^{\perp} \subseteq C \times C^{\perp}$, then $C = G_v$.

*Proof.* Let $g \in G$ be such that $C = gG_{\Upsilon}g^{-1}$. Since $G_v \times G_v^{\perp} \subseteq C \times C^{\perp}$ then by Proposition 2.3 we also have inclusion of their types, namely $\mathrm{star}(v) \subseteq \Upsilon \circ \Upsilon^{\perp}$.

Now let $w \in V\Upsilon$. Since $\Upsilon$ is a clique, we get $\Upsilon \circ \Upsilon^{\perp} \subseteq \mathrm{star}(w)$. In particular $\mathrm{star}(v) \subseteq \mathrm{star}(w)$, but $v$ is assumed to be untransvable therefore $w = v$ and hence $C = gG_vg^{-1}$. Using Proposition 2.3 leads to $g \in G_{\mathrm{star}(v)}$ and therefore $C = G_v$. $\square$

## 2.3 Extension graphs, right-angled buildings and their automorphisms

In this section, we first review the definitions of the extension graph $\Gamma_G^e$ and of the right-angled building $\mathbb{D}_G$ associated to a graph product $G$. We then establish some results regarding the comparison of their automorphism groups.

### 2.3.1 Definitions

**Extension graphs.** They were introduced by Kim and Koberda in the context of right-angled Artin groups [KK13]. Their definition naturally extends to graph products, as already observed for instance in [CRKdING21, Definition 3.1].

**Definition 2.10 (Extension graph [KK13])**. Let $G$ be a graph product over a finite simple graph $\Gamma$. The *extension graph* $\Gamma_G^e$ is the graph whose vertices are the conjugates of the vertex groups of $G$, where two distinct vertices are joined by an edge if the corresponding subgroups commute.

Notice that $\Gamma_G^e$ contains a natural subgraph isomorphic to $\Gamma$, spanned by the vertices corresponding to the vertex groups. The action of $G$ by conjugation on the set of conjugates of vertex groups, induces a $G$-action by graph automorphisms on $\Gamma_G^e$. For this action $\Gamma$ is a fundamental domain, as follows using Lemma 2.8.

We will call a *clique* a complete subgraph of $\Gamma$ (possibly empty and in particular not necessarily maximal). A *standard clique subgroup* of $G$ is a parabolic subgroup of $G$ equal to $G_{\Upsilon}$ for some clique subgraph $\Upsilon \subseteq \Gamma$. A *standard clique coset* is a left coset $gG_{\Upsilon}$ of a standard clique subgroup of $G$. Notice that the stabilizer of the standard clique coset $gG_{\Upsilon}$, for the $G$-action by left translation on itself, is equal to $gG_{\Upsilon}g^{-1}$. From this, one deduces that if $\Upsilon_1, \Upsilon_2$ are two clique subgraphs of $G$ and $g_1, g_2 \in G$, then the standard clique cosets $g_1G_{\Upsilon_1}$ and $g_2G_{\Upsilon_2}$ are equal if and only if $\Upsilon_1 = \Upsilon_2$ and $g_1^{-1}g_2 \in G_{\Upsilon_2}$. In particular we can define the *type* of a clique coset as the clique subgraph $\Upsilon \subseteq \Gamma$ associated to the coset.

**Right-angled buildings.** In [Dav98], Davis associated a right-angled building to any graph product, as follows.

As usual, let $G$ be a graph product over a finite simple graph $\Gamma$. Let $\mathfrak{F}$ be the set of all standard clique cosets. Then $\mathfrak{F}$ is a partially ordered set, ordered by inclusion. Given
$F_1 \subseteq F_2$ in $\mathfrak{F}$, the interval $I_{F_1,F_2}$ is defined as

$$I_{F_1,F_2} = \{F \in \mathfrak{F} \mid F_1 \subseteq F \subseteq F_2\}.$$

Every interval in $\mathfrak{F}$ is a Boolean lattice of finite rank, so there exists a unique (up to isomorphism) cube complex $\mathbb{D}_G$ whose poset of cubes is isomorphic to the poset of intervals of $\mathfrak{F}$, see e.g. [AB08, Proposition A.38]. More concretely,

* the vertex set of $\mathbb{D}_G$ is the set of standard clique cosets of $G$;

* there is an edge between any two standard clique cosets of the form $gG_\Lambda, gG_\Upsilon$, where $\Lambda \subseteq \Upsilon$ and $|V\Upsilon \setminus V\Lambda| = 1$;

* for $k \ge 2$ one glues a $k$-cube to every subgraph isomorphic to the 1-skeleton of a $k$-cube, with the obvious face identifications.

**Definition 2.11** (Right-angled building [Dav98]). The cube complex $\mathbb{D}_G$ is called the Davis realization of the right-angled building of $G$.

In the sequel, we will simply say that $\mathbb{D}_G$ is the right-angled building of $G$. We mention that $\mathbb{D}_G$ is a CAT(0) cube complex by [Dav98, Corollary 11.7].

By construction, every vertex $v \in V\mathbb{D}_G$ corresponds to a left coset of the form $gG_\Lambda$, where $\Lambda \subseteq \Gamma$ is a complete subgraph. The rank of $v$ is the cardinality of the clique $\Lambda$ (this is well-defined: if $gG_\Lambda = g'G_{\Lambda'}$, then $\Lambda = \Lambda'$). Notice that rank 0 vertices correspond to elements of $G$.

### 2.3.2 Comparison with right-angled Artin groups

In this section, we will prove that the extension graph and the right-angled building of a graph product of countably infinite groups over a finite simple graph $\Gamma$, are isomorphic to the same objects associated to the right-angled Artin group over $\Gamma$. This will allow us to derive certain facts about these spaces from the analogous facts in the case of right-angled Artin groups.

More generally, let $\Gamma$ be a finite simple graph, and let $G$ and $H$ be two graph products over $\Gamma$ such that for every $v \in V\Gamma$, the vertex groups $G_v$ and $H_v$ have the same cardinality. For every $v \in V\Gamma$, we fix a bijection $\theta_v : G_v \to H_v$ such that $\theta_v(e) = e$.

Through normal forms (see Section 2.1), this induces a bijection $\theta : G \to H$, in the following way. Let $w_g = (g_1, \dots, g_k)$ be a reduced word that represents $g$. For every $i \in \{1, \dots, k\}$, let $v_i \in V\Gamma$ be such that $g_i \in G_{v_i}$. Let $w_H = (\theta_{v_1}(g_1), \dots, \theta_{v_k}(g_k))$. Our condition that $\theta_{v_i}(e) = e$ ensures that $w_H$ is again a reduced word, which represents an element of $H$ denoted $\theta(g)$. Notice that $\theta(g)$ does not depend on the choice of a reduced word that represents $g$, and that $\theta : G \to H$ is indeed a bijection (its inverse is obtained by considering the bijections $\theta_v^{-1}$).

We say that the bijection $\theta$ defined above is the bijection combined from $(\theta_v)_{v \in V\Gamma}$.

**Lemma 2.12.** Let $\Gamma$ be a finite simple graph, and let $G, H$ be two graph products over $\Gamma$. Assume that for every $v \in V\Gamma$, the vertex groups $G_v$ and $H_v$ have the same cardinality, and choose a bijection $\theta_v : G_v \to H_v$ with $\theta_v(e) = e$. Let $\theta : G \to H$ be the bijection combined from $(\theta_v)_{v \in V\Gamma}$.

Then for every $g \in G$ and every $v \in V\Gamma_G$, one has $\theta(gG_vg^{-1}) = \theta(g)H_v\theta(g)^{-1}$; in particular $\theta$ induces a graph isomorphism $\theta^e : \Gamma_G^e \to \Gamma_H^e$.
Proof. Write $g = g_1 \dots g_j g_{j+1} \dots g_k$, where each $g_i$ belongs to a vertex group, and $g_{j+1}, \dots, g_k \in G_v \times G_v^{\perp}$, with $g_j \notin G_v \times G_v^{\perp}$. Then $gG_v g^{-1} = g_1 \dots g_j G_v g_j^{-1} \dots g_1^{-1}$, so

$$ \theta(gG_v g^{-1}) = \theta(g_1) \dots \theta(g_j) H_v \theta(g_j)^{-1} \dots \theta(g_1)^{-1} = \theta(g) H_v \theta(g)^{-1}. $$

The map $\theta^e : V\Gamma_G^e \to V\Gamma_H^e$ defined by $\theta^e(gG_v g^{-1}) = \theta(gG_v g^{-1})$ is a bijection. That it induces a graph isomorphism follows from Lemma 2.8. $\square$

The analogous statement for the right-angled building is as follows.

**Lemma 2.13.** Let $\Gamma$ be a finite simple graph, and let $G, H$ be two graph products over $\Gamma$. Assume that for every $v \in V\Gamma$, the vertex groups $G_v$ and $H_v$ have the same cardinality, and choose a bijection $\theta_v : G_v \to H_v$ with $\theta_v(e) = e$. Let $\theta : G \to H$ be the bijection combined from $(\theta_v)_{v \in V\Gamma}$.

Then for every standard clique coset $gG_{\Upsilon}$, one has $\theta(gG_{\Upsilon}) = \theta(g)H_{\Upsilon}$; in particular $\theta$ induces a cubical isomorphism $\theta_D : D_G \to D_H$.

Proof. Let us first prove that for all $g \in G$ and all $\Upsilon \subseteq \Gamma$ we have $\theta_G(gG_{\Upsilon}) = \theta_G(g)H_{\Upsilon}$. Let $w = (g_1, \dots, g_k)$ be a reduced word that represents $g$, chosen so that there exists $i \in \{1, \dots, k\}$ such that all $g_j$ with $j > i$ belong to $G_{\Upsilon}$, and every $h \in G_{\Upsilon}$ that belongs to the tail of $g$ is one of the $g_j$ with $j > i$. In this way $gG_{\Upsilon} = g_1 \dots g_i G_{\Upsilon}$, and $\theta_G(gG_{\Upsilon}) = \theta_G(g_1) \dots \theta_G(g_i) H_{\Upsilon}$. On the other hand $\theta_G(g) = \theta_G(g_1) \dots \theta_G(g_i) \theta_G(g_{i+1}) \dots \theta_G(g_k)$, and the elements $\theta_G(g_{i+1}), \dots, \theta_G(g_k)$ all belong to $H_{\Upsilon}$. This proves that $\theta_G(gG_{\Upsilon}) = \theta_G(g)H_{\Upsilon}$, as claimed.

In particular, the above proves that the map $gG_{\Upsilon} \mapsto \theta_G(g)H_{\Upsilon}$ is well-defined, and by construction it induces a cubical isomorphism from $D_G$ to $D_H$. $\square$

**Corollary 2.14.** Every automorphism of $D_G$ preserves ranks of vertices.

Proof. This follows from the case of right-angled Artin groups, established in [HH23, Lemma 2.5], together with Lemma 2.13. $\square$

### 2.3.3 Automorphisms of the extension graph and the right-angled building

We equip Isom($\Gamma_G^e, \Gamma_H^e$) and Isom($D_G, D_H$) with the compact-open topology (equivalently, the topology of pointwise convergence on their vertex sets), and with the $(G \times H)$-action by left/right multiplication, i.e. for $f \in \text{Isom}(\Gamma_G^e, \Gamma_H^e)$ and $(g, h) \in G \times H$, we let $(g, h) \cdot f = gfh^{-1}$. And similarly if $f \in \text{Isom}(D_G, D_H)$. The main result of the present section is the following proposition – this was established as [HH23, Lemma 2.6] for right-angled Artin groups, based on work of Huang [Hua17].

**Proposition 2.15.** Let $G$ and $H$ be two graph products with countably infinite vertex groups over a common finite simple graph $\Gamma$ with no transvection and no partial conjugation.

Then there exists a $(G \times H)$-equivariant continuous map from $\text{Isom}(\Gamma_G^e, \Gamma_H^e)$ to $\text{Isom}(D_G, D_H)$.

The main part of the proof is the definition of the map from $\text{Isom}(\Gamma_G^e, \Gamma_H^e)$ to $\text{Isom}(D_G, D_H)$. The crucial step for this is to extend Huang’s [Hua17, Lemma 4.12]
to graph products. So let us start with $\beta \in \text{Isom}(\Gamma_G^e, \Gamma_H^e)$ and let us define its image in $\text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ denoted by $\Psi(\beta)$. First, remark that the map sending a maximal standard clique coset $gG_\Upsilon$ to its stabilizer $gG_\Upsilon g^{-1}$, is a bijection between maximal standard clique cosets and maximal cliques in the extension graph of $G$. So, if $\beta(gG_\Upsilon g^{-1}) = hH_\Lambda h^{-1}$ we define $\Psi(\beta)(gG_\Upsilon) := hH_\Lambda$. This only defines $\Psi(\beta)$ on maximal vertices. The goal is now to extend this definition to an element in $\text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$. To do so, let us define the image of some $g \in G$ viewed as a rank 0 vertex of $\mathbb{D}_G$.

**Lemma 2.16.** Let $\Gamma$ be a finite simple graph with no transvection and no partial conjugation, not reduced to one vertex. Let $G$ and $H$ be graph products over $\Gamma$ with countably infinite vertex groups. Let $\beta \in \text{Isom}(\Gamma_G^e, \Gamma_H^e)$. Let $g \in G$, and let $B_1, \dots, B_n$ be the maximal standard clique cosets containing $g$.

Then $\Psi(\beta)(B_1) \cap \dots \cap \Psi(\beta)(B_n)$ is a singleton denoted by $\{\Psi(\beta)(g)\}$.

Moreover the map $g \mapsto \Psi(\beta)(g)$ is a bijection from $G$ to $H$ that sends standard clique cosets to standard clique cosets of the same rank.

*Proof.* We start with the first assertion. In the case of right-angled Artin groups, this was proved by Huang [Hua17, Lemmas 4.12 and 4.17], see in particular [Hua17, Equation 4.13]. We explain how to derive the case of arbitrary graph products with countably infinite vertex groups, from the case of right-angled Artin groups. In the following, we denote by $A$ the right-angled Artin group defined over $\Gamma$.

For every $v \in V\Gamma$, we fix bijections $\theta_G^v : G_v \to \mathbb{Z}$ and $\theta_H^v : H_v \to \mathbb{Z}$. Through normal forms, this induces bijections $\theta_G : G \to A$ and $\theta_H : H \to A$ – here $\theta_G$ is combined from $(\theta_G^v)_{v \in V\Gamma}$, and $\theta_H$ is combined from $(\theta_H^v)_{v \in V\Gamma}$, in the sense of the previous section.

Let $\theta_G^e : \Gamma_G^e \to \Gamma_A^e$ and $\theta_H^e : \Gamma_H^e \to \Gamma_A^e$ be the induced bijections provided by Lemma 2.12. Let $\beta_A \in \text{Aut}(\Gamma_A^e)$ be defined by $\beta_A = \theta_H^e \beta(\theta_G^e)^{-1}$ (see also Figure 3).

<figure><img src="image_6.png" /><figcaption>Figure 3: Definition of $\beta_A$</figcaption></figure>

Similarly as in the discussion above the lemma, we can define $\Psi_A(\beta_A)$ as a bijection of the set of maximal standard clique cosets of $A$. Using Lemmas 2.12 and 2.13 and the definitions of $\Psi(\beta)$ and $\Psi_A(\beta_A)$, we get

$$ \Psi(\beta) = \theta_H^{-1} \circ \Psi_A(\beta_A) \circ \theta_G. \qquad (2.1) $$

Let $B = gG_\Upsilon$ be a maximal standard clique coset of $G$, where $g \in G$ and $\Upsilon \subseteq \Gamma$ is a maximal clique. By Lemma 2.13, we have $\theta_G(B) = \theta_G(g)A_\Upsilon$.

The maximal standard clique cosets $B_i$ containing $g$ coincide with the cosets $gG_{\Upsilon_i}$, with $\Upsilon_i$ ranging over the set of maximal cliques of $\Gamma$. But remark that

$$ \theta_G(B_1) \cap \dots \cap \theta_G(B_n) = \theta_G(g)A_{\Upsilon_1} \cap \dots \cap \theta_G(g)A_{\Upsilon_n} = \theta_G(g)A_{\cap_i \Upsilon_i}. $$
Now, since the graph is transvection-free there is no vertex in $\Gamma$ linked to all other vertices of $\Gamma$, thus $\cap_i \Upsilon_i = \emptyset$. Therefore $\theta_G(B_1) \cap \cdots \cap \theta_G(B_n) = \{\theta_G(g)\}$.

The cosets $\theta_G(B_i)$ are exactly the maximal standard clique cosets in $A$ containing $\theta_G(g)$. In the language of [Hua17], these are called the maximal standard flats containing $\theta_G(g)$. It then follows from Huang's aforementioned work [Hua17, Lemma 4.12, Equation 4.13 and Lemma 4.17] that $\Psi_A(\beta_A)(\theta_G(B_1)) \cap \cdots \cap \Psi_A(\beta_A)(\theta_G(B_n))$ is a singleton, denoted by $\{\Psi_A(\beta_A) \circ \theta_G(g)\}$.

Applying $\theta_H^{-1}$ shows that $\theta_H^{-1}\Psi_A(\beta_A)\theta_G(B_1) \cap \cdots \cap \theta_H^{-1}\Psi_A(\beta_A)\theta_G(B_n)$ is a singleton. This by construction (see eq. (2.1)) is exactly $\Psi(\beta)(B_1) \cap \cdots \cap \Psi(\beta)(B_n)$, which proves the first assertion. Notice that, by construction, the map $g \mapsto \Psi(\beta)(g)$ still verifies Equation (2.1).

Let us now show that $g \mapsto \Psi(\beta)(g)$ is a bijection from $G$ to $H$ that sends standard clique cosets to standard clique cosets.

