Second, in Section 6.1.3, we explain how to recognize conjugates of untransverse vertex groups inside a graph product, by iteratively taking maximal direct products and passing to factors. This is the contents of Lemma 6.6 below. This lemma will be a starting point and a guide in the next two sections: proving the Vertex Recognition Property will indeed require establishing a groupoid-theoretic version of Lemma 6.6. Before this, in Section 6.1.2, we provide general definitions and facts regarding products and factors in graph products.

### 6.1.1 Strongly reduced graphs

The following combinatorial condition will be crucial in the present work.

**Definition 6.1 (Strongly reduced graph).** Let $\Gamma$ be a finite simple graph. An induced subgraph $\Lambda \subseteq \Gamma$ is *collapsible* if for any $x, y \in V\Lambda$, one has $\mathrm{lk}(x) \cap (\Gamma\setminus\Lambda) = \mathrm{lk}(y) \cap (\Gamma\setminus\Lambda)$. We refer to Figure 11, page 62 for an illustration.

A graph $\Gamma$ is *strongly reduced* if it does not contain any proper induced subgraph on at least two vertices which is collapsible.

Note that if $\Lambda \subseteq \Gamma$ is collapsible, then every graph product over $\Gamma$ is also a graph product over the graph obtained from $\Gamma$ by collapsing $\Lambda$ to a single vertex $v_\Lambda$, and joining $v_\Lambda$ by an edge to every vertex in the common link of all vertices of $\Lambda$ (see the two graphs $\Gamma$ and $\Gamma'$ of Example 6.2, page 62). Every graph product over a finite simple graph $\Gamma$ which is not reduced to one vertex, can therefore be represented as a graph product over a strongly reduced finite simple graph $\bar{\Gamma}$ not reduced to one vertex.

**Example 6.2.**

* If $\Gamma$ is a join then we can collapse it to an edge (which is strongly reduced).

* If $\Gamma$ is disconnected, then we can collapse it to a disjoint union of two vertices (which is strongly reduced).

* As a more interesting example, the subgraph $\Lambda$ of $\Gamma$ represented in Figure 11 is collapsible. Indeed the set $\mathrm{lk}(x) \cap (\Gamma\setminus\Lambda)$ – represented by the blue vertices in the leftmost picture – coincide with $\mathrm{lk}(y) \cap (\Gamma\setminus\Lambda)$. Collapsing $\Lambda$ into $\bar{\Lambda}$ leads to the graph $\Gamma'$ in the middle of Figure 11. The corresponding strongly reduced finite graph is the pentagon $\bar{\Gamma}$ represented on the right most side.

<figure><img src="image_14.png" /><figcaption>Figure 11: Collapsible and strongly reduced graphs (Example 6.2)</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>The graph Γ on the left is irreducible but contains a collapsible subgraph Λ (in orange). Removing the star of the vertex v (drawn in grey on the right most graph) produces a graph with two different connected components.</figcaption></figure>

<caption>Figure 12: Example of an irreducible graph Γ with partial conjugation</caption>

**Lemma 6.3.** Let Γ be a finite simple graph. If Γ is transvection-free and has no partial conjugation, then either Γ is strongly reduced or Γ is a join.

The idea behind the lemma is the following: if a finite simple graph Γ is irreducible and contains a collapsible subgraph Λ, then either Λ is a clique and in this case Γ has transvections, or Λ is not a clique but in this case, there exists $v \in V\Lambda$ such that $\Gamma\star(v)$ is disconnected (see also Figure 12).

*Proof*. Let us assume that Γ is irreducible and show that Γ is strongly reduced. So assume towards a contradiction that there exists a proper collapsible subgraph Λ ⊆ Γ on at least 2 vertices. Let $w \in V\Lambda$. We observe that Λ ≠ star(w). Otherwise, as Λ is collapsible, for every $v \in V\Lambda$, we would have lk($v$) ⊆ star(w), and as $|V\Lambda| \ge 2$ this would give a transvection in Γ. We can thus choose $u \in V\Lambda\star(w)$.

Let us show that there exists a vertex $u' \in V\Gamma\setminus V\Lambda$ which is not contained in star($w$). By collapsibility of Λ we have star($w$) ⊆ Λ ∘ Λ⊥, therefore Λ ∪ star($w$) = Λ ∘ Λ⊥. But since Λ is proper and Γ assumed to be irreducible, then Λ ∪ star($w$) is a proper subgraph of Γ. Whence the existence of $u'$, as desired.

We claim that star($w$) disconnects $u$ from $u'$, which will contradict the fact that Γ has no partial conjugation – and thus complete our proof. Indeed, let $u = u_1, \dots, u_k = u'$ be a path (i.e. any two consecutive vertices $u_i, u_{i+1}$ are adjacent), and let us prove that one of the vertices along this path is contained in star($w$). Let $i \in \{1, \dots, k-1\}$ be such that $u_i \in V\Lambda$ and $u_{i+1} \notin V\Lambda$. Then $u_{i+1}$ is in lk($u_i$), so by collapsibility it is also in lk($w$), as desired. Hence star($w$) disconnects $u$ from $u'$. □

In the sequel, we will use Lemma 6.3 in the form of the following corollary.
**Corollary 6.4.** Let $\Gamma$ be a finite simple graph with no transvection and no partial conjugation, not reduced to a vertex.

Then $\Gamma$ splits as $\Gamma = \Gamma_1 \circ \cdots \circ \Gamma_k$ (possibly with $k=1$), in such a way that every $\Gamma_j$ is strongly reduced, transvection-free, and not reduced to a vertex.

*Proof.* Write $\Gamma = \Gamma_0 \circ \Gamma_1 \circ \cdots \circ \Gamma_k$, where $\Gamma_0$ is the clique factor of $\Gamma$, and each $\Gamma_j$ with $j \ge 1$ has at least two vertices and does not split non-trivially as a join.

Since $\Gamma$ is transvection-free and not reduced to a vertex, one has $\Gamma_0 = \emptyset$.

In view of Remark 2.1, for every $j \in \{1, \dots, k\}$, the graph $\Gamma_j$ is transvection-free and has no partial conjugation. Therefore Lemma 6.3 shows that $\Gamma_j$ is strongly reduced, which completes our proof. $\square$

### 6.1.2 Products and factors

Let $G$ be a graph product of countably infinite vertex groups over a finite simple graph. A product parabolic subgroup $P \subseteq G$ is a parabolic subgroup which splits as a direct product of two non-trivial parabolic subgroups. We say that $P$ is a maximal product parabolic subgroup if it is maximal for inclusion (among product parabolic subgroups). We say that $P$ is of isolated clique type if $\Gamma$ has a connected component $C$ which is a clique, and $P$ is conjugate to $G_C$.

**Lemma 6.5.** Let $G$ be a graph product with countably infinite vertex groups over a finite simple graph $\Gamma$. Let $P$ be a maximal product parabolic subgroup, and write $P = P_1 \times P_2$, where $P_1$ and $P_2$ are non-trivial parabolic subgroups.