To see that it is a bijection, remark that one can define in an analogous way a map $\Psi'(\beta^{-1}) : H \to G$ and note that it is the inverse of $\Psi(\beta)$.

Using the construction made in [HH23], above Lemma 2.6, we obtain that $\Psi_A(\beta_A)$ sends standard clique cosets to standard clique cosets. Moreover, by induction on the rank, using the bijectivity of $\Psi_A(\beta_A)$ (see again [HH23]) and what precedes, we can show that $\Psi_A(\beta_A)$ preserves the rank. Using eq. (2.1) defining $\Psi(\beta)$, we obtain the wanted assertion. $\square$

**Proof of Proposition 2.15.** Let us start with the definition of our $(G \times H)$-equivariant map

$$ \Psi : \mathrm{Isom}(\Gamma_G^e, \Gamma_H^e) \to \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H). $$

Let $\beta \in \mathrm{Isom}(\Gamma_G^e, \Gamma_H^e)$. Let $g \in G$. Let $B_1, \dots, B_n$ be the maximal standard clique cosets that contain $g$. By Lemma 2.16, the cosets $\Psi(\beta)(B_1) \cap \cdots \cap \Psi(\beta)(B_n)$ intersect in a single point $\Psi(\beta)(g)$, and the map $g \mapsto \Psi(\beta)(g)$ is a bijection that sends every standard clique coset to a standard clique coset. So $\Psi(\beta)$ can be viewed as a map in $\mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$. The assignment $\beta \mapsto \Psi(\beta)$ yields the desired $(G \times H)$-equivariant map

$$ \Psi : \mathrm{Isom}(\Gamma_G^e, \Gamma_H^e) \to \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H). $$

The continuity of $\Psi$ is now a consequence of the following observation: given any two rank 0 vertices $g \in V\mathbb{D}_G$ and $h \in V\mathbb{D}_H$, one has $\Psi(\beta)(g) = h$ if and only if, denoting by $\Upsilon_1, \dots, \Upsilon_k$ the maximal cliques of $\Gamma$, one has $\beta(\{gG_{\Upsilon_i}g^{-1}\}) = \{hH_{\Upsilon_i}h^{-1}\}$. And any vertex of rank at least 1 is determined by finitely many adjacent rank 0 vertices. $\square$

**Remark 2.17.** It is in fact possible to show that the map $\Psi$ constructed in the above proof is a homeomorphism, see [HH23, Section 2.3] where this is explained in the case of right-angled Artin groups. We will however never need this fact in the sequel.

## 2.4 Amenable actions on boundaries of trees

### 2.4.1 Review on Borel amenability of group actions

We now review some basic facts on Borel amenability of group actions. For general background on the topic we refer to [ADR00, Oza06a].
Let $G$ be a countable group, acting by Borel automorphisms on a standard Borel set $\Delta$. The $G$-action on $\Delta$ is *Borel amenable* if there exists a sequence of measurable maps $\mu_n : \Delta \to \text{Prob}(G)$ such that for every $\delta \in \Delta$ and every $g \in G$, one has

$$ \lim_{n \to +\infty} \|\mu_n(g\delta) - g\mu_n(\delta)\|_1 = 0. $$

Here $\text{Prob}(G)$ is the space of probability measures on $G$, equipped with the topology of pointwise convergence. We say that the sequence of maps $\mu_n$ is *asymptotically equivariant*.

For a first example of an amenable action of a non-amenable group, we also refer to [Oza06a, Example 2.2] for the case of an action of $F_2$ on the boundary of a tree. We will use a variation over this argument in our proof of Lemma 2.20 below.

We mention a few elementary facts about Borel amenability of group actions that we will use in the sequel.

* A countable group $G$ is amenable if and only if its action on a point is Borel amenable.

* If $G_1, \dots, G_k$ are countable groups with Borel amenable actions on standard Borel sets $\Delta_1, \dots, \Delta_k$, then the product action of $G_1 \times \dots \times G_k$ on $\Delta_1 \times \dots \times \Delta_k$ is Borel amenable.

* If the $G$-action on $X$ is Borel amenable, then — letting $\mathcal{P}_{\le 2}(X)$ be the standard Borel set consisting of all non-empty subsets of $X$ of cardinality at most 2 — the $G$-action on $\mathcal{P}_{\le 2}(X)$ is Borel amenable.

### 2.4.2 Action on the boundary of the tree $T_v$

Recall that given some $v \in V\Gamma$, we denote by $T_v$ the Bass-Serre tree associated to the splitting $G = G_{\text{star}(v)} *_{G_{\text{lk}(v)}} G_{\Gamma \setminus \{v\}}$ (see Section 2.2.2).

Given a vertex $v \in V\Gamma$, the $G$-action on the tree $T_v$ extends to a $G$-action by homeomorphisms on $\partial_{\infty}T_v$ (equipped with its visual topology). Let $\mathbb{P}_G$ be the (countable) set of all parabolic subgroups of $G$. We will now build a $G$-equivariant Borel map $\theta_v : \partial_{\infty}T_v \to \mathbb{P}_G$.

Given $\xi \in \partial_{\infty}T_v$, we define the *elliptic stabilizer* of $\xi$ as the subgroup of $\text{Stab}_G(\xi)$ consisting of all elements that act elliptically on $T_v$ (i.e. with a fixed point).

**Lemma 2.18.** Let $v \in V\Gamma$ be such that $\Gamma \neq \text{star}(v)$. For every $\xi \in \partial_{\infty}T_v$, the elliptic stabilizer of $\xi$ is a proper parabolic subgroup of $G$.

*Proof.* Let $x \in T_v$ be a vertex, and let $\xi \in \partial_{\infty}T_v$. For every $n \in \mathbb{N}$, let $\gamma_n(x, \xi)$ be the segment in $T_v$ consisting of the union of the first $n$ edges of the ray from $x$ to $\xi$. Let $G_n(x, \xi)$ be the pointwise stabilizer of $\gamma_n(x, \xi)$. Notice that edge stabilizers of $T_v$ are proper parabolic subgroups of $G$. Therefore, for every $n \in \mathbb{N}$, the subgroup $G_n(x, \xi)$ is a proper parabolic subgroup of $G$, being an intersection of proper parabolic subgroups.

As $n$ increases, the subgroups $G_n(x, \xi)$ form a non-increasing chain of proper parabolic subgroups, which is therefore stationary. We denote by $G_{\infty}(x, \xi)$ the ultimate value of this sequence.

Now, if $x_1, x_2, \dots$ form a ray towards $\xi$, then as $n$ increases, the subgroups $G_{\infty}(x_n, \xi)$ form a non-decreasing chain of proper parabolic subgroups of $G$. Therefore it is stationary.
ary, and we denote by $\theta_v(\xi)$ its ultimate value. Notice that the value of $\theta_v(\xi)$ does not depend on the choice of a ray towards $\xi$, because any two such rays eventually coincide.

Finally, note that being in the elliptic stabilizer is equivalent to fixing pointwise a ray towards $\xi$. Therefore $\theta_v(\xi)$ is indeed the elliptic stabilizer of $\xi$. $\square$

We now let $\theta_v : \partial_\infty T_v \to \mathbb{P}_G$ be the map sending $\xi$ to its elliptic stabilizer. Notice that this map is Borel and $G$-equivariant. We let $(\partial_\infty T_v)^{\text{reg}} := \theta_v^{-1}(\{1\})$.

**Lemma 2.19.** Let $v \in V\Gamma$. Then $(\partial_\infty T_v)^{\text{reg}} \neq \emptyset$ if and only if $\Gamma$ is irreducible.

*Proof.* First assume that $\Gamma$ is reducible and decompose it as $\Gamma = \Gamma_1 \circ \Gamma_2$. Assuming without loss of generality that $v \in V\Gamma_1$, the parabolic subgroup $G_{\Gamma_2}$ fixes all the vertices of $T_v$ and therefore is contained in the elliptic stabilizer of any $\xi \in \partial_\infty T_v$.

We now assume that $\Gamma$ is irreducible.

* Let us first show that there exists $g \in G$ that is not contained in any proper parabolic subgroup of $G$.

Let $\{w_1, \dots, w_m\} = V\Gamma$ be an enumeration of $V\Gamma$. For all $i \in \{1, \dots, m\}$ take some $g_i \in G_{w_i} \setminus \{1\}$ and consider $g = g_{w_1} \cdots g_{w_m}$. Using a normal form argument, we now obtain that $g$ is not contained in any proper parabolic subgroup of $G$.

* Let $g$ be as above, then $g$ acts loxodromically on $T_v$ and we let $\xi$ be the attracting fixed point of $g$ in $\partial_\infty T_v$. Let $P = \theta_v(\xi)$. Then $P$ fixes a subray $r$ of the axis of $g$. As in the proof of Lemma 2.18, we can find a compact segment $I \subseteq r$ whose stabilizer is equal to $P$. Then the stabilizer of $gI$ is equal to $gPg^{-1}$, so $P \subseteq gPg^{-1}$. It follows from Proposition 2.3 that $gPg^{-1} = P$, i.e. $g \in P \times P^{-1}$. Since $\Gamma$ is irreducible and $g$ is not contained in any proper parabolic subgroup of $G$, we deduce that $P = \{1\}$. In particular $\xi \in (\partial_\infty T_v)^{\text{reg}}$ and the lemma is proved. $\square$

**Lemma 2.20.** For every $v \in V\Gamma$, the $G$-action on $(\partial_\infty T_v)^{\text{reg}}$ is Borel amenable.

*Proof.* Let $\mathbb{S}$ be the countable set of all oriented segments of $T_v$ which start and end at vertices and have trivial $G$-stabilizer. The $G$-action on $T_v$ induces a $G$-action on $\mathbb{S}$, and stabilizers for the latter action are trivial. Therefore, it is enough to construct an asymptotically equivariant sequence of Borel maps $\mu_n : (\partial_\infty T_v)^{\text{reg}} \to \text{Prob}(\mathbb{S})$ (see e.g. [Oza06b, Proposition 11] or [BGH22, Proposition 2.12]).

Given any vertex $x \in VT_v$ and any $\xi \in (\partial_\infty T_v)^{\text{reg}}$, we denote by $\alpha(x, \xi) \in \mathbb{S}$ the smallest (oriented) segment of the ray $[x, \xi)$ originating at $x$, ending at a vertex, and with trivial stabilizer. This exists, and it is a finite subsegment, by definition of $(\partial_\infty T_v)^{\text{reg}}$. Notice that, for a fixed $x$, the map $\alpha(x, \cdot)$ is continuous.

We now fix a basepoint $o \in VT_v$. Given $\xi \in (\partial_\infty T_v)^{\text{reg}}$ let $\{o = v_1(\xi), v_2(\xi), \dots\}$ be the set of vertices encountered along the ray from $o$ to $\xi$. Given $n \in \mathbb{N}$ and $\xi \in (\partial_\infty T_v)^{\text{reg}}$, we let

$$ \mu_n(\xi) = \frac{1}{n} \sum_{i=1}^{n} \delta_{\alpha(v_i(\xi), \xi)}, $$

a weighted sum of Dirac masses, so $\mu_n(\xi) \in \text{Prob}(\mathbb{S})$, and $\mu_n$ depends measurably on $\xi$. Now remark that $\mu_n(g\xi) = 1/n \sum_{i=1}^n \delta_{\alpha(v_i(g\xi), g\xi)}$ where the $\alpha(v_i(g\xi), g\xi)$ are subintervals of the ray $[o, g\xi)$. Moreover $g\mu_n(\xi) = 1/n \sum_{i=1}^n \delta_{g\alpha(v_i(\xi), \xi)}$ where the $g\alpha(v_i(\xi), \xi)$ are subintervals of the ray $[go, g\xi)$. Since any two rays towards $g\xi$ eventually coincide, this implies that $\mu_n$ is asymptotically equivariant (see also Figure 4). $\square$
<figure><img src="image_7.png" /><figcaption>Figure 4: Illustration of the proof of Lemma 2.20</figcaption></figure>

# 3 Measure and orbit equivalence

This section contains general facts about measure equivalence and orbit equivalence. After reviewing the basic definitions in Section 3.1, we compare measure and orbit equivalence with their discrete counterparts, namely commensurability and strong commensurability (Section 3.2) – this will be useful in Part II to derive commensurability classification theorems for graph products, from their versions in measure equivalence. Finally, in Section 3.3, we establish a lemma regarding how to restrict measure equivalence couplings to subgroups, which will be useful in Part III of our work.

## 3.1 Background and definitions

We recall here some general material about measure equivalence and refer to [Fur11, Gab10] for general surveys. We also refer to Furman’s original article [Fur99b] for more information on the dictionary between measure equivalence and stable orbit equivalence, and the associated cocycles.

### 3.1.1 Measure equivalence and (stable) orbit equivalence

A *standard measure space* $(\Omega, m)$ is a measurable space whose $\sigma$-algebra $\mathcal{B}(\Omega)$ consists of the Borel subsets coming from some Polish (*viz.* separable and completely metrizable) topology on $\Omega$ and equipped with a non-zero $\sigma$-finite measure $m$ on $\mathcal{B}(\Omega)$. One says that an action of a countable group $G$ on a standard measure space $(\Omega, m)$ is *measure-preserving* if the action map $(g, x) \mapsto g * x$ is Borel and if $\mu(g * E) = \mu(E)$ for all $g \in G$ and all Borel subsets $E \subseteq \Omega$. In this setting, a *fundamental domain* for the action of $G$ on $(\Omega, m)$ is a Borel subset $X_G \subseteq \Omega$ which intersects almost every $G$-orbit at exactly one point.

Recall from the introduction that two countable groups $G$ and $H$ are *measure equivalent* if there exists a standard measure space $(\Omega, m)$, equipped with a measure-preserving action of $G \times H$, such that the actions of $G$ and $H$ on $\Omega$ are both free and have finite measure fundamental domains, denoted respectively by $X_G$ and $X_H$. The space $(\Omega, m)$
(sometimes simply denoted $\Omega$) is called a *measure equivalence coupling* between $G$ and $H$. If we need to keep track of the fundamental domains, we will sometimes call “coupling” from $G$ to $H$ the tuple $(\Omega, m, X_G, X_H)$. When additionally one can take $X_G = X_H$ in the above definition, the groups $G$ and $H$ are said to be *orbit equivalent*.

In general, the *compression constant* (from $G$ to $H$) of the measure equivalence coupling $\Omega$ is defined as

$$[G:H]_{\Omega} := \frac{m(X_H)}{m(X_G)}$$

(this does not depend on the choice of fundamental domains). The compression constant of an orbit equivalence coupling is always equal to 1.

By a theorem of Furman [Fur99b] and Gaboriau [Gab05, PME5], two countable groups are orbit equivalent if and only if they admit free, measure-preserving actions on a standard probability space $(X, \mu)$ with the same orbits (on a conull subset). We will call such an $(X, \mu)$ an *orbit equivalence pairing*<fnref n="2" />.

Likewise, measure equivalence is characterized by having *stably orbit equivalent* free, measure-preserving actions on standard probability spaces, in the following sense. Given two free, measure-preserving actions $G \rightsquigarrow (X, \mu)$ and $H \rightsquigarrow (Y, \nu)$ on standard probability spaces, a *stable orbit equivalence* between the two actions is a measure-scaling isomorphism $f: U \to V$ between positive measure subsets $U \subseteq X$ and $V \subseteq Y$ (i.e. $f$ induces a measure space isomorphism after renormalizing $U$ and $V$ to probability spaces) such that for almost every $x \in U$, one has $f((G \cdot x) \cap U) = (H \cdot f(x)) \cap V$. If a map $f$ with these properties exists, the two actions are said to be stably orbit equivalent. The *compression constant* of $f$ is defined as

$$\kappa(f) := \frac{\nu(V)}{\mu(U)}.$$

In fact, two countable groups admit a measure equivalence coupling with compression constant $\kappa$ (from $G$ to $H$) if and only if there is a stable orbit equivalence from a free, measure-preserving $G$-action on a standard probability space $X$, to a measure-preserving $H$-action on a standard probability space $Y$, with compression constant $\kappa$, see [Fur99b, Lemma 3.2 and Theorem 3.3].

### 3.1.2 Cocycles

We now review the notions of measure equivalence and (stable) orbit equivalence cocycles. In the sequel, we will also work with a notion of cocycle in the groupoid theoretic framework – we refer to Section 4.1.3, page 39 for more details on the groupoid viewpoint.

**Measure equivalence cocycles.** A measure equivalence coupling $(\Omega, m, X_G, X_H)$ comes with two induced actions: one of $G$ on the fundamental domain $X_H$ and one of $H$ on $X_G$. We will denote both induced actions by “·” to distinguish them from the action of the groups on the whole space $\Omega$, denoted by “*”. For all $g \in G$ and a.e. $x \in X_H$, the element $g \cdot x$ is defined as the (unique) element of $X_H$ that belongs to the $H$-orbit of $g \ast x$, namely $\{g \cdot x\} = X_H \cap (H \ast (g \ast x))$ (see Figure 1, page 12). In the following we

<footnote n="2">²we use the word *pairing* instead of *coupling* to distinguish it from measure equivalence couplings where the two fundamental domains coincide</footnote>
will (slightly) abuse notations and write $H * g * x$ instead of $H * (g * x)$. Similarly for all $h \in H$ and a.e. $x' \in X_G$ we have $\{h \cdot x'\} = X_G \cap (G * h * x')$.

We thus define $c : G \times X_H \to H$ and $c' : G \times X_H \to H$ the measure equivalence cocycles, by (see also Figure 1)

$$ \forall g \in G, \text{ a.e. } x \in X_H \qquad c(g, x) * g * x = g * x, $$

$$ \forall h \in H, \text{ a.e. } x \in X_G \qquad c'(h, x) * h * x = h * x. $$

They verify the cocycle relation, namely

$$ \forall g, g' \in G, \text{ a.e. } x \in X \qquad c(gg', x) = c(g, g' \cdot x)c(g', x). $$

Two cocycles $c : G \times X_H \to H$ and $c' : G \times X_H \to H$ are said to be cohomologous if there exists a measurable map $\alpha : X_H \to H$ such that for all $g \in G$ and a.e. $x \in X_H$ one has

$$ c'(g, x) = \alpha(g \cdot x)c(g, x)\alpha(x)^{-1}. $$

**Remark 3.1.** Identifying $X_H$ with $\Omega/H$ allows to see the measure equivalence cocycle $c$ as defined over $G \times \Omega/H$. With this identification, changing the fundamental domain $X_H$ amounts to considering cohomologous cocycles. In the following, given two fundamental domains $X_H$ and $X'_H$ and their associated respective cocycles $c : G \times X_H \to H$ and $c' : G \times X'_H \to H$, we will consider $c$ and $c'$ to be cohomologous if they are cohomologous via the natural identifications $X_H \simeq \Omega/H \simeq X'_H$.

**(Stable) orbit equivalence cocycles.** Assume now that $G$ and $H$ are orbit equivalent over some standard probability space $(X, \mu)$. One can define two orbit equivalence cocycles denoted $c : G \times X \to H$ and $c' : H \times X \to G$ by letting $c(g, x)$ (resp. $c'(h, x)$) be the unique element of $H$ (resp. $G$) such that $c(g, x) \cdot x = g \cdot x$ and $c'(h, x) \cdot x = h \cdot x$ (this is well-defined for a.e. $x \in X$). When $\Omega$ is an orbit equivalence coupling between $G$ and $H$, with a common fundamental domain $X = X_G = X_H$, the associated measure equivalence cocycles $c : G \times X \to H$ and $c' : H \times X \to G$ are nothing but the orbit equivalence cocycles associated to the actions $G \curvearrowleft X$ and $H \curvearrowleft X$.

More generally, if $G \curvearrowleft (X, \mu)$ and $H \curvearrowleft (Y, \nu)$ are two free, ergodic, measure-preserving actions on standard probability spaces, and if $f : U \to V$ (with $U \subseteq X$ and $V \subseteq Y$ Borel subsets of positive measure) is a stable orbit equivalence, we say that a cocycle $c : G \times X \to H$ is a stable orbit equivalence (SOE) cocycle associated to $f$ if there exists a Borel map $p : X \to U$ such that

* for almost every $x \in X$, one has $p(x) \in G \cdot x$, and

* for every $g \in G$ and almost every $x \in X$, one has $f \circ p(g \cdot x) = c(g, x) \cdot (f \circ p(x))$. Notice that $p$ can always be chosen so that $p(x) = x$ whenever $x \in U$. Changing $p$ to another projection map yields a cohomologous cocycle.

We also observe that if $f : U \to V$ is a stable orbit equivalence between $G \curvearrowleft (X, \mu)$ and $H \curvearrowleft (Y, \nu)$, then for every positive measure Borel subset $W \subseteq U$, the map $f|_W : W \to f(W)$ is again a stable orbit equivalence between $G \curvearrowleft (X, \mu)$ and $H \curvearrowleft (Y, \nu)$, with the same compression constant. In addition, any SOE cocycle for $f|_W$ is also an SOE cocycle for $f$, and any SOE cocycle for $f$ is cohomologous to an SOE cocycle for $f|_W$.
We also observe that if $P_G \subseteq G$ and $P_H \subseteq H$ are subgroups, such that

• for almost every $x \in U$, one has $f((P_G \cdot x) \cap U) = (P_H \cdot f(x)) \cap V$,

• $P_G$ acts ergodically on $X$,

then the map $p$ can be chosen so that $p(x) \in P_G \cdot x$ for almost every $x \in X$ (by ergodicity). Moreover $f$ is a stable orbit equivalence between the actions $P_G \rightsquigarrow X$ and $P_H \rightsquigarrow Y$. Any SOE cocycle $c: G \times X \to H$ for $f$ is $H$-cohomologous to a cocycle $c'$ such that $c'(P_G \times X) \subseteq P_H$, and $c'$ is an SOE cocycle for $f$, viewed as a stable orbit equivalence between $P_G \rightsquigarrow X$ and $P_H \rightsquigarrow Y$.

### 3.1.3 Quantitative versions

Recall from the introduction (Definition 1.9) that given a measure equivalence coupling $(\Omega, m, X_G, X_H)$ (equipped with fundamental domains $X_G, X_H$) between finitely generated groups $G, H$, and two non-decreasing functions $\varphi$ and $\psi$, one says that the coupling is $(\varphi, \psi)$-integrable (from $G$ to $H$) if for all $g \in G$ and all $h \in H$ we have

$$ \int_{X_H} \varphi(|c(g,x)|_H)dm(x) < +\infty \quad \text{and} \quad \int_{X_G} \psi(|c'(h,x)|_G)dm(x) < +\infty, \quad (3.1) $$

where $c: G \times X_H \to H$ and $c': H \times X_G \to G$ denote the measure equivalence cocycles, and $| \cdot |_{G}, | \cdot |_{H}$ denote word lengths on $G, H$ associated to finite generating sets (the integrability does not depend on the choice of generating sets).

When $\varphi(x) = x^p$ for some $p \in [0, +\infty)$ we will talk about $(L^p, \psi)$-integrable couplings instead of $(\varphi, \psi)$-integrable couplings, to stay consistent with the literature. In particular $L^0$ means that we are not imposing any integrability condition on the corresponding cocycle.

Likewise, an orbit equivalence pairing $(X, \mu)$ between two finitely generated groups $G$ and $H$ (i.e. a standard probability space on which the two groups have the same orbits) is $(\varphi, \psi)$-integrable from $G$ to $H$ if the associated orbit equivalence cocycles $c: G \times X \to H$ and $c': H \times X \to G$ satisfy the integrability conditions (3.1) (with $X = X_G = X_H$).

Using the dictionary between measure equivalence couplings and orbit equivalence pairings, the existence of a $(\varphi, \psi)$-integrable orbit equivalence pairing from $G$ to $H$ is equivalent to the existence of a $(\varphi, \psi)$-integrable measure equivalence coupling of the form $(\Omega, m, X_G, X_H)$ with $X_G = X_H$.

## 3.2 Link with commensurability

Two groups are commensurable if they have isomorphic finite index subgroups. Following [JS01], we say that they are strongly commensurable if they admit isomorphic finite-index subgroups of the same index. The following lemma shows that commensurability and strong commensurability are the respective analogues of measure equivalence and orbit equivalence, when one restricts to discrete (i.e. countable) couplings.

**Lemma 3.2.** Two countable groups $G_1$ and $G_2$ are

1. commensurable if and only if there exists a non-empty countable set $\Omega$ equipped with an action of $G_1 \times G_2$ such that for every $i \in \{1, 2\}$, the action of $G_i$ on $\Omega$ is free and cofinite.
2. strongly commensurable if and only if there exists a non-empty countable set $\Omega$ equipped with an action of $G_1 \times G_2$ such that for every $i \in \{1, 2\}$, the action of $G_i$ on $\Omega$ is free, and the actions of $G_1$ and $G_2$ on $\Omega$ have a common finite fundamental domain.

For comparison, notice also that two countable groups $G_1$ and $G_2$ are isomorphic if and only if there exists a non-empty countable set $\Omega$ equipped with an action of $G_1 \times G_2$, such that for every $i \in \{1, 2\}$, the action of $G_i$ on $\Omega$ is free and transitive. Indeed, the direct implication is clear by considering the left-right action of the group on itself. Conversely, identifying $\Omega$ to $G_1$ and $G_2$ via the orbit maps, gives the needed isomorphism.

*Proof.* Assume first that $G_1$ and $G_2$ are commensurable. Let $H_1 \subseteq G_1$ and $H_2 \subseteq G_2$ be isomorphic finite-index subgroups, and let $\theta : H_1 \to H_2$ be an isomorphism. The group $H_1 \times H_2$ acts on $\tilde{\Omega} = G_1 \times H_1 \times G_2$ via

$$ (h_1, h_2) \cdot (g_1, h, g_2) = (g_1 h_1^{-1}, h_1 h(\theta^{-1}(h_2))^{-1}, h_2 g_2). $$

And the action of $G_1 \times G_2$ on $\tilde{\Omega}$ given by $(g'_1, g'_2) \cdot (g_1, h, g_2) = (g'_1 g_1, h, g_2 (g'_2)^{-1})$ commutes with that of $H_1 \times H_2$, hence descends to an action on the quotient $\Omega = (H_1 \times H_2) \backslash \tilde{\Omega}$. One checks that the actions of $G_1$ and $G_2$ on $\Omega$ are both free. We will denote by $[g_1, h, g_2]$ the class of $(g_1, h, g_2)$ in $\Omega$. Let $X_1 \subseteq G_1$ be a set of representatives of the left cosets of $H_1$ in $G_1$, and $X_2 \subseteq G_2$ be a set of representatives of the right cosets of $H_2$ in $G_2$. Then $\{[e, e, g_2] \mid g_2 \in X_2\}$ is a finite fundamental domain for the $G_1$-action on $\Omega$, and $\{[g_1, e, e] \mid g_1 \in X_1\}$ is a finite fundamental domain for the $G_2$-action on $\Omega$.

In the above, if we further assume that $G_1$ and $G_2$ are strongly commensurable, then $|X_1| = |X_2|$. Let $\alpha : X_1 \to X_2$ be a bijection. Then $\{[g_1, e, \alpha(g_1)] \mid g_1 \in X_1\}$ is a common fundamental domain for the actions of $G_1$ and $G_2$ on $\Omega$.

Conversely, let us now assume that there exists a non-empty countable set $\Omega$ equipped with an action of $G_1 \times G_2$ such that for every $i \in \{1, 2\}$, the action of $G_i$ on $\Omega$ is free and admits a finite fundamental domain $Y_i$. Let $y_2 \in Y_2$, and let $H_1 \subseteq G_1$ be the stabilizer of $y_2$ for the induced action of $G_1$ on $Y_2 \approx G_2 \backslash \Omega$. Then $H_1$ is a finite-index subgroup of $G_1$ which acts freely with finitely many orbits on $G_2 y_2$. Using the fact that the actions of $G_1$ and $G_2$ commute, one checks that $\{g_2 \in G_2 \mid g_2 y_2 \in H_1 y_2\}$ is a subgroup $H_2$ of $G_2$, and that its (finite) index is equal to the number of $H_1$-orbits on $G_2 y_2$. Now $H_1$ and $H_2$ have free and transitive actions on $H_1 y_2$, from which it follows that they are isomorphic.

To get the characterization of strong commensurability, in the above, assume in addition that $Y_1 = Y_2$. Up to changing $\Omega$ to a $(G_1 \times G_2)$-invariant subset (which does not affect the fact that the two actions have a common fundamental domain), we can (and will) assume that the induced actions of $G_1$ on $G_2 \backslash \Omega$ and of $G_2$ on $G_1 \backslash \Omega$ are transitive. Then in the above, we have $[G_1 : H_1] = |Y_2|$. And $G_1$ acts on $\Omega$ with $|Y_1|$ orbits, and each of these orbits meets $G_2 y_2$. So $H_1 = \text{Stab}_{G_1}(y_2)$ acts on $G_2 y_2$ with $|Y_1|$ orbits, which by the above implies that $[G_2 : H_2] = |Y_1|$. So $[G_1 : H_1] = [G_2 : H_2]$, which proves that $G_1$ and $G_2$ are strongly commensurable. $\square$
### 3.3 Restricting couplings to subgroups

**Lemma 3.3.** Let $G$ and $H$ be two countable groups and $M \subseteq G$ and $N \subseteq H$ be subgroups. Let $(\Omega, m, X_G, X_H)$ be a measure equivalence coupling from $G$ to $H$ and denote by $c_1: G \times X_H \to H$ and $c_2: H \times X_G \to G$ the corresponding measure equivalence cocycles. Assume that there exists a positive measure Borel subset $U \subseteq X_G \cap X_H$ such that

* for every $g \in M$ and a.e. $x \in U$, if $gx \in U$, then $c_1(g, x) \in N$;

* for every $h \in N$ and a.e. $x \in U$, if $hx \in U$, then $c_2(h, x) \in M$.

Then there exist an $(M \times N)$-invariant Borel subspace $\Omega' \subseteq \Omega$, and Borel fundamental domains $X_M, X_N$ for the actions of $M, N$ on $\Omega'$, such that

* $(\Omega', m, X_M, X_N)$ is a measure equivalence coupling from $M$ to $N$, and

* $X_M, X_N$ are contained in fundamental domains for the actions of $G, H$ on $\Omega$.

We refer to Remark 4.4 for the groupoid theoretic translation of the above lemma.

*Proof.* We denote by $c_1: G \times X_H \to H$ and $c_2: H \times X_G \to G$ the associated measure equivalence cocycles. Our assumption on $U$ ensures that for a.e. $x, y \in U$, if $g \in M$ sends $x$ to $y$, then $c_1(g, x)$ belongs to $N$. Remark that $M \cdot U \subseteq X_H$ and $N \cdot U \subseteq X_G$. Let $\kappa: M \cdot U \to M$ be a measurable map such that $\kappa(x) \cdot x \in U$ for all $x \in M \cdot U$, and extend it to $X_H$ by letting $\kappa(x) = e_M$ for all $x \in X_H \setminus M \cdot U$. Similarly, define $\lambda: N \cdot U \to N$ a measurable map such that $\lambda(x) \cdot x \in U$ for every $x \in N \cdot U$, and extend it to $X_G$ by letting $\lambda(x) = e_N$ for all $x \in X_G \setminus N \cdot U$. Finally remark that these maps can (and will) be chosen such that for all $x \in U$ we have $\kappa(x) = e_M$ and $\lambda(x) = e_N$.

**Shifted fundamental domains for G and H** First let

$$X'_G := \{c_2(\lambda(x), x) * x \mid x \in X_G\},$$

$$X'_H := \{c_1(\kappa(x), x) * x \mid x \in X_H\}.$$

Remark that $X'_G$ and $X'_H$ are Borel subsets of $\Omega$ and both contain $U$. Now since $X'_G$ and $X'_H$ are fundamental domains for the respective actions of $G$ and $H$ on $\Omega$, we also have an induced action of $G$ (resp. $H$) on $X'_H$ (resp. $X'_G$). We will denote these actions by “•” (see also eq. (3.2) below).

**Cocycles** Now define for all $g \in M$ and a.e. $x \in M \cdot U$

$$c'_1(g, x) := c_1(\kappa(g \cdot x), g \cdot x)c_1(g, x)c_1(\kappa(x), x)^{-1}, \\ = c_1(\kappa(g \cdot x)g\kappa(x)^{-1}, \kappa(x) \cdot x).$$

In other words $c'_1(g, x)$ verifies $c'_1(g, x) \cdot (\kappa(x) \cdot x) = \kappa(g \cdot x) \cdot (g \cdot x)$ (see Figure 5). But by definition of $\kappa$, the product $\kappa(g \cdot x)g\kappa(x)^{-1}$ belongs to $M$ and both $\kappa(x) \cdot x$ and $\kappa(q \cdot x) \cdot (g \cdot x)$ belong to $U$. Therefore, $c'_1(g, x) \in N$. Similarly, define for all $h \in N$ and all $x \in N \cdot U$

$$c'_2(h, x) := c_2(\lambda(h \cdot x), h \cdot x)c_2(h, x)c_2(\lambda(x), x)^{-1}.$$

As above $c'_2$ verifies $c'_2(h, x) \cdot (\lambda(x) \cdot x) = \lambda(h \cdot x) \cdot (h \cdot x)$ and $c'_2(h, x) \in M$.
<figure><img src="image_8.png" /><figcaption>The action of $c'_1(g, x)$ is represented by the dotted arrow, it corresponds to the element of $H$ sending $(\kappa(x) \cdot x)$ to $\kappa(g \cdot x) \cdot (g \cdot x)$. Actions of elements of $G$ (in orange) are represented by the other arrows.</figcaption></figure>

<caption>Figure 5: Definition of $c'_1$</caption>

Remark that, if we denote by $x' = c_1(\kappa(x), x) * x$ an element of $X'_H$, for all $g \in M$ and a.e. $x \in M \cdot U$ we have (see also Figure 6)

$$g \cdot x' = c'_1(g, x) * g * x'. \qquad (3.2)$$

And similarly for $c'_2$: for all $h \in N$ and a.e. $y \in N \cdot U$, if we let $y' := c_2(\lambda(y), y) * y$, we have $h \cdot y' = c'_2(h, y) * h * y'$.

<figure><img src="image_9.png" /><figcaption>Figure 6: Definition of $g \cdot x'$ (see eq. (3.2)). Here $X'_H$ is represented wavy to emphasize that it is not a translate of $X_H$.</figcaption></figure>

**Definition of the space** Let us now define the following two Borel spaces

$$\Omega_{M,N} := N * (M \cdot U), \quad \Omega_{N,M} := M * (N \cdot U).$$
Remark that $\Omega_{M,N}$ (resp. $\Omega_{N,M}$) is $N$-invariant (resp. $M$-invariant). In addition $N \cdot U$ and $M \cdot U$ are contained in the fundamental domains $X'_G$ and $X'_H$ respectively. Thus, $M \cdot U$ (resp. $N \cdot U$) is a Borel fundamental domain of finite measure for the $N$-action on $\Omega_{M,N}$ (resp. for the $M$-action on $\Omega_{N,M}$). Therefore, to obtain a measure equivalence coupling between $M$ and $N$, it only remains to prove that $\Omega_{M,N} = \Omega_{N,M}$: indeed, the lemma then follows by letting $\Omega' = \Omega_{M,N}$, and letting $X_M = N \cdot U$ and $X_N = M \cdot U$, which are respectively contained in $X'_G$ and $X'_H$. By symmetry, it is enough to prove the inclusion $\Omega_{M,N} \subseteq \Omega_{N,M}$.