Then either $P$ is of isolated clique type, or else $P_1$ or $P_2$ contains a non-abelian free subgroup.

*Proof.* We first assume that the type $\Lambda$ of $P$ is not a clique. We can assume without loss of generality that the type $\Lambda_1$ of $P_1$ contains two non-adjacent vertices $v, w$. Being infinite, the groups $G_v$ and $G_w$ both contain either an infinite order element, or else a finite subgroup of order at least 3. The free product $G_v * G_w$ (which has a conjugate contained in $P_1$) therefore contains a non-abelian free subgroup.

We now assume that the type $\Lambda$ of $P$ is a clique, and aim to prove that it is isolated. Otherwise, $\Lambda$ is a proper subgraph of a connected component of $\Gamma$. We can therefore find $v \in V\Lambda$ such that there exists a vertex $v' \in lk(v) \setminus V\Lambda$. Then $G_v \times G_v^{\perp}$ splits as a product of infinite parabolic subgroups and contains $P$ properly. This contradicts that $P$ is a maximal product parabolic subgroup. $\square$

Given a finite simple graph $\Gamma$, and an induced subgraph $\Lambda \subseteq \Gamma$, we define the *clique factor* of $\Lambda$ as the subgraph spanned by all vertices that are joined by an edge to every other vertex of $\Lambda$. If $G$ is a graph product over $\Gamma$, and if $P = gG_{\Lambda}g^{-1}$, we define the *clique factor* of $P$ as the parabolic subgroup $gG_{\Lambda_0}g^{-1}$, where $\Lambda_0$ is the clique factor of $\Lambda$.

Let $P$ be a parabolic subgroup, and write $P = C_0 \times F_1 \times \cdots \times F_n$, where $C_0$ is the clique factor of $P$, and no $F_j$ splits as a direct product. We say that a subgroup $F \subseteq Q$ is a *factor* if it is equal to $C_0$ or to one of the subgroups $F_j$ of the above decomposition.
### 6.1.3 Recognizing untrans vectorable vertices

Recall that given a finite simple graph $\Gamma$, a vertex $v \in V\Gamma$ is untrans vectorable if there does not exist any vertex $w \in V\Gamma$ distinct from $v$ such that $\mathrm{lk}(v) \subseteq \mathrm{star}(w)$. The goal of this section is to give a combinatorial/algebraic characterization of untrans vectorable vertices, which will be our starting point for proving the Vertex Recognition Property in later sections.

**Lemma 6.6.** Let $\Gamma$ be a finite simple graph, and let $G$ be a graph product over $\Gamma$. A vertex $v \in V\Gamma$ is untrans vectorable if and only if there exists a chain of parabolic subgroups

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

such that

1. for every $j \in \{1, \dots, n\}$, $P_j$ is a maximal product parabolic subgroup of $F_{j-1}$ and not of isolated clique type;

2. for every $j \in \{1, \dots, n-1\}$, the clique factor of $P_j$ is trivial, and $F_j$ is a factor of $P_j$, and

3. $F_n$ is the clique factor of $P_n$.

At first sight, it might not seem intuitive why the conditions given in the lemma characterize untrans vectorable vertices and their conjugates. One useful observation is that if there exist two vertices $v, w \in V\Gamma$ such that $\mathrm{lk}(v) \subseteq \mathrm{star}(w)$, then any maximal product containing $v$ will also contain $w$ (so our process cannot pinpoint a trans vectorable vertex). Before reading the proof, the reader is invited to work out the conditions on an example, see Example 6.7 below.

*Proof.* We first assume that $v$ is untrans vectorable, and construct parabolic subgroups $P_j, F_j$ as in the statement.

Starting from $F_0 = G$, our goal is to define inductively two finite sequences of parabolic subgroups $(P_j)_j = (G_{\Lambda_j})_j$ and $(F_j)_j = (G_{\Upsilon_j})_j$, such that if we have defined $P_j$ and $F_j$ for some $j \in \mathbb{N}$ then

1. Either $F_j = G_v$ and we stop.

2. Or else

(a) We can choose $P_{j+1}$ a maximal product parabolic subgroup of $F_j$ containing $G_{\mathrm{star}(v)} \cap F_j$ which is not of isolated clique type;

(b) And either $G_v$ is the clique factor of $P_{j+1}$ or $P_{j+1}$ has trivial clique factor;

(c) And we denote by $F_{j+1}$ the factor of $P_{j+1}$ containing $G_v$;

(d) Either $F_{j+1} = G_v$, or $v$ is untrans vectorable in $\Upsilon_{j+1}$ and in particular it is not contained in an isolated clique of $\Upsilon_{j+1}$.

The process has to stop eventually as we are constructing a chain of proper inclusions of parabolic subgroups (except perhaps $F_0 = P_1$ if $F_0$ splits as a product).

So let $j \in \mathbb{N}$ and assume that we have defined $F_j$ verifying that $v$ is untrans vectorable in $\Upsilon_j$. If $F_j = G_v$, we stop.

Now assume that $F_j \neq G_v$. In particular $v$ is not isolated in $F_j$, so we can choose a maximal product parabolic subgroup $P_{j+1}$ of $F_j$ that contains $G_{\mathrm{star}(v)} \cap F_j$. Without loss of generality, we will assume that $P_{j+1} = G_{\Lambda_{j+1}}$ for some induced subgraph $\Lambda_{j+1} \subseteq \Upsilon_j$.
Being an untrans vector of $\Upsilon_j$, the vertex $v$ cannot be contained in an isolated
clique of $\Upsilon_j$, so $P_{j+1}$ is not of isolated clique type. Using again that $v$ is untrans vector,
we get that either $P_{j+1} = G_{\text{star}(v)}$ and $G_v$ is the clique factor of $P_{j+1}$ (in which case
we define $F_{j+1} = G_v$), or else $P_{j+1}$ has trivial clique factor. We assume that the latter
holds, and let $F_{j+1}$ be the factor of $P_{j+1}$ that contains $G_v$.