So let $h \in N$ and $u \in U$ and $g \in M$ and consider $h \ast (g \cdot u) \in \Omega_{M,N}$. Since $g \in M$ we have $c'_1(g, u) \in N$ and therefore $h' := hc'_1(g, u)$ also belongs to $N$. Remark that $h'$ is the element of $N$ sending $g \ast u$ to $h \ast (g \cdot u)$: indeed using eq. (3.2) (see also Figure 7) we obtain

$$h' \ast g \ast u = hc'_1(g, u) \ast g \ast u = h \ast (g \cdot u). \qquad (3.3)$$

Note moreover that since $h'$ belongs to $N$ then $c'_2(h', u)$ is in $M$. Then, using eq. (3.3) and that the actions of $H$ and $G$ commute we get (see also Figure 7)

$$ \begin{aligned} h \ast (g \cdot u) &= h' \ast g \ast u, \\ &= g \ast h' \ast u, \\ &= g \ast c'_2(h', u)^{-1} \ast (h' \cdot u) \in M \ast (N \cdot U). \end{aligned} $$

<figure><img src="image_10.png" /><figcaption>Figure 7: Showing that $h \ast (g \cdot u)$ belongs to $\Omega_{N,M} = M \ast (N \cdot U)$</figcaption></figure>

Therefore $\Omega_{M,N} = \Omega_{N,M}$. Hence denoting again $m$ the measure induced by $m$ on $\Omega_{M,N}$ we obtain that $(\Omega_{M,N}, m, N \cdot U, M \cdot U)$ is a measure equivalence coupling from $M$ to $N$. $\square$

# 4 Measured groupoids

Since the work of Kida on the measure equivalence rigidity of mapping class groups [Kid10], proofs of measure equivalence classification and rigidity theorems have often taken advantage of the language of measured groupoids. In this section, we review
the relevant notions for the present work and emphasize the relationship to measure equivalence. In particular, we finish this section by providing a general framework, phrased in the language of measured groupoids, that encompasses Kida's strategy, and will serve as a blueprint in Part II of the present work (Section 4.4). General references about measured groupoids include [AD13, Section 2.1], [Kid09] or [GH21b, Section 3].

## 4.1 Measured groupoids

In this section we recall some necessary material about measured groupoids and give numerous examples coming from the framework of groups acting on measure spaces.

A Borel groupoid is a standard Borel space $\mathcal{G}$ endowed with two Borel maps $s, r : \mathcal{G} \to X$ towards a standard Borel space $X$ and equipped with a measurable (partially defined) composition law, a measurable inverse map, and a unit element for all $x \in X$ denoted by $e_x$. We call $X$ the base space of $\mathcal{G}$ and if $g \in \mathcal{G}$ we call $s(g)$ the source and $r(g)$ the range of $g$. Two elements $g, h \in \mathcal{G}$ are composable (i.e. one can form the product $gh$) if and only if $r(h) = s(g)$. Let $B_1, B_2 \subseteq \mathcal{G}$, we will denote by $B_1B_2$ the set

$$B_1B_2 = \{\mathbf{b}_1\mathbf{b}_2 : \mathbf{b}_1 \in B_1, \mathbf{b}_2 \in B_2, r(\mathbf{b}_2) = s(\mathbf{b}_1)\}.$$

All Borel groupoids considered in this article are discrete, that is to say there are at most countably many elements with a given source or range.

**Example 4.1.** Let $G$ be a countable group acting on a standard probability space $(X, \mu)$. Then $G \times X$ has a natural structure of Borel groupoid over $X$. The associated source and range maps are then defined by $s(g, x) = x$ and $r(g, x) = g \cdot x$, the unit elements are given by $e_x := (e_G, x)$ and the inverse map by $(g, x)^{-1} := (g^{-1}, g \cdot x)$. We denote this groupoid by $G \times X$.

Let $U \subseteq X$ be a Borel subset. The restriction of $\mathcal{G}$ to $U$ is defined as

$$\mathcal{G}|_U := \{\mathbf{g} \in \mathcal{G} : s(\mathbf{g}), r(\mathbf{g}) \in U\},$$

which is naturally a Borel groupoid over the base space $U$.

A Borel subset $B \subseteq \mathcal{G}$ is called a bisection if $s|_B$ and $r|_B$ are Borel isomorphisms to Borel subsets $s(B), r(B) \subseteq X$ respectively.

In the sequel of the paper we will work with measured groupoids, defined as follows. A measured groupoid over a standard measure space $(X, \mu)$ is a discrete Borel groupoid $\mathcal{G}$ over $X$ for which the measure $\mu$ is quasi-invariant in the following sense: for every bisection $B \subseteq \mathcal{G}$, one has $\mu(r(B)) = 0$ if and only if $\mu(s(B)) = 0$.

The example of interest for us is when $G$ acts measure preservingly on a probability space $X$: the groupoid $G \times X$ defined in Example 4.1 is then a measured groupoid.

We mention that there is a natural notion of measured subgroupoid of $\mathcal{G}$, as well as of groupoid isomorphism and refer to [GH21b, Section 3] for more details.

Let us now illustrate how to construct measured groupoids from measure or orbit equivalence couplings.

**Example 4.2.** • If $G$ and $H$ are orbit equivalent over some probability space $(X, \mu)$, then the corresponding measured groupoids $G \times X$ and $H \times X$ are isomorphic. In
the following, when $G$ and $H$ are orbit equivalent, we will therefore identify $G \times X$
and $H \times X$ to the same measured groupoid.

* If two countable groups $G$ and $H$ are measure equivalent over some measure space $(\Omega, m)$, one can choose the fundamental domains $X_G$ and $X_H$ such that $m(X_G \cap X_H) > 0$. For such a choice, let $X := X_G \cap X_H$ and consider $(G \times X_H)_{|X}$ and $(H \times X_G)_{|X}$. Remark that the orbit equivalence relations on $X$ induced by the $G$-action on $X_H$ and $H$-action on $X_G$ coincide. In fact the two measured groupoids are isomorphic, via the identification $(g, x) \mapsto (c(g, x), x)$, where $c: G \times X_H \to H$ is the associated measure equivalence cocycle.

4.1.1 Finite index and infinite type groupoids

The index of a measured subgroupoid was defined by Kida [Kid14, Section 3.2], gener-
alizing a notion of Feldman-Sutherland-Zimmer for equivalence relations [FSZ89]. Let
$\mathcal{H} \subseteq \mathcal{G}$ be a measured subgroupoid and define the equivalence relation $\sim_{\mathcal{H}}$ on $\mathcal{G}$ as
$g_1 \sim_{\mathcal{H}} g_2$ if and only if $g_2 g_1^{-1} \in \mathcal{H}$.

* One says that $\mathcal{H}$ has *finite index* in $\mathcal{G}$ if for a.e. $x \in X$ there are only finitely many equivalence classes modulo $\sim_{\mathcal{H}}$ on $s^{-1}(x)$.

* One says that $\mathcal{G}$ is of infinite type if there does not exist any positive measure Borel subset $U \subseteq X$ such that the trivial subgroupoid has finite index in $\mathcal{G}_{|U}$, that is to say if for every Borel subset $U \subseteq X$ of positive measure and a.e. $x \in U$ there are infinitely many $g \in \mathcal{G}_{|U}$ such that $s(g) = x$.

4.1.2 Stably something groupoids

In the present paper we will often work with properties verified *up to some subdivision of the base space*. If A denotes some adjective, we will say that the groupoid is *stably A* if there exist a conull Borel subset $X^* \subseteq X$ and a partition $X^* = \sqcup_{i \in I} X_i$ into at most countably many Borel subsets such that $G_{|X_i}$ is A. For example:

* We say that $\mathcal{G}$ is *stably trivial* if there exist a conull Borel subset $X^* \subseteq X$ and a partition $X^* = \sqcup_{i \in I} X_i$ into at most countably many Borel subsets such that $G_{|X_i} = \{e_x : x \in X_i\}$ for every $i \in I$.

* Let $\mathcal{H}, \mathcal{H}'$ be two measured subgroupoids of $\mathcal{G}$. We say that $\mathcal{H}$ is *stably contained* in $\mathcal{H}'$ (resp. *stably equal to* $\mathcal{H}'$) if there exist a conull Borel subset $X^* \subseteq X$ and a partition $X^* = \sqcup_{i \in I} X_i$ into at most countably many Borel subsets such that $\mathcal{H}_{|X_i} \subseteq \mathcal{H}'_{|X_i}$ (resp. $\mathcal{H}_{|X_i} = \mathcal{H}'_{|X_i}$) for all $i \in I$.

We will also work with the notion of *stably normalized subgroupoid* (see Section 4.2.1).

4.1.3 Cocycles

We now introduce the notion of cocycle for measured groupoids. In order to avoid confusion with measure and orbit equivalence cocycles defined above, we will denote by $c$ the latter and by $\rho$ a cocycle defined on a groupoid. The link between the different notions is made in the examples below.

Let $\mathcal{G}$ be a measured groupoid over some standard measure space $X$ and $\mathcal{G}$ be a
countable group. A map $\rho : \mathcal{G} \to \mathcal{G}$ is called a *cocycle* if there exists a conull Borel
subset $X^* \subseteq X$ such that for all $g_1, g_2 \in G_{|X^*}$ such that $s(g_1) = r(g_2)$, it verifies $\rho(g_1g_2) = \rho(g_1)\rho(g_2)$. Up to taking a conull subset, we can (and will) always assume that the equality holds everywhere.

**Example 4.3.** • Let $G$ and $X$ be as in Example 4.1. The groupoid $G \times X$ comes with a cocycle $\rho: G \times X \to G$ defined as $\rho(g, x) = g$. We call it the *natural cocycle* associated to the action.

* Let $G$ and $H$ be two groups that are orbit equivalent over some probability space $X$. Recall that we have an orbit equivalence cocycle $c: G \times X \to H$ where $c(g, x)$ is defined as the (unique) element of $H$ such that $c(g, x) \cdot x = g \cdot x$. Therefore this map $c$ induces a cocycle $\rho$ from the groupoid $G = G \times X$ to the group $H$, defined by $\rho(g, x) = c(g, x)$. Note that since $G \times X$ and $H \times X$ are isomorphic, if $c': H \times X \to G$ denotes the other cocycle associated to the orbit equivalence, one can also define a cocycle $\rho': G \to G$ by letting $\rho'(h, x) = c'(h, x)$.

* Similarly, assume that $G$ and $H$ are two measure equivalent groups over some measure space $\Omega$ and $X$ is as in the second point of Example 4.2. One can thus consider $G := (G \times X_H)|_X$ and define a cocycle $\rho: G \to H$ by letting $\rho(g, x) := c(g, x)$ where $c(g, x)$ denotes here the measure equivalence cocycle associated to the coupling $(\Omega, m, X_G, X_H)$.

**Remark 4.4.** For future use, notice that the assumption made in Lemma 3.3 can be reformulated in terms of measured subgroupoids. Indeed, let $G$ be the measured groupoid on $X_G \cap X_H$ naturally associated to the measure equivalence coupling (as in the second point of Example 4.2), and let $\rho_G: G \to G$ and $\rho_H: G \to H$ be the natural cocycles. Then our assumption on the cocycles $c_1$ and $c_2$ in Lemma 3.3 is equivalent to requiring that $\rho_G^{-1}(M)|_U = \rho_H^{-1}(N)|_U$.

The kernel of $\rho: G \to G$ is the set of all $g \in G$ such that $\rho(g) = e_G$. We say that $\rho$ has trivial kernel if $\rho^{-1}(e_G)$ only consists of units of $G$. A cocycle $\rho$ is called nowhere trivial if there does not exist any Borel subset $U \subseteq X$ of positive measure such that $\rho(G|_U) = \{e_G\}$. The cocycle $\rho$ is called stably trivial if there exist a conull Borel subset $X^* \subseteq X$ and a Borel partition $X^* = \bigsqcup_{i \in I} X_i$ into at most countably many Borel subsets such that $\rho(G|_{X_i}) = \{e_G\}$ for all $i \in I$.

**Equivariant maps** Let $\rho: G \to G$ be a cocycle and assume that the group $G$ acts on a standard Borel space $\Delta$ by Borel automorphisms. One says that a map $\Phi: X \to \Delta$ is $(G, \rho)$-equivariant if there exists a conull Borel subset $X^* \subseteq X$ such that for all $g \in G|_{X^*}$ one has

$$ \Phi(r(g)) = \rho(g)\Phi(s(g)). $$

**Example 4.5.** Let $G$ be a countable group acting measure-preservingly on a standard measure space $X$. Denote by $G := G \times X$ the associated groupoid and $\rho: G \to G$ the natural cocycle. If $G$ acts by measurable automorphisms on a standard Borel space $\Delta$, then any $G$-equivariant map from $X$ to $\Delta$ is $(G, \rho)$-equivariant.
## 4.2 Normalization and amenable groupoids

A crucial tool in our proofs is the study of amenable subgroupoids and their normalizers (see for example Section 6.2.2). We recall in this section the main definitions and properties needed regarding the aforementioned two notions.

### 4.2.1 Normalization

A notion of normal subgroupoid of a measured groupoid was formalized by Kida [Kid10, Section 2.4], building upon a similar notion for measured equivalence relations due to Feldman-Sutherland-Zimmer [FSZ89]. We recall here the reformulation given in [GH21b, Section 3.2].

Let $\mathcal{H}$ be a measured subgroupoid of $\mathcal{G}$. Given a Borel subset $B \subseteq \mathcal{G}$, one says that $\mathcal{H}$ is $B$-invariant if there exists a conull Borel subset $X^* \subseteq X$ such that for all $\mathbf{g}_1, \mathbf{g}_2 \in B \cap \mathcal{G}_{|X^*}$ and all $\mathbf{h} \in \mathcal{G}_{|X^*}$ verifying $s(\mathbf{h}) = s(\mathbf{g}_1)$ and $r(\mathbf{h}) = s(\mathbf{g}_2)$, one has $\mathbf{h} \in \mathcal{H}$ if and only if $\mathbf{g}_2\mathbf{h}\mathbf{g}_1^{-1} \in \mathcal{H}$.

Given two measured subgroupoids $\mathcal{H}, \mathcal{N}$, one says that $\mathcal{H}$ is *normalized* by $\mathcal{N}$ if there exists a covering of $\mathcal{N}$ by countably many Borel subsets $B_n \subseteq \mathcal{G}$ so that $\mathcal{H}$ is $B_n$-invariant for all $n \in \mathbb{N}$.

**Example 4.6.** Let $\rho : \mathcal{G} \to \mathcal{G}$ be a cocycle towards a countable group $\mathcal{G}$. If $H$ and $N$ are two subgroups of $\mathcal{G}$ such that $H$ is normalized by $N$ then $\rho^{-1}(H)$ is normalized by $\rho^{-1}(N)$.

Note that similarly as in Section 4.1.2, one can define the notion of *stably normalized* subgroupoids.

### 4.2.2 Amenability

Generalizing Zimmer's notion of amenability of a group action [Zim78], Kida introduced the notion of amenable measured groupoid. We refer to [Kid09, Section 4] for the formal definition and only recall here the properties needed for our proofs. As an example, if $(X, \mu)$ is a standard probability space equipped with a measure-preserving action of a countable group $\mathcal{G}$, then the groupoid $\mathcal{G} \times X$ is amenable if and only if the $\mathcal{G}$-action on $X$ is amenable in the sense of Zimmer.

**Lemma 4.7** (see [GH21b, Corollary 3.39]). Let $\mathcal{G}$ be a measured groupoid, and $\mathcal{G}$ be an amenable countable group. If there exists a cocycle $\mathcal{G} \to \mathcal{G}$ with trivial kernel, then $\mathcal{G}$ is amenable.

In the following, if $K$ is a compact metrizable space we denote by $\text{Prob}(K)$ the space of Borel probability measures on $K$ equipped with the weak-$\*$ topology.

**Proposition 4.8** ([Kid09, Proposition 4.14]). Let $\mathcal{G}$ be an amenable measured groupoid over some base space $X$ and $\rho : \mathcal{G} \to \mathcal{G}$ be a cocycle toward $\mathcal{G}$ a countable group. Assume that $\mathcal{G}$ acts by homeomorphisms on a compact metrizable space $K$. Then there exists a $(\mathcal{G}, \rho)$-invariant Borel map from $X$ to $\text{Prob}(K)$.

One says that $\mathcal{G}$ is *everywhere non-amenable* if for every Borel subset $U \subseteq X$ of positive measure, the restriction $\mathcal{G}|_U$ is not amenable.
## 4.3 Action-like cocycles

We will work with the following definition.

**Definition 4.9 (Action-like cocycle).** Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $G$ be a countable group. A cocycle $\rho : \mathcal{G} \to G$ is action-like if

1. $\rho$ has trivial kernel;

2. whenever $H_1 \subseteq H_2$ is an infinite index inclusion of subgroups of $G$, for every positive measure Borel subset $U \subseteq X$, the subgroupoid $\rho^{-1}(H_1)|_U$ does not have finite index in $\rho^{-1}(H_2)|_U$;

3. for every nonamenable subgroup $H \subseteq G$, the groupoid $\rho^{-1}(H)$ is everywhere non-amenable.

**Remark 4.10.** We warn the reader that, although we are using the same terminology, our notion is slightly different from the one in [HH22b, page 53], where the third requirement was dropped – it is also slightly more restrictive than the notion of *action-type* cocycle from [GH21b, Definition 3.20].

**Remark 4.11.** We observe that if $\rho : \mathcal{G} \to G$ is action-like, then for every positive measure Borel subset $U \subseteq X$, the restriction $\rho : \mathcal{G}|_U \to G$ is again action-like. Also, if $H \subseteq G$ is a subgroup, then the restricted cocycle $\rho^{-1}(H) \to H$, viewed as a cocycle towards $H$, is action-like.

Building on Example 4.3, the main example of importance to us is the following.

**Lemma 4.12.** Let $(X, \mu)$ be a standard probability space, equipped with a measure-preserving action of a countable group $G$. Let $\mathcal{G} = G \times X$ be the corresponding measured groupoid.

Then the natural cocycle $\rho : \mathcal{G} \to G$ is action-like.

*Proof.* The first two items of the definition are checked in [HH22b, Lemma B.3] (we emphasize here, as recalled in Remark 4.10, that the notion of an action-like cocycle used in [HH22b] only had the first two items from Definition 4.9). We now prove that if $H \subseteq G$ is a nonamenable subgroup, then $\rho^{-1}(H)$ is everywhere nonamenable. So let $U \subseteq X$ be a positive measure Borel subset, and assume that $\rho^{-1}(H)|_U$ is amenable. Letting $H \cdot U$ be the saturation of $U$ under the $H$-action, it follows from [Kid09, Theorem 4.16(iii)] that $\rho^{-1}(H)|_{H \cdot U}$ is amenable. But since $H \cdot U$ is an $H$-invariant positive measure subset on which $H$ acts in a measure-preserving way, then [Zim84, Proposition 4.3.3] implies that $H$ is amenable, which is a contradiction. $\square$

We will often use the following observation.

**Lemma 4.13.** Let $\mathcal{G}$ be a measured groupoid, let $G$ be a countable group, and let $\rho : \mathcal{G} \to G$ be an action-like cocycle. Let $H, H'$ be two subgroups of $G$. If $\rho^{-1}(H)$ is stably contained in $\rho^{-1}(H')$, then $H$ has a finite index subgroup contained in $H'$.

*Proof.* Let $U \subseteq X$ be a positive measure Borel subset such that $\rho^{-1}(H)|_U \subseteq \rho^{-1}(H')|_U$. Then $\rho^{-1}(H \cap H')|_U = \rho^{-1}(H)|_U$, and it follows from the definition of an action-like cocycle that $H \cap H'$ has finite index in $H$. $\square$
As a consequence, when an action-like cocycle takes its values in a graph product $G$, one can relate the inclusion of two parabolic subgroups of $G$ (see Section 2.2 for the definition) with the inclusion of groupoids.

**Lemma 4.14.** Let $G$ be a graph product of countably infinite groups over a finite simple graph $\Gamma$. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$, and let $U \subseteq X$ be a positive measure Borel subset. Let $P_1, P_2 \subseteq G$ be two parabolic subgroups.

If $\rho^{-1}(P_1)|_U \subseteq \rho^{-1}(P_2)|_U$, then $P_1 \subseteq P_2$.

*Proof.* As $\rho^{-1}(P_1)|_U \subseteq \rho^{-1}(P_2)|_U$, Lemma 4.13 (applied to the groupoid $\mathcal{G}|_U$) shows that $P_1$ has a finite-index subgroup contained in $P_2$. The conclusion then follows from Lemma 2.5, page 21. $\square$

### 4.3.1 Support

In this section specifically, we consider measured groupoids equipped with cocycles towards a graph product, and review the notion of support introduced in [HH22a, Section 3.3]. It was defined in a more general context than graph products in [HH22a], but we will specify to this setting since this will be the only one of interest to us in the present paper.

**Tight support** Let $G$ be a graph product of countable groups over a finite simple graph. Recall that $\mathbb{P}_G$ denotes the set of parabolic subgroups of $G$. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho : \mathcal{G} \to G$ be a cocycle. Following [HH22a, Section 3.3], given a parabolic subgroup $P \in \mathbb{P}_G$, we say that $(\mathcal{G}, \rho)$ is *tightly* $P$-supported if

1. there exists a conull Borel subset $X^* \subseteq X$ such that $\rho(\mathcal{G}|_{X^*}) \subseteq P$, and

2. for every proper parabolic subgroup $Q \subsetneq P$, there does not exist any positive measure Borel subset $U \subseteq X$ such that $\rho(\mathcal{G}|_U) \subseteq Q$.

Given a pair $(\mathcal{G}, \rho)$, if there exists some parabolic subgroup $P \in \mathbb{P}_G$ such that $(\mathcal{G}, \rho)$ is tightly $P$-supported then such a $P$ is unique. Notice also that if $(\mathcal{G}, \rho)$ is tightly $P$-supported, then for every positive measure Borel subset $U \subseteq X$, the pair $(\mathcal{G}|_U, \rho)$ is tightly $P$-supported. The following lemma provides examples of tightly supported pairs.

**Lemma 4.15.** Let $G$ be a graph product of countably infinite groups over a finite simple graph. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$. Let $P \subseteq G$ be a parabolic subgroup.

Then $(\rho^{-1}(P), \rho)$ is tightly $P$-supported.

*Proof.* The first part of the definition obviously holds. So it is enough to prove that if $Q \subsetneq P$ is a parabolic subgroup, and $U \subseteq X$ is a positive measure Borel subset, we have $\rho^{-1}(P)|_U \not\subseteq \rho^{-1}(Q)|_U$. This is a consequence of Lemma 4.14. $\square$

The next lemma formalizes the fact that up to a countable partition of the base space, we can work with tightly supported pairs. It follows from [HH22a, Lemma 3.7], by noticing that the collection $\mathbb{P}_G$ of parabolic subgroups of $G$ is admissible in the sense of [HH22a] (it is countable, stable under conjugation, intersections, and every descending chain of parabolic subgroups is finite).
**Lemma 4.16** ([HH22a, Lemma 3.7]). Let $G$ be a graph product of countably infinite groups over a finite simple graph. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be a cocycle.

Then there exist a partition $X = \cup_n X_n$ into at most countably many Borel subsets, and for every $n$, a parabolic subgroup $P_n \subseteq G$, such that $(\mathcal{G}_{|X_n}, \rho)$ is tightly $P_n$-supported.

**Tight support and normalizers** We will often make use of the following lemma from [HH22a].

**Lemma 4.17** ([HH22a, Lemma 3.8]). Let $G$ be a graph product of countably infinite groups over a finite simple graph. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be a cocycle. Let $\mathcal{A}, \mathcal{H}$ be measured subgroupoids of $\mathcal{G}$, with $\mathcal{A}$ normalized by $\mathcal{H}$.

If $(\mathcal{A}, \rho)$ is tightly $P$-supported for some parabolic subgroup $P \subseteq G$, then there exists a conull Borel subset $X^* \subseteq X$ such that $\rho(\mathcal{H}_{|X^*}) \subseteq P \times P^\perp$.

**Definition 4.18.** Let $G$ be a graph product over a finite simple graph, let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be a cocycle.

We say that $(\mathcal{G}, \rho)$ is properly $\mathbb{P}_G$-supported if there exist a partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets, and for every $i \in I$, a proper parabolic subgroup $P_i \subsetneq G$ such that $\rho(\mathcal{G}_{|X_i}) \subseteq P_i$.

We refer to page 18 for the definition of irreducible graph.

**Lemma 4.19.** Let $G$ be a graph product of countably infinite groups over a finite irreducible simple graph. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\rho: \mathcal{G} \to G$ be a cocycle. Let $\mathcal{A}, \mathcal{H} \subseteq \mathcal{G}$ be measured subgroupoids, with $\mathcal{A}$ normalized by $\mathcal{H}$.

If $(\mathcal{A}, \rho)$ is properly $\mathbb{P}_G$-supported and $\rho|_{\mathcal{A}}$ is nowhere trivial, then $(\mathcal{H}, \rho)$ is properly $\mathbb{P}_G$-supported.

*Proof.* Up to a countable Borel partition of $X$, we can assume that $(\mathcal{A}, \rho)$ is tightly $P$-supported for some proper parabolic subgroup $P \subsetneq G$, and aim to show that there exists a conull Borel subset $X^* \subseteq X$ such that $\rho(\mathcal{H}_{|X^*})$ is contained in a proper parabolic subgroup of $G$.

As $\rho|_{\mathcal{A}}$ is nowhere trivial, we have $P \neq \{1\}$. By Lemma 4.17, there exists a conull Borel subset $X^* \subseteq X$ such that $\rho(\mathcal{H}_{|X^*}) \subseteq P \times P^\perp$. As $G$ is irreducible and $P$ is neither equal to $\{1\}$ nor to $G$, the parabolic subgroup $P \times P^\perp$ is proper, which completes our proof. $\square$

## 4.4 Kida's blueprint

In this section, we review Kida's general strategy towards measure equivalence classification and rigidity theorems. This originates from his work on mapping class groups [Kid08a, Kid10] and was also used in subsequent measure equivalence rigidity theorems. We extract an abstract setting from Kida's work, that we will follow in Part II. We start with a very brief review of the mapping class group case that will serve as a motivating example – we refer to the references below for basic definitions and background, but will never need anything about mapping class groups in the sequel of the present work.
**Review.** Recall that the mapping class group of a finite-type orientable surface Σ is the group Mod(Σ) of isotopy classes of orientation-preserving homeomorphisms of Σ. It acts by automorphisms on the curve graph C(Σ), defined as follows: vertices of C(Σ) are isotopy classes of essential simple closed curves on Σ, and edges correspond to the existence of disjoint representatives. An important theorem of Ivanov [Iva97] asserts that the automorphism group of C(Σ) is exactly the extended mapping class group Mod$^{±}$(Σ), where orientation-reversing homeomorphisms are allowed.

A key step in Kida's proof of the measure equivalence rigidity of the mapping class group Mod(Σ) of a finite-type surface [Kid10], was to establish that every self-coupling Ω of Mod(Σ), factors through Aut(C(Σ)) ≃ Mod$^{±}$(Σ). From there, a standard argument allows to derive the measure equivalence rigidity of Mod(Σ), from this rigidity statement for self-couplings.

Kida's strategy for proving this statement about self-couplings was in part inspired by earlier work of Ivanov [Iva97]. Ivanov had proved that the abstract commensurator of Mod(Σ) is equal to Mod$^{±}$(Σ): this means that every isomorphism between finite-index subgroups of Mod(Σ), is given by the conjugation by an element of Mod$^{±}$(Σ). Ivanov's proof goes as follows: he showed that, if $H_1, H_2 \subseteq \text{Mod}(\Sigma)$ are two finite-index subgroups of Mod(Σ), then any isomorphism $f: H_1 \to H_2$ sends curve stabilizers (in $H_1$) to curve stabilizers (in $H_2$), and sends stabilizers of disjoint curves (up to isotopy) to the stabilizers of disjoint curves. In other words $f$ induces an automorphism of the curve graph. Using that $\text{Aut}(C(\Sigma)) \simeq \text{Mod}^{\pm}(\Sigma)$, this precisely says that $f$ is the conjugation by an element of Mod$^{±}(\Sigma)$.

Kida's strategy can be viewed as a groupoid-theoretic version of Ivanov's argument. Given a measured groupoid over a standard probability space with two action-like cocycles towards Mod(Σ), he proves that subgroupoids that correspond (in an appropriate sense) to curve stabilizers for the first cocycle, also correspond to curve stabilizers for the second cocycle. Through the dictionary between measure equivalence couplings and measured groupoids, this establishes the desired factorization of self-couplings of Mod(Σ).

Kida's strategy has later been used in several contexts to establish new measure equivalence rigidity theorems (see for example [CK15, Kid11, HH21, HH20, GH21b]). We now formulate it in an abstract setting, which will be applied to the extension graph of the graph product in Part II of the present work.

**Preliminary tool: from measured groupoids to couplings.**

**Lemma 4.20.** Let $G, H$ be two countable groups. Let $\Delta$ be a Polish space, equipped with an action of $G \times H$ by measurable automorphisms. Assume that for any measured groupoid $G$ over a standard probability space $X$, and any action-like cocycles $\rho_G: G \to G$ and $\rho_H: G \to H$, there exists a $(\rho_G, \rho_H)$-equivariant Borel map $\theta: X \to \Delta$, i.e. such that (up to restricting $G$ to a conull Borel subset) for every $g \in G$, one has

$$ \theta(r(g)) = \rho_H(g)\theta(s(g))\rho_G(g)^{-1}. $$

Then for every measure equivalence coupling $\Omega$ between $G$ and $H$, there exists a $(G \times H)$-equivariant Borel map $\Omega \to \Delta$.

As usual, the equivariance of the map $\Omega \to \Delta$ is understood almost everywhere.
Proof. The argument of this proof comes from the work of Kida, see the demonstration of [Kid10, Theorem 5.6].

Recall that we denote by $*$ the actions of $G$ and $H$ on $\Omega$, and by $\cdot$ the induced actions on the fundamental domains.

Let $X_G$ and $X_H$ be fundamental domains for the actions of $G$ and $H$ respectively. We can choose $X_G$ and $X_H$ such that their intersection $X := X_G \cap X_H$ verifies $(G \times H) * X = \Omega$ up to a null set, see e.g. [Kid09, Lemma 2.27]. Note that this last equality implies in particular that $X$ has positive measure.

Let $\mathcal{G}$ be the measured groupoid over $X$ given by the restrictions of the $G$- and $H$-actions on $X_H$ and $X_G$ respectively, as in the second point of Example 4.2. We thus have two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$ (see Example 4.3 for their definition, and Lemma 4.12 and Remark 4.11 for the fact that they are action-like). So by assumption, we have a $(\rho_G, \rho_H)$-equivariant Borel map $\theta : X \to \Delta$. We extend it to a map $\bar{\theta} : \Omega \to \Delta$ by defining $\bar{\theta}((g, h) * x) = h\theta(x)g^{-1}$, for a.e. $x \in X$ and all $(g, h) \in G \times H$.

It only remains to verify that the definition of $\bar{\theta}$ does not depend on the choice of the representatives $g, h$ and $x$ – its equivariance then follows from its definition. So let us show that for all $g_1, g_2 \in G$, all $h_1, h_2 \in H$ and a.e. $x_1, x_2 \in X$ verifying $(g_1, h_1) * x_1 = (g_2, h_2) * x_2$, we have $h_1\theta(x_1)g_1^{-1} = h_2\theta(x_2)g_2^{-1}$. Since the actions of $G$ and $H$ commute, our assumption that $(g_1, h_1) * x_1 = (g_2, h_2) * x_2$ rewrites as $(h_2^{-1}h_1) * (g_2^{-1}g_1) * x_1 = x_2$. This translates in terms of actions on the fundamental domains as

$$ (h_2^{-1}h_1) \cdot x_1 = x_2 = (g_2^{-1}g_1) \cdot x_1. $$

The $(\rho_G, \rho_H)$-equivariance of $\theta$ thus gives $\theta(x_2) = h_2^{-1}h_1\theta(x_1)g_1^{-1}g_2$ almost surely, and hence $\bar{\theta}$ is well defined. $\square$

**Abstract setting.** Let $G$ be a countable group, let $\mathcal{C}_G$ be a conjugacy-invariant countable set of subgroups of $G$, and let $\mathcal{I}_G$ be a graph with vertex set $\mathcal{C}_G$, such that the conjugation action of $G$ on $\mathcal{C}_G$ extends to a graph automorphism of $\mathcal{I}_G$.

**Remark 4.21.** In Kida's setting, $\mathcal{C}_G$ is the set of all stabilizers in Mod($\Sigma$) of isotopy classes of essential simple closed curves on $\Sigma$, and $\mathcal{I}_G$ is the curve graph –adjacency corresponds to the disjointness of the curves up to isotopy, or more algebraically to the fact that the centers of their stabilizers commute. In our setting $\mathcal{C}_G$ will be the set of conjugates of vertex groups, and $\mathcal{I}_G$ will be the extension graph.

Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with a cocycle $\rho : \mathcal{G} \to G$. Let $\mathcal{H} \subseteq \mathcal{G}$ be a measured subgroupoid.

We say that $(\mathcal{H}, \rho)$ is stably $\mathcal{C}_G$ if there exists a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets, such that for every $i \in I$, there exists $C_i \in \mathcal{C}_G$ such that $\mathcal{H}_{|X_i} = \rho^{-1}(C_i)|_{X_i}$.

Let now $\mathcal{H}_1, \mathcal{H}_2$ be two measured subgroupoids such that both $(\mathcal{H}_1, \rho)$ and $(\mathcal{H}_2, \rho)$ are stably $\mathcal{C}_G$. Then (after refining the two partitions given by the above definition) there exists a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets, such that for every $i \in I$, there exist $C_{1,i}, C_{2,i} \in \mathcal{C}_G$ such that $(\mathcal{H}_1)_{|X_i} = \rho^{-1}(C_{1,i})|_{X_i}$ and $(\mathcal{H}_2)_{|X_i} = \rho^{-1}(C_{2,i})|_{X_i}$.
We say that $(\mathcal{H}_1, \rho)$ and $(\mathcal{H}_2, \rho)$ are $\mathcal{I}_G$-adjacent if the above partition and subgroups can be chosen such that for every $i \in I$ such that $X_i$ has positive measure, the groups $C_{1,i}$ and $C_{2,i}$ are distinct and adjacent in $\mathcal{I}_G$.

**Definition 4.22 (ME-witness).** Let $G, H$ be two countable groups, with countable sets of subgroups $\mathcal{C}_G, \mathcal{C}_H$, and graphs $\mathcal{I}_G, \mathcal{I}_H$ with vertex sets $\mathcal{C}_G, \mathcal{C}_H$. We say that $(\mathcal{I}_G, \mathcal{I}_H)$ is an ME-witness for $G, H$ if for every measured groupoid $\mathcal{G}$ over a standard probability space $X$, any two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$, and any measured subgroupoids $C_1, C_2 \subseteq \mathcal{G}$, the following two properties hold:

1. $(C_1, \rho_G)$ is stably $\mathcal{C}_G$ if and only if $(C_2, \rho_H)$ is stably $\mathcal{C}_H$, and

2. if $(C_1, \rho_G)$ and $(C_2, \rho_G)$ are stably $\mathcal{C}_G$, then they are $\mathcal{I}_G$-adjacent if and only if $(C_1, \rho_H)$ and $(C_2, \rho_H)$ are $\mathcal{I}_H$-adjacent.