Let $\Upsilon_{j+1} \subseteq \Lambda_{j+1}$ be the induced subgraph such that $F_{j+1} = G_{\Upsilon_{j+1}}$. Let $\Upsilon_{j+1}^{\perp}$ be the orthogonal of $\Upsilon_{j+1}$ inside $\Lambda_{j+1}$. We are left with showing that $v$ is an untrans vectorable vertex of $\Upsilon_{j+1}$. So let $w \in V\Upsilon_{j+1}$ be such that $\text{lk}_{\Upsilon_{j+1}}(v) \subseteq \text{star}_{\Upsilon_{j+1}}(w)$. We aim to prove that $\text{lk}_{\Upsilon_j}(v) \subseteq \text{star}_{\Upsilon_j}(w)$ which will imply $w = v$. As $G_{\text{star}(v)} \cap F_j \subseteq P_{j+1}$, we have

$$
\mathrm{star}_{\Upsilon_j}(v) := \mathrm{star}(v) \cap \Upsilon_j \subseteq \Lambda_{j+1}.
$$

Since $\Lambda_{j+1} \subseteq \Upsilon_j$ we thus have $\mathrm{star}_{\Lambda_{j+1}}(v) = \mathrm{star}_{\Upsilon_j}(v)$. Therefore, using that $P_{j+1}$ is a product, we have

$$
\mathrm{lk}_{\Upsilon_j}(v) = \mathrm{lk}_{\Lambda_{j+1}}(v) = \mathrm{lk}_{\Upsilon_{j+1}}(v) \circ \Upsilon_{j+1}^{\perp}.
$$

We deduce that

$$
\mathrm{lk}_{\Upsilon_j}(v) \subseteq \mathrm{star}_{\Upsilon_{j+1}}(w) \circ \Upsilon_{j+1}^{\perp} \subseteq \mathrm{star}_{\Upsilon_j}(w).
$$

As $v$ is untrans vectorable when viewed as a vertex of $\Upsilon_j$, it follows that $w = v$, as desired.

This finishes our inductive construction, and completes the proof of one implication
of the lemma.

Conversely, let us assume that there exist two sequences of parabolic subgroups $(F_j)_j$
and $(P_j)_j$ as in the statement. Up to a global conjugation, we can assume that all $F_j$
and $P_j$ are of the form $G_{\Upsilon_j}, G_{\Lambda_j}$ for some induced subgraphs $\Upsilon_j, \Lambda_j \subseteq \Gamma$.

Let $w \in V\Gamma$ be such that $\mathrm{lk}(w) \subseteq \mathrm{star}(w)$. We aim to prove that $w = v$.

We first prove inductively that $w \in V\Upsilon_j$ and $w \in V\Lambda_{j+1}$ for every $j \in \{0, \dots, n-1\}$,
as follows.

- First, $w \in V\Upsilon_0 = V\Gamma$.

* Assuming that $w \in V\Upsilon_j$, we prove that $w \in V\Lambda_{j+1}$. Indeed, assume towards a contradiction that $w \notin V\Lambda_{j+1}$. Write

$$
\Lambda_{j+1} = \Lambda_{j+1}^{1} \circ \cdots \circ \Lambda_{j+1}^{k}
$$

as a maximal join, with $v \in V\Lambda_{j+1}^1$ (namely, $\Lambda_{j+1}^1 = \Upsilon_{j+1}^1$). Since $\text{lk}(v) \subseteq \text{star}(w)$,
we have

$$
\Lambda_{j+1}^2 \circ \dots \circ \Lambda_{j+1}^k \subseteq \mathrm{star}(w).
$$

So let $\langle \Lambda_{j+1}^1, w \rangle$ be the subgraph of $\Upsilon_j$ induced by $\Lambda_{j+1}^1$ and $w$. Then

$$
\langle \Lambda_{j+1}^{1}, w \rangle \circ \Lambda_{j+1}^{2} \circ \dots \circ \Lambda_{j+1}^{k}
$$

is again a join, which properly contains $P_{j+1}$. This contradicts the maximality
of $P_{j+1}$.

* Assuming that $w \in V\Lambda_{j+1}$ with $j \in \{0, \dots, n-2\}$, we prove that $w \in V\Upsilon_{j+1}$.
Indeed, write $\Lambda_{j+1} = \Lambda_{j+1}^1 \circ \dots \circ \Lambda_{j+1}^k$ as a maximal join, with $v \in V\Lambda_{j+1}^1 = V\Upsilon_{j+1}$.
By contradiction, if $w \in V\Lambda_{j+1}^i$ with $i \ge 2$, then using that $\mathrm{lk}(v) \subseteq \mathrm{star}(w)$, we see
that $\Lambda_{j+1}^i \subseteq \mathrm{star}(w)$, and therefore $\Lambda_{j+1} \subseteq \mathrm{star}(w)$. This contradicts the fact that $P_{j+1}$ has trivial clique factor.

In particular we have proved that $w \in V\Lambda_n$. The same argument as in the last point shows that $w$ belongs to the clique factor of $P_n$. Since the clique factor of $P_n$ is $G_v$ we thus obtain that $w = v$. $\square$

<figure><img src="image_16.png" /><figcaption>Figure 13: Graph for Example 6.7</figcaption></figure>

**Example 6.7.** This example illustrates the statement of Lemma 6.6.

Consider $\Gamma$ to be the graph in Figure 13 and $G$ a graph product over $\Gamma$. The only untrans vector in $\Gamma$ is the one drawn in dark orange on the leftmost part of the graph. Let us describe the possible chains of parabolic subgroups in $G$ verifying the conditions of Lemma 6.6.

Up to conjugation, there are exactly two maximal product parabolic subgroups in $G$, so there are a priori two possible choices for $P_1$.

* If $P_1$ is taken to be a maximal product with type encompassed by the brown dots, then $P_1$ has a clique factor $F_1$ equal to a conjugate of a vertex group. This vertex group corresponds to the dark orange vertex, which is untrans vector. Hence here taking $F_1 = F_n$ gives the wanted sequence.

* On the other hand, if $P_1$ is taken to be the maximal product of $G$ with type encompassed by the orange dashes, then $P_1$ can be decomposed as $P_1 = F_1 \times F'_1$ where $F_1$ is the factor whose type is induced by the three blue vertices and $F'_1$ has type induced by the two (dark and light) orange vertices. Then $P_1$ has trivial clique factor and neither $F_1$ nor $F'_1$ contains a product parabolic subgroup. In particular, we can not find a maximal product $P_2$ in any of the factors of $P_1$ verifying point 1 of Lemma 6.6.

Hence, up to conjugation, the only possible sequence of parabolic subgroups verifying the conditions of Lemma 6.6 is $G = F_0 \supseteq P_1 \supseteq F_1 = F_n$, where $P_1$ is a maximal product parabolic subgroup as in the first point.

## 6.2 Groupoid-theoretic tools

We now establish a few statements regarding how to exploit the amenability of a groupoid $\mathcal{A}$ equipped with a cocycle $\rho$ towards a graph product, and in particular get information on the normalizer of $\mathcal{A}$. The arguments developed in this section are largely inspired by work of Adams [Ada94] regarding the indecomposability of equivalence relations generated by probability measure-preserving actions of Gromov hyperbolic groups. Adams's strategy was later developed by Kida in the context of mapping class groups [Kid08a] and has already found many applications towards measure equivalence classification/rigidity results in many contexts.
In Section 6.2.1, we introduce the notion of elementarity of $(\mathcal{A}, \rho)$, an analogue for groupoids to the elementarity of a group action on a tree (i.e. having a fixed point or a finite orbit at infinity). See e.g. [HH22b, Section 5.2] for a similar treatment in a slightly different context. This notion will only be used in Section 8.

Section 6.2.2 contains a lemma in a similar spirit, that will only be used in the proof of Lemma 7.4, and is inspired from the case of right-angled Artin groups [HH22a, Lemma 3.10].

## 6.2.1 Elementary groupoids with a cocycle towards a graph product

Let $\Gamma$ be a finite simple graph. For every $v \in V\Gamma$, let $T_v$ be the Bass–Serre tree of the splitting of $G$ as $G = G_{\text{star}(v)} *_{G_{\text{lk}(v)}} G_{\Gamma\setminus\{v\}}$, as in Section 2.2.2. We equip $\partial_{\infty}T_v \cup V(T_v)$ with the observers’ topology, for which connected components of complements of points in $T_v$ form a subbasis of open sets. For this topology $\partial_{\infty}T_v \cup V(T_v)$ is compact and metrizable, and $\partial_{\infty}T_v$ and $V(T_v)$ are Borel subsets. The induced topology on $\partial_{\infty}T_v$ is nothing but the visual topology.

Given a standard Borel space $Z$, we denote by $\mathcal{P}_{<\infty}(Z)$ the set of all non-empty finite subsets of $Z$, equipped with its natural Borel structure. We denote by $\mathcal{P}_{\le 2}(Z)$ the set of all non-empty subsets of $Z$ of cardinality at most 2.

We also recall from page 29 that $(\partial_{\infty}T_v)^{\text{reg}}$ is the set of elements in $\partial_{\infty}T_v$ having trivial elliptic stabilizer. We also refer to Definition 4.18, page 44 for the definition of properly $\mathbb{P}_G$-supported pairs.

**Definition 6.8 (Boundary) $G$-elementary.** Let $G$ be a graph product of countably infinite groups over a finite simple graph $\Gamma$, and let $\mathcal{G}$ be a measured groupoid over a standard probability space, equipped with a strict cocycle $\rho : \mathcal{G} \to G$.

We say that $(\mathcal{G}, \rho)$ is *boundary $G$-elementary* if for every $v \in V\Gamma$, there exists a stably $(\mathcal{G}, \rho)$-equivariant Borel map $X \to \mathcal{P}_{\le 2}((\partial_{\infty}T_v)^{\text{reg}})$.

We say that $(\mathcal{G}, \rho)$ is *$G$-elementary* if there exists a Borel partition $X = X_1 \sqcup X_2$ such that $(\mathcal{G}|_{X_1}, \rho)$ is properly $\mathbb{P}_G$-supported, and $(\mathcal{G}|_{X_2}, \rho)$ is boundary $G$-elementary.

**Remark 6.9.** When $(\mathcal{G}, \rho)$ is $G$-elementary, there exists a Borel partition $X = X'_1 \sqcup X'_2$ such that $(\mathcal{G}|_{X'_1}, \rho)$ is properly $\mathbb{P}_G$-supported and $(\mathcal{G}|_{X'_2}, \rho)$ is tightly $G$-supported and boundary $G$-elementary.

Here is a basic example of a non-elementary pair $(\mathcal{G}, \rho)$.

**Lemma 6.10.** Let $G$ be a graph product of countably infinite groups over a finite simple graph $\Gamma$ which is not a clique. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\rho : \mathcal{G} \to G$ be an action-like cocycle.

Then $(\mathcal{G}, \rho)$ is not $G$-elementary.

*Proof.* By Lemma 4.15, page 43, $(\rho^{-1}(G), \rho)$ is tightly $G$-supported, thus since $\mathcal{G} = \rho^{-1}(G)$ we obtain that for every positive measure Borel subset $U \subseteq X$, the pair $(\mathcal{G}|_U, \rho)$ is tightly $G$-supported. It is therefore enough to prove that $(\mathcal{G}, \rho)$ is not boundary $G$-elementary. So assume towards a contradiction that for every $v \in V\Gamma$, there exists a stably $(\mathcal{G}, \rho)$-equivariant Borel map $X \to \mathcal{P}_{\le 2}((\partial_{\infty}T_v)^{\text{reg}})$. By Lemma 2.20, page 29, the $G$-action on $(\partial_{\infty}T_v)^{\text{reg}}$ is Borel amenable, and therefore so is the $G$-action on $\mathcal{P}_{\le 2}((\partial_{\infty}T_v)^{\text{reg}})$.
As $\rho$ has trivial kernel, it follows from [GH21b, Proposition 3.38] (recasting [Kid08a, Proposition 4.33]) that $\rho^{-1}(G)$ is amenable. Since $\Gamma$ is not a clique, $G$ is not amenable, and we obtain a contradiction to the fact that $\rho$ is action-like (third point of Definition 4.9, page 42). $\square$

In contrast, Lemma 6.11 and Remark 6.12 below will show that any amenable groupoid is $G$-elementary, with respect to any cocycle.

Recall that whenever $K$ is a compact metrizable space, we equip $\text{Prob}(K)$ with the weak-* topology, coming from the duality with the space of continuous real-valued functions on $K$ given by the Riesz-Markov-Kakutani theorem. The reader is refered to [Kec95, Chapter 17] for a detailed analysis of the induced Borel structure on $\text{Prob}(K)$, which will justify the measurability of all maps appearing in the proof of the following lemma.

**Lemma 6.11.** Let $G$ be a graph product of countably infinite groups over a finite irreducible simple graph $\Gamma$ not reduced to a vertex. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be a cocycle.

If for all $v \in V\Gamma$ there exists a $(\mathcal{G}, \rho)$-equivariant Borel map $X \to \text{Prob}(V(T_v) \cup \partial_\infty T_v)$, then $(\mathcal{G}, \rho)$ is $G$-elementary.

**Remark 6.12.** In view of Proposition 4.8, page 41 the assumption of the lemma regarding the existence of a $(\mathcal{G}, \rho)$-equivariant Borel map $X \to \text{Prob}(V(T_v) \cup \partial_\infty T_v)$ is satisfied whenever $\mathcal{G}$ is amenable.

*Proof.* Let $X_1 \subseteq X$ be a Borel subset of maximal measure such that $(\mathcal{G}|_{X_1}, \rho)$ is properly $\mathbb{P}_G$-supported. Let $X_2 = X \setminus X_1$. Then $(\mathcal{G}|_{X_2}, \rho)$ is tightly $G$-supported. We will prove that $(\mathcal{G}|_{X_2}, \rho)$ is boundary $G$-elementary. Let $v \in V\Gamma$.

**Step 1** Let us prove that for all positive measure Borel subsets $U \subseteq X_2$, there does not exist any $(\mathcal{G}|_U, \rho)$-equivariant Borel map $U \to \text{Prob}(V(T_v))$.

Assume towards a contradiction that there exist a positive measure Borel subset $U \subseteq X_2$, and a $(\mathcal{G}|_U, \rho)$-equivariant Borel map $U \to \text{Prob}(V(T_v))$. We first build a $(\mathcal{G}|_U, \rho)$-equivariant Borel map from $U$ to $V(T_v)$, the construction is summed up in Figure 14.