**Remark 4.23.** Kida shows that the curve graph is an ME-witness for an appropriate subgroup of Mod($\Sigma$). We will prove in Part II that the extension graphs of graph products of countably infinite groups defined over transvection-free graphs with no partial conjugations, are also ME-witnesses.

In the following proposition, and throughout the paper, we equip the set $\text{Isom}(\mathcal{I}_G, \mathcal{I}_H)$ of all graph isomorphisms between $\mathcal{I}_G$ and $\mathcal{I}_H$, with the compact-open topology, or equivalently the topology of pointwise convergence. The following proposition is inspired from [Kid10, Section 5].

**Proposition 4.24.** Let $G, H$ be two countable groups. Let $\mathcal{C}_G, \mathcal{C}_H$ be conjugacy-invariant countable sets of pairwise non-commensurable subgroups, and let $\mathcal{I}_G, \mathcal{I}_H$ be graphs with vertex sets $\mathcal{C}_G, \mathcal{C}_H$, for which the actions of $G$ and $H$ by conjugation extend to graph automorphisms. Assume that $(\mathcal{I}_G, \mathcal{I}_H)$ is an ME-witness for $G, H$.

Then for every measure equivalence coupling $\Omega$ between $G$ and $H$, there exists a $(G \times H)$-equivariant measurable map $\Omega \to \text{Isom}(\mathcal{I}_G, \mathcal{I}_H)$.

*Proof.* In view of Lemma 4.20, it is enough to prove that for every measured groupoid $\mathcal{G}$ over a standard probability space $X$, and any two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$, there exists a Borel map $\theta : X \to \text{Isom}(\mathcal{I}_G, \mathcal{I}_H)$ which is $(\rho_G, \rho_H)$-equivariant in the following sense: there exists a conull Borel subset $X^* \subseteq X$ such that for every $g \in \mathcal{G}|_{X^*}$, one has $\theta(y) = \rho_H(g)\theta(x)\rho_G(g)^{-1}$, where $x = s(g)$ and $y = r(g)$. We will now prove this fact.

**Observation** We start by showing that if $C_1, C_2 \in \mathcal{C}_G$ verify $\rho_G^{-1}(C_1)|_U = \rho_G^{-1}(C_2)|_U$ for some Borel subset $U \subseteq X$ of positive measure, then $C_1 = C_2$.

For this, observe that the measured subgroupoid $\rho_G^{-1}(C_1 \cap C_2)|_U$ is equal to both $\rho_G^{-1}(C_1)|_U$ and to $\rho_G^{-1}(C_2)|_U$. As $\rho_G$ is action-like, it follows that $C_1 \cap C_2$ has finite index in both $C_1$ and $C_2$. Thus $C_1$ and $C_2$ are commensurable, which by assumption on $\mathcal{C}_G$, implies that $C_1 = C_2$, as desired.

The same observation also holds for $C'_1, C'_2 \in \mathcal{C}_H$, with respect to the cocycle $\rho_H$.

**Definition of $\theta$** Fix $C \in V\mathcal{I}_G = \mathcal{C}_G$ and let $\mathcal{C} := \rho_G^{-1}(C)$. Note that in particular $(C, \rho_G)$ is stably $\mathcal{C}_G$.

Since $(\mathcal{I}_G, \mathcal{I}_H)$ is an ME-witness, $(C, \rho_H)$ is of type $\mathcal{C}_H$. This means that there exists a partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most
countably many Borel subsets, such that for all $i \in I$ there exists $C'_i \in V\mathfrak{J}_H = \mathfrak{C}_H$ such that $C_{|X_i} = \rho_H^{-1}(C'_i)|_{X_i}$ (and by the above observation, $C'_i$ is unique whenever $X_i$ has positive measure). We then define for a.e. $x \in X$

$$ \theta(x) : C \in V\mathfrak{J}_G \mapsto C'_i \quad \text{if } x \in X_i. \qquad (4.1) $$

We now prove that, up to null sets, our definition of $\theta(x)(C)$ does not depend on the choice of partition. So assume that, starting from $C$, we are given two partitions as above. Let $X^* = \cup_{i \in I} X_i$ be a common refinement, with associated subgroups $C'_{1,i}$ and $C'_{2,i}$. By construction, for every $i \in I$, we have $\rho_H^{-1}(C'_{1,i})|_{X_i} = \rho_G^{-1}(C)|_{X_i} = \rho_H^{-1}(C'_{2,i})|_{X_i}$. By the observation made at the beginning of the proof $C'_{1,i} = C'_{2,i}$.

**Injectivity of $\theta(x)$** Let us now show, for a.e. $x \in X$, the injectivity of $\theta(x)$. It is enough to prove that for every positive measure Borel subset $U \subseteq X$, and any two $C_1, C_2 \in V\mathfrak{J}_G$, if $\theta(x)(C_1) = \theta(x)(C_2)$ for every $x \in U$, then $C_1 = C_2$. Let $U, C_1, C_2$ be as above. Up to restricting to positive measure Borel subset of $U$, we will assume that $\theta(x)(C_1) = \theta(x)(C_2)$ is constant on $U$, with value denoted by $C'$. We can assume that the partitions of $X$ given by $C_1$ and $C_2$ coincide and denote this common partition $X^* = \cup_{i \in I} X_i$. Now let $i \in I$ such that $U_i := U \cap X_i$ has positive measure. By our definition of $\theta$, we have

$$ \rho_G^{-1}(C_1)|_{U_i} = \rho_H^{-1}(C')|_{U_i} = \rho_G^{-1}(C_2)|_{U_i}. $$

By the observation made at the beginning of the proof, this implies that $C_1 = C_2$.

**Surjectivity of $\theta(x)$** Take a vertex $C'$ in $\mathfrak{J}_H$. Since $(\mathfrak{J}_G, \mathfrak{J}_H)$ is an ME-witness, there exist a conull Borel subset $X^* \subseteq X$ and a partition $X^* = \cup_{j \in J} Y_j$ into at most countably many Borel subsets such that for all $j \in J$ there exists $C_j \in V\mathfrak{J}_G$ such that $\rho_H^{-1}(C')|_{Y_j} = \rho_G^{-1}(C_j)|_{Y_j}$. Now, for a.e. $x \in X$ there exists $j \in J$ such that $x$ belong to $Y_j$. Therefore, by the above definition of $\theta$, it verifies $\theta(x)(C_j) = C'$ almost surely. Hence the surjectivity.

**Adjacency** We now prove that for a.e. $x \in X$ the map $\theta(x)$ is a graph isomorphism. So consider $C_1, C_2 \in V\mathfrak{J}_G$. Up to taking a common refinement of the two partitions, we can assume that the partitions of (a conull Borel subset of) $X$ given by $C_1$ and $C_2$ coincide. We denote it by $X^* = \cup_{i \in I} X_i$. Now let $C_1 := \rho_G^{-1}(C_1)$ and $C_2 := \rho_G^{-1}(C_2)$. For $i \in I$, denote by $C'_{1,i}$ and $C'_{2,i}$ the vertices of $\mathfrak{J}_H$ such that $(C_1)|_{X_i} = \rho_H^{-1}(C'_{1,i})|_{X_i}$ and $(C_2)|_{X_i} = \rho_H^{-1}(C'_{2,i})|_{X_i}$. In other words $C'_{1,i} = \theta(x)(C_1)$ and $C'_{2,i} = \theta(x)(C_2)$ whenever $x \in X_i$. Since $(\mathfrak{J}_G, \mathfrak{J}_H)$ is an ME-witness, we get that $C'_{1,i}$ and $C'_{2,i}$ are distinct and adjacent if and only if $C_1$ and $C_2$ are distinct and adjacent, which is the desired conclusion.

**Equivariance of $\theta$** First for all $g \in G$ and all $h \in H$, let

$$ B_{g,h} := \{ \mathbf{g} \in \mathcal{G} \mid \rho_G(\mathbf{g}) = g, \rho_H(\mathbf{g}) = h \}, $$

and remark that $\mathcal{G} = \cup_{g \in G, h \in H} B_{g,h}$. Now consider $g \in G$ and $h \in H$ and denote $B := B_{g,h}$. Now define $U := s(B)$ and $V := r(B)$, which are Borel subsets of $X$ (see [Kec95, Theorem 18.10]). Since $\rho_G$ is action-like, the map $f := r \circ s^{-1}$ is a Borel isomorphism from $U$ to $V$. To prove the equivariance on $B_{g,h}$, we need to
show that $\theta(r(\mathbf{g})) = \rho_H(\mathbf{g})\theta(s(\mathbf{g}))\rho_G(\mathbf{g})^{-1}$, that is to say for a.e. $x \in U$ and for all $C \in V\mathcal{J}_G$

$$ \theta(f(x))(gCg^{-1}) = h(\theta(x)(C))h^{-1}. \qquad (4.2) $$

We now fix $C \in V\mathcal{J}_G$. Recall that the definition of $\theta$ does not depend on the choice of partition. So we can partition $U = \sqcup_{i \in I} U_i$ and $V = \sqcup_{i \in I} V_i$ into Borel subsets, such that for all $i \in I$, we have $f(U_i) = V_i$, and the maps $x \mapsto \theta(x)(C)$ and $x \mapsto \theta(f(x))(gCg^{-1})$ are constant a.e. on $U_i$. We denote by $C'_{U,i}, C'_{V,i} \in V\mathcal{J}_H$ their respective values. Hence by definition of $\theta$, we have

$$ \rho_G^{-1}(C)|_{U_i} = \rho_H^{-1}(C'_{U,i})|_{U_i} \quad \text{and} \quad \rho_G^{-1}(gCg^{-1})|_{V_i} = \rho_H^{-1}(C'_{V,i})|_{V_i}. \qquad (4.3) $$

Now, to prove eq. (4.2) we need to show that $C'_{V,i} = hC'_{U,i}h^{-1}$ for all $i$. Using the definition of $B_{g,h}$ for the first and third equalities and eq. (4.3) for the second one, we have

$$ \begin{aligned} \rho_G^{-1}(gCg^{-1})|_{V_i} &= B\rho_G^{-1}(C)|_{U_i}B^{-1} = B\rho_H^{-1}(C'_{U,i})|_{U_i}B^{-1} \\ &= \rho_H^{-1}(hC'_{U,i}h^{-1})|_{V_i}. \end{aligned} $$

Combining the above two equations, we deduce that $\rho_H^{-1}(C'_{V,i})|_{V_i} = \rho_H^{-1}(hC'_{U,i}h^{-1})|_{V_i}$. The first observation of the proof thus yields $C'_{V,i} = hC'_{U,i}h^{-1}$ as desired. Hence the equivariance. $\square$

<figure><img src="image_11.png" /><figcaption>Figure 8: Illustration for the proof of the equivariance of $\theta$</figcaption></figure>
## Part II

# Measure equivalence classification of graph products

The goal of this part is to prove the following theorem, which recalls Theorem 1.2 from the introduction.

**Theorem 1.2.** Let $\Gamma_G, \Gamma_H$ be two finite simple graphs, not reduced to one vertex, with no transvection and no partial conjugation. Let $G, H$ be graph products of countably infinite groups over $\Gamma_G, \Gamma_H$, respectively. Then the following assertions are equivalent.

1. The groups $G$ and $H$ are measure equivalent.

2. The groups $G$ and $H$ are orbit equivalent.

3. There exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, the groups $G_v$ and $H_{\sigma(v)}$ are orbit equivalent.

## Strategy of the proof

Let us describe the idea of the proof. The implication $3 \Rightarrow 2$ was proved in [HH22a] and the implication $2 \Rightarrow 1$ is obvious. We explain the implication $1 \Rightarrow 2$, and give a hint about $2 \Rightarrow 3$.

So let $G$ and $H$ be as in the theorem, and assume that $G$ and $H$ are measure equivalent. Let $\Omega$ be a measure equivalence coupling between $G$ and $H$, and let $\mathcal{G}$ be a measured groupoid associated to $\Omega$ as in Example 4.2, coming with two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$.

**General idea and main tools** The proof that $G$ and $H$ are orbit equivalent is decomposed in two main steps.

* In the first step of the proof, we work with the extension graphs $\Gamma_G^e$ and $\Gamma_H^e$, associated to $G$ and $H$. Following Kida's blueprint recalled in Section 4.4, our goal in this step is to show that $(\Gamma_G^e, \Gamma_H^e)$ is an ME-witness between $G$ and $H$ in the sense of Definition 4.22.

Recall that vertices of $\Gamma_G^e, \Gamma_H^e$ are the conjugates of the vertex groups $G_v, H_w$. Thus, the key point is to show that a subgroupoid $\mathcal{V} \subseteq \mathcal{G}$ verifies $\mathcal{V} = \rho_G^{-1}(G_v)$ for some $v \in V\Gamma_G$ if and only if (up to a countable partition of the base space) there exist $w \in V\Gamma_H$ and $h \in H$ such that $\mathcal{V} = \rho_H^{-1}(hH_ww^{-1})$. This property is what we formalize under the name of Vertex Recognition Property in Definition 5.1 below. Such a subgroupoid $\mathcal{V}$ is called of vertex type, with respect to either $\rho_G$ or $\rho_H$ (see Section 5.1 for the formal definition). The proof of this Vertex Recognition Property goes through a characterization of subgroupoids of vertex type which is independent from the cocycle – in other words, we show that there exists $v \in V\Gamma_G$ such that $\mathcal{V} = \rho_G^{-1}(G_v)$ if and only if $\mathcal{V}$ verifies some purely groupoid theoretic conditions. This is done in Sections 7 and 8; we will say more about this below.

* In the second step of the proof, we use the actions of $G, H$ on their right-angled buildings $\mathbb{D}_G, \mathbb{D}_H$ (see Section 2.3 for the definition). These are used to build a
fundamental domain of $G$ and $H$ on $\Omega$ (showing that $G$ and $H$ are orbit equivalent),
and also induce orbit equivalence couplings at the level of the vertex groups, in the
following way.

First, through Proposition 4.24, page 47, the previous step gives us a measurable
$(G \times H)$-equivariant map $\Omega \to \text{Isom}(\Gamma_G^e, \Gamma_H^e)$, where $G, H$ act by pre- and post-
composition on $\text{Isom}(\Gamma_G^e, \Gamma_H^e)$. In particular $\Gamma_G^e$ and $\Gamma_H^e$ are isomorphic. This
automatically implies that $\Gamma_G$ and $\Gamma_H$ are isomorphic, using that $\Gamma_G$ and $\Gamma_H$ have
no transvection and no partial conjugation [Hua17].

Now, in view of the comparison between $\text{Isom}(\Gamma_G^e, \Gamma_H^e)$ and $\text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ made
in Proposition 2.15, page 25, we deduce a measurable $(G \times H)$-equivariant map $\Theta$
from $\Omega$ to $\text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$.

Notice that $G, H$ act transitively on the sets of rank 0 vertices of $\mathbb{D}_G, \mathbb{D}_H$. Thus,
the subset $Y \subseteq \text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ consisting of all isomorphisms that send $\{e_G\}$ to
$\{e_H\}$ (viewed as rank 0 vertices) is a fundamental domain for both the $G$- and the
$H$-action on $\text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$. So $\Theta^{-1}(Y)$ is a common fundamental domain for the
$G$- and $H$-actions on $\Omega$. This shows that $G$ and $H$ are orbit equivalent.

We finally say a word about the implication $2 \Rightarrow 3$. Starting with an isomorphism
$\sigma' : \Gamma_G \to \Gamma_H$, the idea is to identify $v$ and $\sigma'(v)$ to rank 1 vertices of $\mathbb{D}_G, \mathbb{D}_H$,
and let $\Omega'_v := \Theta^{-1}(\{f|f(v) = \sigma'(v)\})$. We show that there exists a choice of $\sigma'$ for
which $\Omega'_v$ has positive measure for all $v \in V\Gamma_G$. For this choice, the set $\Omega'_v$ (which
is $(G_v \times H_{\sigma'(v)})$-invariant) is in fact the desired orbit equivalence coupling between
$G_v$ and $H_{\sigma'(v)}$. We refer to Proposition 5.10, page 57 for the details.

**More on the Vertex Recognition Property: A zoom-in process** As we explained
in the description of the first step of the proof, in order to prove the Vertex Recognition
Property, we need to give a characterization of subgroupoids of $\mathcal{G}$ of vertex type that
is independent from the cocycle. However, since we do not have any hypothesis on the
vertex groups – besides the fact that they are countably infinite – we cannot hope to
obtain a characterization via their *intrinsic* properties. Instead, we rely on how they
behave *inside* the bigger groupoid $\mathcal{G}$, to get the needed characterization.

The central combinatorial idea behind our proof is a *zoom-in* process: starting from $G$,
vertex groups are obtained by successively passing to maximal (parabolic) product sub-
groups, and then restricting to factors. More precisely, given a vertex group $G_v$, we can
find two finite sequences $(F_j)_j$ and $(P_j)_j$ of parabolics subgroups of $G$ such that

• $G = F_0 \supseteq P_1 \supseteq F_1 \supseteq \dots \supseteq P_n \supseteq F_n = G_v;$

• And $P_{j+1}$ is a maximal product inside $F_j$;

• And $F_j$ is a factor of $P_j$ (and $F_n$ is just the clique factor of $P_n$).

We refer to Lemma 6.6, page 65 for a precise statement. Conversely, we manage to
use this zoom-in process as a starting point towards an algebraic characterization of
(conjugates of) vertex groups.