As $V(T_v)$ is countable, there exists a $G$-equivariant Borel map $\text{Prob}(V(T_v)) \to \mathcal{P}_{<\infty}(V(T_v))$, sending a probability measure $\mu$ to the nonempty finite set consisting of all vertices with maximal $\mu$-measure. Using that every bounded set of a CAT(0) space has a unique circumcenter [BH99, Proposition II.2.7], we have a $G$-equivariant map $\mathcal{P}_{<\infty}(V(T_v)) \to T_v$. As the action of $G$ on $T_v$ is without edge inversion, there is also a $G$-equivariant map $T_v \to V(T_v)$: one can indeed choose a $G$-invariant orientation of the edges of $T_v$, send every vertex $v$ to itself, and every point in the interior of an edge $e$ to the origin of $e$. Altogether, we obtain a Borel $(\mathcal{G}|_U, \rho)$-equivariant map $U \to V(T_v)$.

Let now $U' \subseteq U$ be a positive measure Borel subset where this map is constant (this exists because $V(T_v)$ is countable). As no vertex of $\Gamma$ is joined to all other vertices, vertex stabilizers of $T_v$ are proper parabolic subgroups of $G$ (see page 21). Therefore $\rho(\mathcal{G}|_{U'})$ is contained in a proper parabolic subgroup of $G$, which contradicts the fact that $(\mathcal{G}|_{X_2}, \rho)$ is tightly $G$-supported. This completes Step 1.

By assumption we have a $(\mathcal{G}, \rho)$-equivariant Borel map $X \to \text{Prob}(V(T_v) \cup \partial_\infty T_v)$. Therefore, by Step 1, up to replacing $X_2$ by a conull Borel subset, we have a $(\mathcal{G}|_{X_2}, \rho)$-equivariant Borel map $\nu: X_2 \to \text{Prob}(\partial_\infty T_v)$.
<figure><img src="image_17.png" /><figcaption>Figure 14: Construction of the equivariant map from $U \subseteq X_2$ to $V(T_v)$ in Step 1 of the proof of Lemma 6.11.</figcaption></figure>

**Step 2** We prove that for almost every $x \in X_2$, the probability measure $\nu(x)$ gives full measure to $(\partial_\infty T_v)^{\text{reg}}$.

Let $(\partial_\infty T_v)^{\text{para}} := \partial_\infty T_v \setminus (\partial_\infty T_v)^{\text{reg}}$. Recall that there exists a $G$-equivariant Borel map $\theta_v : (\partial_\infty T_v)^{\text{para}} \to \mathbb{P}_G \setminus \{1\}$, where $\mathbb{P}_G$ is the set of parabolic subgroups of $G$, sending every $\xi \in (\partial_\infty T_v)^{\text{para}}$ to its elliptic stabilizer (see above Lemma 2.19, page 29). Since $\Gamma$ is irreducible we can apply Lemma 2.18, to obtain that $\theta_v(\xi) \neq G$ for all $\xi$.

Assume towards a contradiction that there exists a positive measure Borel subset $U \subseteq X_2$ such that $\nu(x)((\partial_\infty T_v)^{\text{para}}) > 0$ for all $x \in U$. We now restrict each of the probability measures $\nu(x)$ to $(\partial_\infty T_v)^{\text{para}}$, renormalize it to get a probability measure, and push it forward through the map $\theta_v : (\partial_\infty T_v)^{\text{para}} \to \mathbb{P}_G \setminus \{1, G\}$. This yields a Borel $(G|_U, \rho)$-equivariant map $U \to \text{Prob}(\mathbb{P}_G \setminus \{1, G\})$. As $\mathbb{P}_G$ is countable, as above, we deduce a Borel $(G|_U, \rho)$-equivariant map $U \to \mathcal{P}_{<\infty}(\mathbb{P}_G \setminus \{1, G\})$. Therefore, there exist a positive measure Borel subset $U' \subseteq U$ and a non-empty finite set $Q = \{Q_1, \dots, Q_k\}$ of proper non-trivial parabolic subgroups of $G$ such that, denoting by $H$ the setwise stabilizer of $Q$ (for the conjugation action of $G$ on $\mathbb{P}_G$), one has $\rho(G|_{U'}) \subseteq H$. Let $H^0$ be the finite-index subgroup of $H$ that fixes each $Q_i$ (again, for the conjugation action of $H$ on $\mathbb{P}_G$). Thus $H^0$ normalizes $Q_1$, i.e. $H^0 \subseteq Q_1 \times Q_1^\perp$. As $\Gamma$ is irreducible, $Q_1 \times Q_1^\perp$ is a proper parabolic subgroup of $G$. Lemma 2.4, page 21 thus ensures that $H$ is contained in a proper parabolic subgroup $Q$ of $G$. Thus $\rho(G|_{U'}) \subseteq Q$, contradicting the fact that $(G|_{X_2}, \rho)$ is tightly $G$-supported. This completes Step 2.

**Step 3** We now prove that for almost every $x \in X_2$, the support of $\nu(x)$ has cardinality at most 2.

Otherwise, let $U \subseteq X_2$ be a positive measure Borel subset such that for every $x \in U$, the support of $\nu(x)$ has cardinality at least 3. Then for every $x \in U$, the probability measure $\nu(x) \otimes \nu(x) \otimes \nu(x)$ on $(\partial_\infty T_v)^3$ gives positive measure to the subset $(\partial_\infty T_v)^{(3)}$ consisting of pairwise distinct triples. After renormalizing these measures, we obtain a Borel $(G|_U, \rho)$-equivariant map $U \to \text{Prob}((\partial_\infty T_v)^{(3)})$. Pushing this map forward through the continuous barycenter map $(\partial_\infty T_v)^{(3)} \to V(T_v)$, we derive a Borel $(G|_U, \rho)$-equivariant map $U \to \text{Prob}(V(T_v))$, which is a contradiction to Step 1. This completes Step 3.

By considering the support of $\nu(x)$, we deduce a Borel $(G|_{X_2}, \rho)$-equivariant map $X_2 \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$, as desired. $\square$
**Lemma 6.13.** Let $\mathcal{G}$ be a graph product of countably infinite groups over a finite irreducible simple graph, not reduced to one vertex. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with a cocycle $\rho: \mathcal{G} \to \mathcal{G}$. Let $\mathcal{A} \subseteq \mathcal{G}$ be a measured subgroupoid, and assume that $(\mathcal{A}, \rho)$ is boundary $\mathcal{G}$-elementary and tightly $\mathcal{G}$-supported.

Then there exists an essentially unique stably $(\mathcal{A}, \rho)$-equivariant measurable map $\theta_{v,\max}: X \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$ such that for every stably $(\mathcal{A}, \rho)$-equivariant measurable map $\theta: X \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$ and almost every $x \in X$, one has $\theta(x) \subseteq \theta_{v,\max}(x)$.

If in addition $\mathcal{H} \subseteq \mathcal{G}$ is a measured subgroupoid that stably normalizes $\mathcal{A}$, then the map $\theta_{v,\max}$ is also stably $(\mathcal{H}, \rho)$-equivariant.

The argument for the existence and essential uniqueness of $\theta_{v,\max}$ comes from the work of Adams [Ada94, Lemmas 3.2 and 3.3]; we review it here in our setting for convenience, following the presentation from [GH21b, Lemma 12.12]. The argument for its invariance under a subgroupoid that stably normalizes $\mathcal{A}$ comes from [Ada94, Lemma 3.4], we will follow [GH21b, Corollary 12.13].

*Proof.* The essential uniqueness is clear: if $\theta_{v,\max}^1$ and $\theta_{v,\max}^2$ are two such maps, then for almost every $x \in X$, one has $\theta_{v,\max}^1(x) \subseteq \theta_{v,\max}^2(x)$ and $\theta_{v,\max}^2(x) \subseteq \theta_{v,\max}^1(x)$.

For the existence, we first notice that if $\theta, \theta': X \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$ are two $(\mathcal{G}, \rho)$-equivariant Borel maps, then for almost every $x \in X$, the subset $\theta(x) \cup \theta'(x)$ of $(\partial_\infty T_v)^{\text{reg}}$ has cardinality at most 2. Indeed otherwise, the barycenter argument as in Step 3 of the previous proof would yield a contradiction to the fact that $(\mathcal{G}, \rho)$ is tightly $\mathcal{G}$-supported.

Let now $U \subseteq X$ be a Borel subset of maximal measure such that there exists a stably $(\mathcal{G}|_U, \rho)$-equivariant Borel map $\varphi: U \to \mathcal{P}_{=2}((\partial_\infty T_v)^{\text{reg}})$, where $\mathcal{P}_{=2}((\partial_\infty T_v)^{\text{reg}})$ is the set of all subsets of $(\partial_\infty T_v)^{\text{reg}}$ of cardinality exactly 2. Furthermore, by boundary $\mathcal{G}$-elementarity of $(\mathcal{A}, \rho)$, there also exists a stably $(\mathcal{G}|_{X\setminus U}, \rho)$-equivariant Borel map $\psi: X\setminus U \to (\partial_\infty T_v)^{\text{reg}}$. We let $\theta_{v,\max}$ be the map that coincides with $\varphi$ on $U$ and with $\psi$ on $X\setminus U$, and we claim that it satisfies the first conclusion of the lemma. Indeed, let $\theta: X \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$ be a stably $(\mathcal{A}, \rho)$-equivariant map. Then $\theta \cup \theta_{v,\max}$ is again $(\mathcal{A}, \rho)$-equivariant, and therefore $\theta(x) \cup \theta_{v,\max}(x)$ must have cardinality 2 on $U$ and cardinality 1 on $X\setminus U$ (almost everywhere). It follows that $\theta(x) \subseteq \theta_{v,\max}(x)$ almost everywhere.

For future use, we also make the following observation.

**Observation** For every positive measure Borel subset $U \subseteq X$, the map $(\theta_{v,\max})|_U$ is also the essentially unique maximal stably $(\mathcal{A}|_U, \rho)$-equivariant Borel map $U \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$.

Let now $\mathcal{H} \subseteq \mathcal{G}$ be a measured subgroupoid that stably normalizes $\mathcal{A}$. Using the above observation, we can (and will) assume up to a countable partition of $X$ that $\mathcal{H}$ normalizes $\mathcal{A}$.

By definition of normalization between measured subgroupoids, we can write $\mathcal{H}$ as a countable union of Borel subsets $B_n$ such that for every $n \in \mathbb{N}$,

* $B_n$ induces a Borel isomorphism $f_n: U_n \to V_n$ between its source $U_n := s(B_n)$ and its range $V_n := r(B_n)$, and

* for every $\mathbf{a} \in \mathcal{G}$ and every $\mathbf{b}_1, \mathbf{b}_2 \in B_n$, if the composition $\mathbf{b}_1\mathbf{a}\mathbf{b}_2^{-1}$ is well-defined, then $\mathbf{a} \in \mathcal{A}$ if and only if $\mathbf{b}_1\mathbf{a}\mathbf{b}_2^{-1} \in \mathcal{A}$.
Up to subdividing $B_n$ if necessary, we will also assume without loss of generality that the cocycle $\rho$ takes a single value on $B_n$, say $\rho(B_n) = \{g_n\}$.

Let now $n \in \mathbb{N}$ be such that $U_n$ has positive measure. Then the map

$$ \theta'_{v,\max} : \begin{array}{rcl} U_n & \to & \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}}) \\ x & \mapsto & g_n^{-1} \theta_{v,\max}(f_n(x)) \end{array} $$

is a maximal stably $(\mathcal{A}|_{U_n}, \rho)$-equivariant Borel map. So by the above observation, it coincides with $\theta_{v,\max}$ on a conull Borel subset of $U_n$. As this is true for every $n \in \mathbb{N}$ such that $U_n$ has positive measure, and the corresponding sets $B_n$ cover $\mathcal{H}|_{X*}$ for a conull Borel subset $X^* \subseteq X$, it follows that the map $\theta_{v,\max}$ is (stably) $(\mathcal{H}, \rho)$-equivariant. $\square$

**Lemma 6.14.** 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 $X$, equipped with a strict cocycle $\rho: \mathcal{G} \to \mathcal{G}$. Let $P \subseteq G$ be a parabolic subgroup whose type is irreducible, and let $\rho_P: \mathcal{G} \to P$ be the cocycle obtained by postcomposing $\rho$ by the natural retraction $G \to P$. Let $\mathcal{A}, \mathcal{H} \subseteq \mathcal{G}$ be measured groupoids, with $\mathcal{A}$ normalized by $\mathcal{H}$.

If $(\mathcal{A}, \rho_P)$ is $P$-elementary and $(\rho_P)|_{\mathcal{A}}$ is nowhere trivial, then $(\mathcal{H}, \rho_P)$ is $P$-elementary.

*Proof.* Since $(\mathcal{A}, \rho_P)$ is $P$-elementary, there exists a Borel partition $X = X_1 \sqcup X_2$ such that $(\mathcal{A}|_{X_1}, \rho_P)$ is properly $\mathbb{P}_P$-supported and $(\mathcal{A}|_{X_2}, \rho_P)$ is tightly $P$-supported and boundary $P$-elementary (see Remark 6.9).

Since $\mathcal{A}$ is normalized by $\mathcal{H}$ and the type of $P$ is irreducible, and $(\rho_P)|_{\mathcal{A}}$ is nowhere trivial, Lemma 4.19 ensures that $(\mathcal{H}|_{X_1}, \rho_P)$ is properly $\mathbb{P}_P$-supported.