One important feature here is that the properties we use in the process (being a
maximal product, being a factor) can in fact be completely characterized just in terms
of amenability of certain subgroups, and inclusion or normalization of one subgroup by
another – this allows for a translation to the groupoid-theoretic setting, using the notions
of amenability and normality recalled in Section 4. More precisely, we characterize –
independently from the cocycle $\rho_G$ – subgroupoids of the form $\mathcal{P} = \rho_G^{-1}(P)$ for $P$ some maximal product in $G$, using properties of their behavior inside of $G$. Likewise, we then characterize subgroupoids of the form $\mathcal{F} = \rho_G^{-1}(F)$ where $F$ is a factor of $P$. Iterating this process on the sequences of parabolics given by the above zoom-in process, we are then able to characterize vertex type groupoids and show our Vertex Recognition Property.

The details of the arguments used to “recognize” maximal products and factors are explained in the introductory part of Section 7, to which the reader is refered. Readers less familiar with measured groupoids can also read appendix A where we prove the analogue of Theorem 1.2 with regard to isomorphism classification. The proof in this appendix can serve as a guideline (written in the group theoretic framework) through Section 7.

# Structure of Part II

Our proof of Theorem 1.2 occupies Sections 5 to 8 – Section 9 is not needed for the main theorem, and is only a slight variation over the Vertex Recognition Property, in preparation for Part III of the present work. Examples illustrating Theorem 1.2 and variations obtained along the way, are provided in Section 10.

In Section 5, we formulate the Vertex Recognition Property, and explain how to prove Theorem 1.2 assuming this property has been established. In other words, Section 5 carries Step 2 from the above proof outline, which exploits the right-angled buildings of the graph products. Section 5 also contains the proofs of the commensurability classification theorem (Theorem 1.6), as well as the application to fundamental groups of associated von Neumann algebras (Corollary 5.15, page 61), both derived from the statement in measure equivalence.

The next three sections are devoted to the proof of the Vertex Recognition Property. Section 6 contains preparatory tools of two sorts. First, it contains some combinatorial lemmas, namely:

* we establish the combinatorial lemma that provides the zoom-in process described above (Section 6.1.3);

* we show in Section 6.1.1 that if $\Gamma$ is transvection-free and has no partial conjugation, then either $\Gamma$ splits as a join (and $G$ splits as a product), or else $\Gamma$ is strongly reduced in the sense that it provides a minimal decomposition of $G$ as a graph product. Our proof of the Vertex Recognition Property will be split into this two cases in later sections.

Section 6 also contains some groupoid-theoretic tools in preparation for later sections.

Our proof now follows the dichotomy provided by the second combinatorial lemma. Namely, in Section 7, we establish the Vertex Recognition Property when the defining graph is strongly reduced. And in Section 8, we extend this to include graph products over graphs that split as a join.

Finally Section 10 provides some examples, illustrating the main results and the different cases treated in the present part.
# 5 From vertex recognition to rigidity

In the following section we formalize the aforementioned Vertex Recognition Property and show how to use it to obtain our main classification theorem (Theorem 1.2). We start by applying Kida's blueprint (detailed in Section 4.4) to factorize our coupling through the extension graphs (see Section 5.2) and prove our main theorem in Section 5.3. We then turn in Section 5.4 to applications of the Vertex Recognition Property to the classifications up to commensurability and isomorphisms and to the study of fundamental groups of equivalence relations and associated von Neumann algebras.

## 5.1 The Vertex Recognition Property

Let $G$ be a graph product. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with a cocycle $\rho: \mathcal{G} \to G$, and let $\mathcal{V} \subseteq \mathcal{G}$ be a measured subgroupoid. We say that $(\mathcal{V}, \rho)$ is of vertex type if there exists a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$, and for every $i \in I$, a parabolic subgroup $P_i \subseteq G$ that is conjugate to a vertex group, such that $\mathcal{V}|_{X_i} = \rho^{-1}(P_i)|_{X_i}$.

A crucial tool in our proof of Theorem 1.2 is a recognition technique phrased in the language of measured groupoids.

**Definition 5.1 (Vertex Recognition Property).** A class $\mathcal{C}$ of graph products satisfies the *Vertex Recognition Property* if for every $G, H \in \mathcal{C}$, every measured groupoid $\mathcal{G}$ over a standard probability space, equipped with action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$, and every measured subgroupoid $\mathcal{V} \subseteq \mathcal{G}$, the pair $(\mathcal{V}, \rho_G)$ is of vertex type if and only if $(\mathcal{V}, \rho_H)$ is of vertex type.

**Remark 5.2.** Note that the Vertex Recognition Property for the class $\mathcal{C}$ implies that for every $G \in \mathcal{C}$, every automorphism $\varphi$ of $G$ sends every vertex group to a conjugate of a vertex group: indeed, this is proved by taking $G = H$, $\mathcal{G} = G$, $\rho_G = \text{id}$ and $\rho_H = \varphi$. In fact our notion is a groupoid-theoretic extension of this algebraic property.

The Vertex Recognition Property holds in the context of interest to us.

**Proposition 5.3.** The class of all graph products of countably infinite groups over finite simple graphs not reduced to one vertex, which are transvection-free and have no partial conjugation, has the Vertex Recognition Property.

The proof of Proposition 5.3 is postponed to Sections 7 (where it is proved in the irreducible case) and 8 (where it is proved in the reducible case). In the present section, we explain how we derive our classification theorem (Theorem 1.2) from the Vertex Recognition Property. In other words, we carry out the second point from the strategy of the proof described at the beginning of Part II of the present work.

With the language and notations from Section 4.4 (Kida's blueprint), we will let $\mathcal{C}_G$ be the set of all conjugates of vertex groups of $G$, and let $\mathfrak{J}_G = \Gamma_G^e$ be the extension graph of $G$, a graph acted upon by $G$ whose vertex set is precisely $\mathcal{C}_G$. In this language, a subgroupoid of vertex type is exactly a subgroupoid which is stably $\mathcal{C}_G$. The first step in our proof of Theorem 1.2, carried in Section 5.2, consists in showing that $(\mathfrak{J}_G, \mathfrak{J}_H)$ is an ME-witness for $G, H$ (see page 47 for the definition). This requires showing that
adjacency in the extension graph is visible at the groupoid level. The second step, carried in Section 5.3, is where we depart from Kida's strategy for measure equivalence classification: we will use the right-angled buildings of the graph products as a tool for keeping track of volumes and passing from measure equivalence to orbit equivalence.

## 5.2 Factoring the self-coupling

The purpose of this section is to apply Kida's blueprint (Section 4.4). The first step is to obtain a characterization of adjacency in the extension graph $\Gamma_G^e$ (i.e. commutation) in terms of measured groupoids. More precisely, given two groupoids we are able to recognize when their image by a cocycle will give adjacent parabolic subgroups *independently from the cocycle*.

**Lemma 5.4.** Let $G$ be a graph product of countably infinite groups over a finite simple graph. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be an action-like cocycle. Let $P, Q$ be two parabolic subgroups that are conjugate to vertex groups. Let $\mathcal{P} = \rho^{-1}(P)$ and $\mathcal{Q} = \rho^{-1}(Q)$. The following assertions are equivalent.

1. The groups $P$ and $Q$ are distinct and commute.

2. The groupoid $\mathcal{Q}$ normalizes $\mathcal{P}$ and is different from $\mathcal{P}$.

3. There exists a positive measure Borel subset $U \subseteq X$ such that $\mathcal{Q}|_U$ normalizes $\mathcal{P}|_U$, and $\mathcal{Q}|_U \neq \mathcal{P}|_U$.

*Proof.* The assertion 1 $\Rightarrow$ 2 follows from Example 4.6, page 41 (for the fact that $\mathcal{Q}$ normalizes $\mathcal{P}$), and from Lemma 4.14, page 43 (for the fact that $\mathcal{Q} \neq \mathcal{P}$). The assertion 2 $\Rightarrow$ 3 is clear. For 3 $\Rightarrow$ 1, notice first that $(\mathcal{Q}, \rho)$ is tightly $\mathcal{Q}$-supported (Lemma 4.15), and therefore $(\mathcal{Q}|_U, \rho)$ is tightly $\mathcal{Q}$-supported. As $\mathcal{Q}|_U$ normalizes $\mathcal{P}|_U$, Lemma 4.17 implies that (up to restricting to a conull Borel subset) $\mathcal{Q}|_U \subseteq \rho^{-1}(P \times P^{\perp})|_U$. It thus follows that $\mathcal{Q} \subseteq P \times P^{\perp}$. As $\mathcal{Q}$ is conjugate to a vertex group, this implies that $\mathcal{Q}$ is either equal to $P$ or commutes with $P$. The fact that $\mathcal{Q}|_U \neq \mathcal{P}|_U$ implies that $P \neq \mathcal{Q}$, using Lemma 4.14. $\square$

For the following lemma, we recall that $\text{Isom}(\Gamma_G^e, \Gamma_H^e)$ is equipped with the topology of pointwise convergence, and with the action of $G \times H$ given by pre- and post-composition.

**Lemma 5.5.** Let $\mathcal{C}$ be a class of graph products that satisfies the Vertex Recognition Property. Let $G, H \in \mathcal{C}$.

For every measure equivalence coupling $\Omega$ between $G$ and $H$, there exists a $(G \times H)$-equivariant Borel map $\Omega \to \text{Isom}(\Gamma_G^e, \Gamma_H^e)$. (In particular, if $G$ and $H$ are measure equivalent, then $\Gamma_G^e$ and $\Gamma_H^e$ are isomorphic.)

*Proof.* Let $\mathcal{C}_G$ be the set of conjugates of vertex groups of $G$, and let $\mathcal{I}_G = \Gamma_G^e$ be the extension graph of $G$. Define $\mathcal{C}_H$ and $\mathcal{I}_H$ in the same way. The Vertex Recognition Property, together with the recognition of adjacency given by Lemma 5.4, ensure that $(\mathcal{I}_G, \mathcal{I}_H)$ is an ME-witness for $G, H$ (see Definition 4.22). The conclusion thus follows from Proposition 4.24. $\square$
In the sequel, it will be useful to have a version of the above lemma at the level of the right-angled buildings of the graph products (see Section 2.3 for definitions). Similarly to Isom($\Gamma_G^e, \Gamma_H^e$), we will always equip Isom($\mathbb{D}_G, \mathbb{D}_H$) with the compact-open topology, and with the action of $G \times H$ by pre- and post-composition.

**Corollary 5.6.** Let $\mathcal{C}$ be a class of graph products that satisfies the Vertex Recognition Property. Let $G, H \in \mathcal{C}$.

For every measure equivalence coupling $\Omega$ between $G$ and $H$, there exists a $(G \times H)$-equivariant Borel map $\Omega \to \text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$.

*Proof.* This is a consequence of Lemma 5.5, and the existence of a $(G \times H)$-equivariant Borel map $\text{Isom}(\Gamma_G^e, \Gamma_H^e) \to \text{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ (Proposition 2.15). $\square$

We now deduce another consequence of Lemma 5.4. We mention that this will not be used to prove Theorem 1.2, but will allow us to obtain classification results beyond the case covered by the latter theorem (see for example Section 10) as well as quantitative statements (see Section 12).

**Corollary 5.7.** Let $\mathcal{C}$ be a class of graph products that satisfies the Vertex Recognition Property. Let $G, H \in \mathcal{C}$ defined over $\Gamma_G$ and $\Gamma_H$ respectively.

If $G$ and $H$ are measure equivalent, then there exists a graph homomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for all $v \in V\Gamma_G$, the group $G_v$ is measure equivalent to $H_{\sigma(v)}$.

The proof uses the Vertex Recognition Property to define the wanted map $\sigma$, and Lemma 5.4 to show that the latter is a graph homomorphism.

*Proof.* Let $\Omega$ be a measure equivalence coupling between $G$ and $H$. As in Examples 4.2 and 4.3, let $X \subseteq \Omega$ be a Borel subset of finite positive measure, and $\mathcal{G}$ be a measured groupoid over $X$, coming with action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$.

* Let us first show that there exists a Borel subset $U \subseteq X$ of positive measure such that for all $v \in V\Gamma_G$ there exist $h_v \in H$ and $\sigma(v) \in V\Gamma_H$ such that

$$ \rho_G^{-1}(G_v)|_U = \rho_H^{-1}(h_v H_{\sigma(v)} h_v^{-1})|_U. $$

Since $\mathcal{C}$ has the Vertex Recognition Property, for all $v \in V\Gamma_G$ there exist a countable Borel partition $X^*(v) = \cup_{i \in I(v)} X_i(v)$ of a conull Borel subset $X^*(v) \subseteq X$ (depending a priori on $v$), and for every $i \in I(v)$, a parabolic subgroup $P_i(v) \subseteq H$ that is conjugate to a vertex group, such that

$$ \rho_G^{-1}(G_v)|_{X_i(v)} = \rho_H^{-1}(P_i(v))|_{X_i(v)}. \qquad (5.1) $$

Since $V\Gamma_G$ is finite, up to taking the common refinement of all the partitions $(X_i(v))_i$ and the intersection of all the subsets $X^*(v)$, we can obtain a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^*$ such that eq. (5.1) is verified and that does not depend on $v$. Remark that since $X^*$ has full measure, there thus exists $i \in I$ such that $\mu(X_i) > 0$. For such an $i \in I$, for all $v \in V\Gamma_G$ there exist (by eq. (5.1)) an element $h_v \in H$ and $\sigma(v) \in V\Gamma_H$ such that

$$ \rho_G^{-1}(G_v)|_{X_i} = \rho_H^{-1}(h_v H_{\sigma(v)} h_v^{-1})|_{X_i}. \qquad (5.2) $$
Taking $U = X_i$ thus leads to the wanted assertion.

* The above eq. (5.1) moreover implies that $G_v$ and $h_v H_{\sigma(v)} h_v^{-1}$ are measure equiv- alent for all $v \in V\Gamma_G$ in view of Lemma 3.3, page 35 and Remark 4.4, page 40. We can thus define a measure equivalence between $G_v$ and $H_{\sigma(v)}$ for all $v \in V\Gamma_G$.

* Let us now prove that $\sigma$ is a graph homomorphism.

If $v, w \in V\Gamma_G$ are adjacent in $\Gamma_G$, then $G_v$ and $G_w$ are distinct and commute and therefore by Lemma 5.4 and eq. (5.2), $h_v H_{\sigma(v)} h_v^{-1}$ and $h_w H_{\sigma(w)} h_w^{-1}$ are also distinct and commute. In particular Lemma 2.8, page 22 implies that $\sigma(v)$ and $\sigma(w)$ are adjacent in $\Gamma_G$. Hence $\sigma$ is a graph homomorphism. $\square$

**Remark 5.8.** The class of groups covered by Corollary 5.7 is larger than those covered by our main theorem (Theorem 1.2): indeed, it applies to all (direct products of) graph products of countably infinite groups over transvection-free strongly reduced<sup>3</sup> graphs (not reduced to a vertex), without the necessity to require that the graphs have no partial conjugation (see Proposition 8.1, page 90).

However, the conclusion is weaker in two ways. First, the homomorphism $\sigma$ need not be an isomorphism. Second, we only get a measure equivalence between the vertex groups, not an orbit equivalence.

In fact, the conclusion of Corollary 5.7 cannot be strengthened (even if we had assumed $G$ and $H$ to be orbit equivalent to start with). Here is an example, illustrated in Figure 9. The groups $G_1$ and $G_2$ are two index 2 subgroups of $G$. Indeed $G_1$ is isomorphic to the kernel of the homomorphism $G \to \mathbb{Z}/2\mathbb{Z}$, sending the first generator $a$ of $G_v \simeq F_2$ to 1 and the second generator $b$ to 0, and sending all other vertex groups to $\{0\}$. And $G_2$ is isomorphic to the kernel of the homomorphism $G \to \mathbb{Z}/2\mathbb{Z}$ sending the generator of $G_w \simeq \mathbb{Z}$ to 1, and sending all other vertex groups to $\{0\}$. That $G_1$ and $G_2$ are indeed as described can be checked, for instance, by using the Reidemeister–Schreier method to obtain a presentation of a finite-index subgroup of $G$, see e.g. [LS77, Proposition II.4.1].

As a consequence of the second point of Lemma 3.2, page 33, being subgroups of the same finite index in $G$, the groups $G_1$ and $G_2$ are orbit equivalent. And they satisfy the conclusion of Corollary 5.7, however the homomorphism $\sigma$ that appear cannot be taken to be an isomorphism, and the vertex groups $F_2$ and $F_3$ are measure equivalent, but not orbit equivalent [Gab00].

In Example 10.2, we will further illustrate Corollary 5.7 by obtaining new examples of right-angled Artin groups that can be distinguished in measure equivalence.

## 5.3 Proof of the measure equivalence classification theorem

We now turn to the proof of Theorem 1.2 and start with the following statement.

**Proposition 5.9.** Let $G$ and $H$ be graph products of countably infinite vertex groups over finite simple graphs with no transvection and no partial conjugation, not reduced to one vertex.

Then every measure equivalence coupling $\Omega$ between $G$ and $H$ is an orbit equivalence coupling between $G$ and $H$, i.e. $G$ and $H$ have a common fundamental domain on $\Omega$.

<footnote n="3">³see Definition 6.1, page 62.</footnote>
<figure><img src="image_12.png" /><figcaption>Figure 9: Groups from Remark 5.8</figcaption></figure>

*Proof.* First note that by Proposition 5.3, the groups $G$ and $H$ belong to a same class $C$ that verifies the Vertex Recognition Property.

By Corollary 5.6, there exists a $(G \times H)$-equivariant Borel map $\Omega \to \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$.

Recall from Corollary 2.14 that every isomorphism from $\mathbb{D}_G$ to $\mathbb{D}_H$ sends rank 0 vertices to rank 0 vertices. Additionally the respective actions of $G$ and $H$ on rank 0 vertices of $\mathbb{D}_G$ and $\mathbb{D}_H$ are transitive. Therefore, identifying the neutral elements $e_G$ and $e_H$ with the associated rank 0 vertices of $\mathbb{D}_G$ and $\mathbb{D}_H$, we deduce that

$$Y := \{f \in \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H) : f(e_G) = e_H\}$$

is a fundamental domain for both the $G$- and the $H$-actions on $\mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$. Hence, by equivariance of $\Theta$, the set $X := \Theta^{-1}(Y)$ is a common Borel fundamental domain for the actions of $G$ and $H$ on $\Omega$. This shows that $\Omega$ is an orbit equivalence coupling. $\square$

We now prove the following statement which shows the implication $1 \Rightarrow 3$ of our main theorem.

**Proposition 5.10.** Let $G, H$ be graph products of countably infinite vertex groups over finite simple graphs $\Gamma_G, \Gamma_H$ with no transvection and no partial conjugation, not reduced to one vertex. Let $(\Omega, m)$ be a measure equivalence coupling between $G$ and $H$.

Then there exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, there exists a $(G_v \times H_{\sigma(v)})$-invariant Borel subset $\Omega_v \subseteq \Omega$, which is an orbit equivalence coupling between $G_v$ and $H_{\sigma(v)}$.

**Remark 5.11.** We will see that the coupling we construct satisfies $m(g\Omega_v \cap \Omega_v) = 0$ for every $g \in (G \setminus G_v) \cup (H \setminus H_{\sigma(v)})$. In particular, every Borel fundamental domain for the $G_v$-action (resp. $H_{\sigma(v)}$-action) on $\Omega_v$ is contained in a Borel fundamental domain for the $G$-action (resp. $H$-action) on $\Omega$.

*Proof.* Let $\Omega$ be a measure equivalence coupling between $G$ and $H$. Proposition 5.9 implies that $\Omega$ is in fact an orbit equivalence coupling. As $G$ and $H$ are measure equivalent, the extension graphs $\Gamma_G^e$ and $\Gamma_H^e$ are isomorphic (Lemma 5.5). As the graphs $\Gamma_G$ and $\Gamma_H$ are transvection-free and have no partial conjugations, it follows from [Hua17, Corollary 4.16 and Lemma 4.17] that $\Gamma_G$ and $\Gamma_H$ are isomorphic – so $\mathrm{Isom}(\Gamma_G, \Gamma_H)$ is non-empty.
As in the previous proof, we identify the neutral elements $e_G, e_H$ with rank 0 vertices of $\mathbb{D}_G, \mathbb{D}_H$, and recall that the set

$$Y := \{f \in \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H) : f(e_G) = e_H\}$$

is a fundamental domain for both the G- and the H-action on $\mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$. We also identify every vertex $v \in V\Gamma_G$ (resp. $w \in V\Gamma_H$) with the rank 1 vertex of $\mathbb{D}_G$ (resp. $\mathbb{D}_H$) corresponding to the coset $G_v$ (resp. $H_w$). Now, for all $\sigma \in \mathrm{Isom}(\Gamma_G, \Gamma_H)$ and all $v \in V\Gamma_G$ let

$$Y_{\sigma}^{v} := \{f \in \mathrm{Isom}(\mathbb{D}_{G}, \mathbb{D}_{H}) : f(v) = \sigma(v), f(e_{G}) = e_{H}\},$$ 
$$\mathrm{Isom}_{v \to \sigma(v)}(\mathbb{D}_{G}, \mathbb{D}_{H}) := \{f \in \mathrm{Isom}(\mathbb{D}_{G}, \mathbb{D}_{H}) : f(v) = \sigma(v)\},$$ 
$$\Omega_{\sigma}^{v} := \Theta^{-1}(\mathrm{Isom}_{v \to \sigma(v)}(\mathbb{D}_{G}, \mathbb{D}_{H})),$$ 
$$X_{\sigma}^{v} := \Theta^{-1}(Y_{\sigma}^{v}).$$

We claim that there exists $\sigma_0 \in \mathrm{Isom}(\Gamma_G, \Gamma_H)$ such that for all $v \in V\Gamma_G$, one has $m(X_{\sigma_0}^v) > 0$. Indeed, for $\sigma \in \mathrm{Isom}(\Gamma_G, \Gamma_H)$, let

$$Y_{\sigma} := \{f \in \mathrm{Isom}(\mathbb{D}_{G}, \mathbb{D}_{H}) : f(v) = \sigma(v), \forall v \in V\Gamma_{G}\}.$$

Observe that $Y = \sqcup_{\sigma} Y_{\sigma}$. Indeed, we have $\sqcup_{\sigma} Y_{\sigma} \subseteq Y$ because $e_G$ (resp. $e_H$) is the unique rank 0 vertex adjacent to all rank 1 vertices corresponding to cosets of the form $G_v$ (resp. $H_w$). For the reverse inclusion, notice that any isomorphism $f \in \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ induces a bijection $\sigma$ from the set of rank 1 vertices adjacent to $e_G$, to the set of rank 1 vertices adjacent to $f(e_G)$. Now if $f$ belongs to $Y$, namely $f(e_G) = e_H$, we view $\sigma$ as a bijection from $V\Gamma_G$ to $V\Gamma_H$. Observe that this bijection preserves adjacency and non-adjacency: this follows from the fact that $f$ preserves ranks of vertices (Corollary 2.14), after observing that two rank one vertices $v_1, v_2$ correspond to adjacent vertices of $\Gamma_G$ if and only if they are adjacent to a common rank 2 vertex in $\mathbb{D}_G$. This proves that $Y = \sqcup_{\sigma} Y_{\sigma}$.

Since $Y$ is a fundamental domain for the G- and H-actions on $\mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ and $\Theta$ is equivariant, the Borel set $\Theta^{-1}(Y)$ is a fundamental domain for the G- and H-actions on $\Omega$. Therefore, there exists $\sigma_0 \in \mathrm{Isom}(\Gamma_G, \Gamma_H)$ such that $m(\Theta^{-1}(Y_{\sigma_0})) > 0$. Noting that for all $v \in V\Gamma_G$ we have $Y_{\sigma_0} \subset Y_{\sigma_0}^v$, we obtain that $m(X_{\sigma_0}^v) > 0$ for all $v \in V\Gamma_G$, which proves our claim.

Notice that for every $v \in V\Gamma_G$, the set $\Omega_{\sigma_0}^v$ is invariant under $G_v \times H_{\sigma_0(v)}$. Let us now prove that $X_{\sigma_0}^v$ is a common fundamental domain for the actions of $G_v$ and $H_{\sigma_0(v)}$ on $\Omega_v := \Omega_{\sigma_0}^v$. By equivariance of $\Theta$, it is enough to verify that $Y_{\sigma_0}^v$ is a fundamental domain for both actions on $\mathrm{Isom}_{v \to \sigma_0(v)}(\mathbb{D}_G, \mathbb{D}_H)$. First, since the action of $H$ on the set of rank 0 vertices of $\mathbb{D}_H$ is free, the $H_{\sigma_0(v)}$-translates of $Y_{\sigma_0}^v$ are pairwise disjoint (the translate of $Y_{\sigma_0}^v$ by an element $h$ consists of isomorphisms $f \in \mathrm{Isom}(\mathbb{D}_G, \mathbb{D}_H)$ sending $e_G$ to $h$). And likewise for their $G_v$-translates. Second, pick $f \in \mathrm{Isom}_{v \to \sigma_0(v)}(\mathbb{D}_G, \mathbb{D}_H)$, then $f(v) = \sigma_0(v)$. Since $e_G$ and $v$ are adjacent (as vertices of $\mathbb{D}_G$), so are $f(e_G)$ and $f(v) = \sigma_0(v)$. Therefore $f(e_G)$ being of rank 0, it corresponds to an element – denoted by $h$ – of $H_{\sigma_0(v)}$ (see Figure 10, page 59). Hence $h^{-1} \cdot f$ belongs to $Y_{\sigma_0}^v$ and $Y_{\sigma_0}^v$ is thus a
<figure><img src="image_13.png" /><figcaption>Figure 10: Illustration of the end of the proof of Proposition 5.10 for $\Gamma_G$ and $\Gamma_H$ pentagons</figcaption></figure>

fundamental domain for the action of $H_{\sigma_0(v)}$. Similarly, for the $G_v$-action, since $e_H$ and $\sigma_0(v)$ are adjacent so are $f^{-1}(e_H)$ and $f^{-1}(\sigma_0(v)) = v$. Therefore $f^{-1}(e_H)$ is an element of $G_v$ (see again Figure 10) and as above, we get that $Y_{\sigma_0}^v$ is a fundamental domain for the $G_v$-action.

Thus, for all $v \in V\Gamma_G$, the subspace $\Omega_v$ is an orbit equivalence coupling between $G_v$ and $H_{\sigma_0(v)}$ (and by construction it also satisfies the requirement from Remark 5.11). $\square$

We can finally complete the proof of our main measure equivalence classification theorem.

*Proof of Theorem 1.2.* The implication $2 \Rightarrow 1$ is immediate and $3 \Rightarrow 2$ comes from [HH22a, Proposition 4.2]. Finally $1 \Rightarrow 3$ is the contents of Proposition 5.10. $\square$

## 5.4 Applications: Commensurability and von Neumann algebras

We now prove Theorem 1.6 regarding the commensurability classification. We also give applications to fundamental groups and von Neumann algebras.

### 5.4.1 Application to commensurability

By restricting to discrete measure equivalence couplings (i.e. $\Omega$ is countable), we complete our proof of the commensurability classification theorem that we recall below.

**Theorem 1.6.** Let $\Gamma_G, \Gamma_H$ be two finite simple graphs, not reduced to one vertex, with no transvection and no partial conjugation. Let $G, H$ be graph products of countably infinite groups over $\Gamma_G, \Gamma_H$, respectively. Then the following assertions are equivalent.

1. The groups $G$ and $H$ are commensurable.

2. The groups $G$ and $H$ are strongly commensurable.

3. There exists a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for every $v \in V\Gamma_G$, the groups $G_v$ and $H_{\sigma(v)}$ are strongly commensurable.

*Proof.* The implication $2 \Rightarrow 1$ is obvious, and the implication $3 \Rightarrow 2$ was proved by Januszkiewicz-Świątkowski [JS01, Theorem 1]. For $1 \Rightarrow 3$, since $G$ and $H$ are commensurable, there exists a discrete measure equivalence coupling $\Omega$ between $G$ and $H$ (Lemma 3.2). Proposition 5.10 thus yields a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$ such
that for every $v \in V\Gamma_G$, there exists a $(G_v \times H_{\sigma(v)})$-invariant subset $\Omega_v \subseteq \Omega$ which is a (discrete) orbit equivalence coupling between $G_v$ and $H_{\sigma(v)}$, i.e. $G_v$ and $H_{\sigma(v)}$ have a common (finite) fundamental domain. This exactly says that $G_v$ and $H_{\sigma(v)}$ are strongly commensurable in view of Lemma 3.2. $\square$

**Remark 5.12.** Similarly as in Corollary 5.7 we can also show that if $G$ and $H$ are commensurable and belong to a same class of graph products having the Vertex Recognition Property, then there exists a graph homomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that $G_v$ is commensurable to $H_{\sigma(v)}$ for all $v \in V\Gamma_G$.

**Remark 5.13.** By restricting to countable couplings $\Omega$ on which the actions of $G$ and $H$ are free and transitive, we can derive that the same conclusion as in Theorem 1.6 holds after replacing “commensurable” and “strongly commensurable” by “isomorphic”. However, for isomorphisms, the statement can be made more general, in particular vertex groups do not need to be assumed countably infinite, and we can consider more general classes of graphs $\Gamma_G, \Gamma_H$. For this reason, we postpone the proof of the analogous isomorphic classification theorem to the appendix of this work (Corollary A.2).

**Remark 5.14.** Another algebraic consequence of Proposition 5.9 is the following. Let $G$ be a graph product with countably infinite vertex groups over a finite simple graph with no transvection and no partial conjugation (and not reduced to one vertex). Then any two isomorphic finite-index subgroups of $G$ have the same index. Indeed, if $G_1$ and $G_2$ are two isomorphic finite-index subgroups, then for every $i \in \{1, 2\}$, the coupling between $G$ and $G_i$ by left-right multiplication has coupling index $[G : G_i]$. By composition of couplings, one can construct a self measure equivalence coupling of $G$ whose coupling index equals $[G : G_2]/[G : G_1]$, see e.g. Item c on page 300 of [Fur11]. Proposition 5.9 therefore implies that $[G : G_1] = [G : G_2]$.

This phenomenon (any two isomorphic finite index subgroups have the same index) was called *finite index rigidity* by Lazarovich, who established it for all non-elementary hyperbolic groups [Laz23]. Finite index rigidity also occurs for all groups having at least one positive $\ell^2$-Betti number, since these are multiplicative with the index. In the case of graph products, $\ell^2$-Betti numbers were computed by Davis-Okun [DO12], so in many cases the statement was known (though our work does cover new cases).

### 5.4.2 Fundamental groups and von Neumann algebras

Recall that to every free, ergodic, probability measure-preserving action $G \curvearrowleft X$ of a countable group $G$ on a standard probability space $X$, one associates a von Neumann algebra $L(G \curvearrowleft X)$, in fact a $\Pi_1$ factor, via Murray and von Neumann’s group measure space construction [MvN36].

The *fundamental group* of an ergodic measured equivalence relation $\mathcal{R}$ over a standard probability space $X$ (resp. a $\Pi_1$ factor $M$) is the subgroup of $\mathbb{R}_+^*$ consisting of all $t > 0$ such that the amplification $\mathcal{R}^t$ (resp. $M^t$) is isomorphic to $\mathcal{R}$ (resp. $M$). In particular $t < 1$ belongs to the fundamental group of $\mathcal{R}$ if for any Borel subset $U \subseteq X$ of measure $t$, the restriction $\mathcal{R}|_U$ is isomorphic to $\mathcal{R}$ (up to rescaling the measure). And if $L(\mathcal{R})$ is the $\Pi_1$ factor associated to $\mathcal{R}$, then $t$ belongs to the fundamental group of $L(\mathcal{R})$ if for any $U$ as above, $L(\mathcal{R}|_U)$ is isomorphic to $L(\mathcal{R})$.
This notion was introduced by Murray and von Neumann in [MvN43] as an invariant to distinguish between $\Pi_1$ factors, and they proved that the fundamental group of the hyperfinite $\Pi_1$ factor is $\mathbb{R}_+^*$, but besides this example no computations were available for an extended period of time. Connes famously proved that the fundamental group of any $\Pi_1$ factor with Property (T) is countable [Con80]. It is only much later than the first example of a $\Pi_1$ factor with trivial fundamental group was exhibited by Popa [Pop06a], solving a long-standing open problem of Kadison, in the framework of his deformation/rigidity theory.

Theorem 1.2, combined with a Cartan rigidity theorem established for graph products by Chifan and Kunnawalkam Elayavalli [CKE21, Theorem 1.3] in the framework of Popa's deformation/rigidity theory, yields the following corollary.

**Corollary 5.15.** Let $G$ be a graph product of countably infinite vertex groups over a finite simple graph $\Gamma$ with no transvection and no partial conjugation. Let $G \stackrel{\cdot}{\to} X$ be a free, ergodic, measure-preserving action of $G$ by Borel automorphisms on a standard probability space $X$. Let $\mathcal{R}(G \stackrel{\cdot}{\to} X)$ be the associated orbit equivalence relation, and $L(G \stackrel{\cdot}{\to} X)$ be the associated von Neumann algebra.

Then $\mathcal{R}(G \stackrel{\cdot}{\to} X)$ and $L(G \stackrel{\cdot}{\to} X)$ have trivial fundamental groups.

*Proof.* Let $\mathcal{R} = \mathcal{R}(G \stackrel{\cdot}{\to} X)$. We first prove that the fundamental group of $\mathcal{R}$ is trivial. So let $t \le 1$ such that $\mathcal{R}$ and $\mathcal{R}^t$ are isomorphic. This yields a stable orbit equivalence with compression constant $t$ between two free, ergodic, measure-preserving actions of $G$ on standard probability spaces. By [Fur99b, Theorem 3.3], this in turn gives a self measure equivalence coupling $\Omega$ of $G$ such that the ratio of the fundamental domains of the two actions of $G$ is equal to $t$. Proposition 5.9 implies that $t = 1$ shows that the fundamental group of $\mathcal{R}(G \stackrel{\cdot}{\to} X)$ is trivial.

We now prove that the fundamental group of $L(G \stackrel{\cdot}{\to} X) = L(\mathcal{R})$ is trivial. Let $t \le 1$, and assume that $L(\mathcal{R})^t = L(\mathcal{R}^t)$ is isomorphic to $L(\mathcal{R})$. Let $U \subseteq X$ be a Borel subset of measure $t$, so that $\mathcal{R}^t$ is isomorphic to the restriction of $\mathcal{R}$ to $U$. By [CKE21, Theorem 1.3], up to unitary conjugacy, $L^\infty(X)$ is the unique Cartan subalgebra of $L(\mathcal{R})$. Therefore any isomorphism from $L(\mathcal{R})$ to $L(\mathcal{R}^t)$ can be unitarily conjugated so as to send $L^\infty(X)$ to $L^\infty(U)$. It thus follows from [FM77] that $\mathcal{R}$ and $\mathcal{R}^t$ are isomorphic, and by the above $t = 1$. □

# 6 Combinatorial and groupoid-theoretic tools

The proof of the Vertex Recognition Property (Proposition 5.3) occupies the next two sections. This section contains preparatory tools.

## 6.1 Combinatorial tools

This subsection contains several combinatorial tools that will be crucial in the proof of the Vertex Recognition Property. First, in Section 6.1.1, we introduce the notion of a strongly reduced graph, and show that every finite simple graph with no transvection and no partial conjugation is either strongly reduced or splits non-trivially as a join (Lemma 6.3 below). This will allow us to decompose the proof of the Vertex Recognition Property into these two cases, that will be treated separately in Sections 7 and 8 below.