Let $\Lambda \subseteq \Gamma$ be the type of $P$, and let $v \in V\Lambda$. Viewing $P$ as a graph product over $\Lambda$, we let $T_v^P$ be the splitting of $P$ associated to $v$ (see Section 2.2.2). Let $\theta_{v,\max}: X_2 \to \mathcal{P}_{\le 2}((\partial_\infty T_v^P)^{\text{reg}})$ be the map given by Lemma 6.13. Since $\mathcal{A}$ is normalized by $\mathcal{H}$, the map $\theta_{v,\max}$ is stably $(\mathcal{H}|_{X_2}, \rho_P)$-equivariant. This shows that $(\mathcal{H}|_{X_2}, \rho_P)$ is boundary $P$-elementary, which concludes our proof. $\square$

### 6.2.2 Exploiting normal amenable subgroupoids

**Lemma 6.15.** 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 $X$, equipped with a cocycle $\rho: \mathcal{G} \to \mathcal{G}$. Let $\mathcal{A}, \mathcal{N}$ be measured subgroupoids of $\mathcal{G}$. Let $A, N \subseteq G$ be parabolic subgroups. Assume that

* $\mathcal{A}$ is amenable and normalized by $\mathcal{N}$, and $\mathcal{N}$ is non-amenable,

* $(\mathcal{A}, \rho)$ is tightly $A$-supported, and $(\mathcal{N}, \rho)$ is tightly $N$-supported, with $N \subseteq A$,

* $\rho|_{\mathcal{N}}$ has trivial kernel.

Then the clique factor $C_0$ of $\mathcal{A}$ is non-amenable, and in fact $N \cap C_0$ is non-amenable.

*Proof.* Up to a global conjugation of the cocycle $\rho$, we will assume without loss of generality that $A = G_\Lambda$ for some induced subgraph $\Lambda \subseteq \Gamma$.

Consider a join decomposition $\Lambda = \Lambda_0 \circ \Lambda_1 \circ \dots \circ \Lambda_n$, where $\Lambda_0$ is the clique factor of $\Lambda$, and $\Lambda_1, \dots, \Lambda_n$ do not admit any non-trivial join decomposition. Remark that for
<figure><img src="image_18.png" /><figcaption>Figure 15: Definition of the map $\rho_j$</figcaption></figure>

all $j \ge 1$, the graph $\Lambda_j$ contains at least 2 vertices (otherwise it could be included in the clique factor $\Lambda_0$). This join decomposition of $\Lambda$ induces a direct product decomposition $A = C_0 \times F_1 \times \dots \times F_n$. Here $C_0$ is the clique factor of $A$.

Let $j \in \{1, \dots, n\}$. Up to replacing $X$ by a conull Borel subset, we can (and will) assume that $\rho(\mathcal{A}) \subseteq \mathcal{A}$. Let $\rho_j : \rho^{-1}(\mathcal{A}) \to F_j$ be the cocycle obtained by postcomposing $\rho$ with the $j$th projection (see Figure 15). Since $N \subseteq \mathcal{A}$, the cocycle $\rho_j$ is defined on $\mathcal{A}$ and on $\mathcal{N}$. Let $v \in V\Lambda_j$, and let $T_v$ be the Bass-Serre tree of the splitting of $F_j$ associated to $v$. Since $\mathcal{A}$ is amenable, there exists an $(\mathcal{A}, \rho_j)$-equivariant Borel map $\nu : X \to \text{Prob}(\partial_\infty T_v \cup V(T_v))$, see Remark 6.12.

Since $(\mathcal{A}, \rho)$ is tightly $\mathcal{A}$-supported, necessarily $(\mathcal{A}, \rho_j)$ is tightly $F_j$-supported. Therefore, Lemma 6.11 ensures that $(\mathcal{A}, \rho_j)$ is boundary $F_j$-elementary. And Lemma 6.13 ensures that there is an essentially unique maximal Borel stably $(\mathcal{A}, \rho_j)$-equivariant map $\theta_{v,\max}^j : X \to \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$, i.e. such that for any such map $\theta$ and almost every $x \in X$, one has $\theta(x) \subseteq \theta_{v,\max}^j(x)$. Since $\mathcal{A}$ is normalized by $\mathcal{N}$, Lemma 6.13 also ensures that $\theta_{v,\max}^j$ is stably $(\mathcal{N}, \rho_j)$-equivariant.

Combining the maps $\theta_{v,\max}^j$ for all $j \in \{1, \dots, n\}$ and all $v \in V\Lambda_j$, we get a Borel stably $(\mathcal{N}, \rho)$-equivariant map

$$X \to \prod_{j,v} \mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}}) \qquad (6.1)$$

By Lemma 2.20, page 29, for every $j \in \{1, \dots, n\}$, the action of $F_j$ on the non-empty set $(\partial_\infty T_v)^{\text{reg}}$ is Borel amenable, and therefore so is its action on $\mathcal{P}_{\le 2}((\partial_\infty T_v)^{\text{reg}})$. So the action of $F_1 \times \dots \times F_n$ on the above product is Borel amenable.

We now prove that $N \cap C_0$ is not amenable, so assume towards a contradiction that it is. Then $N \subseteq (N \cap C_0) \times F_1 \times \dots \times F_n$, and as $N \cap C_0$ is amenable it follows that the action of $N$ on the product in eq. (6.1) is again Borel amenable. Since $\rho|_N$ has trivial kernel, [GH21b, Proposition 3.38] (which recasts [Kid08a, Proposition 4.33]) ensures that $\mathcal{N}$ is amenable, a contradiction. $\square$

# 7 The irreducible case

In this section and the next, we prove the key Proposition 5.3 establishing the Vertex Recognition Property for the class of graph products $G$ with countably infinite vertex groups over finite simple graphs $\Gamma$ with no transvection and no partial conjugation. In the present section, we only consider the case where $\Gamma$ is irreducible, in other words $G$ does not split as a direct product non-trivially. And in the next section, we will provide the extra arguments to allow for $\Gamma$ to be reducible. When $\Gamma$ is irreducible, Lemma 6.3
ensures that $\Gamma$ is strongly reduced in the sense of Definition 6.1, that is, it is not possible
to obtain a new decomposition of $G$ as a graph product by collapsing a proper subgraph
of $\Gamma$. The main goal of the present section is to prove the following proposition.

**Proposition 7.1.** The class of all graph products of countably infinite groups over transvection-free strongly reduced finite simple graphs (not reduced to one vertex) satisfies the Vertex Recognition Property.

Note that here we allow graphs to have partial conjugations.

**7.1 Overview of the proof**

Let $G$ be a graph product of countably infinite vertex groups over a transvection-free strongly reduced finite simple graph, not reduced to a vertex. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\rho : \mathcal{G} \to G$ be an action-like cocycle. Given a measured subgroupoid $\mathcal{H} \subseteq \mathcal{G}$, our goal is to “recognize” when $(\mathcal{H}, \rho)$ is of vertex type (see Section 5.1, page 53) using a property that is purely phrased in terms of the structure of $\mathcal{H}$, i.e. with no reference to the cocycle $\rho$. In this way, if $\mathcal{G}$ comes equipped with two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$, then subgroupoids of vertex type with respect to $\rho_G$, are also of vertex type with respect to $\rho_H$, and vice versa.

The group-theoretic analogue of this problem is to recognize conjugates of vertex
groups of $G$ using a purely group-theoretic property. Here we will formulate our strategy
in the group-theoretic setting; in a sense, most of the technical work in the sequel of
the present section will consist in “translating” this strategy to the groupoid-theoretic
framework. With this in mind, we will formulate our characterization using amenability
and normalization of subgroups, as these notions have their counterparts for groupoids.

Our starting point is the combinatorial statement provided by Lemma 6.6, page 65:
starting from $G$, vertex groups and their conjugates are obtained by successively passing
to maximal product parabolic subgroups and zooming in their factors (and at the last
step, taking the clique factor). Thus, our goal decomposes into two tasks:

1. Characterize maximal product parabolic subgroups of $G$.

2. When $P$ is a product parabolic subgroup, characterize either

* the factors of $P$, if $P$ has trivial clique factor, or

* the clique factor of $P$, if it is non-trivial.

**Task 1: Characterizing maximal products.** This task is carried in Section 7.2,
following a strategy that was already used in [HHI23] in the context of right-angled
Artin groups.

The idea is the following. If $P = C_0 \times F_1 \times \cdots \times F_k$ is a maximal product (where $C_0$ is the clique factor, and the $F_j$ are irreducible for all $j$), letting $A \subseteq F_j$ be an amenable subgroup and $N = C_0 \times F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$ (where we have just removed the $j$th factor), then $A$ is normalized by $N$, and $N$ is normal in $P$. It turns out that conversely, maximal product parabolic subgroups of $G$ are characterized as those maximal subgroups $P$ of $G$ for which there exists an infinite amenable subgroup $A$ of $P$, and a non-amenable subgroup $N \subseteq P$, such that $A$ is normalized by $N$, and $N \trianglelefteq P$. This is proved in Lemma 7.4, at the groupoid-theoretic level (Property ($P_{\text{prod}}$) below is the groupoid-theoretic translation of the above property).
**Task 2: Characterizing (clique) factors.** This task is carried in Section 7.3. Let $P$ be a product parabolic subgroup.

As a warm-up, let us assume that we know in advance that $P = F_1 \times \cdots \times F_k$ has
trivial clique factor. Then the co-factors $F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$ are characterized as
the maximal normal subgroups $N \subset P$ that normalize an infinite amenable subgroup
(namely, any $A \subseteq F_j$ infinite amenable). And the factors $F_j$ are then characterized as
the minimal non-trivial intersections of co-factors.

This approach is in general a bit too naive for two reasons:

* It does not work if the clique factor $C_0$ of $P$ is non-trivial (indeed $C_0$ could contain a normal infinite amenable subgroup $A$, and then $N = P$ in the above would work).

* This would also necessit to be able to characterize in advance the existence or not of a non-trivial clique factor.

We follow a slightly different approach. This necessits to introduce some extra terminol-
ogy. We write $P = F_1 \times \cdots \times F_k$, where each $F_j$ is irreducible (in particular the clique
factor has been decomposed as a product of conjugates of vertex groups).

A subgroup $S \subseteq P$ is *special* if $S \trianglelefteq P$ and $N_G(S) \not\subseteq N_G(P)$. One should typically think that certain subproducts of $P$ (i.e. products of only some of the subgroups $F_j$) could be special. See Figure 16, page 76 for an illustration, where the special subgroups are exactly the *orange* vertex groups in the clique factor. In fact in general, if the type of the clique factor is a complete graph on at least two vertices, then the subgroups $F_j$ whose type is reduced to one vertex are automatically special, using that $\Gamma$ is transvection-free.

Now we say that an infinite subgroup $B \subseteq P$ is *appropriate* if it satisfies the following
two properties:

* $N_G(B) \not\subseteq N_G(P)$;

* if $B$ is contained in a special subgroup $S \trianglelefteq P$, then $N_G(B) \not\subseteq N_G(S)$.

We make two important observations:

* For every $j \in \{1, \dots, k\}$, there exists an appropriate subgroup $B \subseteq F_j$, which can be taken to be a conjugate of a vertex group. This is a consequence of our crucial assumption that $\Gamma$ is strongly reduced: we can find $B$ whose normalizer is not contained in $F_j \times F_j^{\perp}$, so regardless of whether $F_j$ is special or not, the above conditions are satisfied. See Figure 16, page 76 for an illustration, where the *brown* vertex groups are appropriate.

* On the other hand, the clique factor $C_0$ does not contain any appropriate subgroup: indeed, if $B \subseteq C_0$ satisfies $N_G(B) \not\subseteq P \times P^{\perp}$, then the smallest subclique $C_0' \subseteq C_0$ containing $B$ also satisfies $N_G(C_0') \not\subseteq P \times P^{\perp}$. So $C_0'$ is special, and $N_G(B) \subseteq N_G(C_0')$.

Armed with these observations, we now deduce that *clique-inclusive cofactors* of $P$ (i.e. subgroups of the form $C_0 \times F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$) are characterized as follows: they are the maximal subgroups $N \subset P$ that normalize an appropriate infinite subgroup of $P$.

And the clique factor is then characterized as the intersection of all clique-inclusive
cofactors. (If $C_0 = \{1\}$, then factors are characterized as the minimal non-trivial inter-
section of clique-inclusive cofactors.)

**Conclusion.** We finish this overview by mentioning that in Section 7.4, we complete our proof of Proposition 7.1 by combining the combinatorial zooming lemma (Lemma 6.6)
<figure><img src="image_19.png" /><figcaption>Figure 16: The orange vertex groups are special in P, and the brown vertex groups are appropriate.</figcaption></figure>

with the previously obtained characterizations of maximal product parabolic subgroups and their factors.

Recall that appendix A, page 143 provides a group theoretic analogue of Proposition 7.1 and can be read as a warm up for the proof of the latter proposition.

## 7.2 Recognizing maximal products

Let now $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho: \mathcal{G} \to \mathcal{G}$. Let $\mathcal{H}$ be a measured subgroupoid of $\mathcal{G}$. We say that $(\mathcal{H}, \rho)$ is of parabolic type 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 parabolic subgroup $P_i \subseteq G$ such that $\mathcal{H}_{|X_i} = \rho^{-1}(P_i)|_{X_i}$. We say that $(\mathcal{H}, \rho)$ is of product type (resp. of maximal product type, resp. of clique type, resp. of isolated clique type if in the above, one can take $P_i$ to be a product parabolic subgroup (resp. a maximal product parabolic subgroup, resp. a parabolic subgroup whose type is a clique, resp. a parabolic subgroup whose type is an isolated clique) for every $i \in I$. We say that $(\mathcal{G}, \rho)$ is nowhere of clique type (resp. nowhere of isolated clique type) if there does not exist any positive measure Borel subset $U \subseteq X$ such that $(\mathcal{G}|_U, \rho)$ is of clique type (resp. of isolated clique type).

Our goal is now to give a partial characterization of pairs $(\mathcal{P}, \rho)$ of maximal product type with no reference to the cocycle $\rho$ (see Remark 7.5 below for why we only reach a partial characterization). For this we introduce the following property.

**Definition 7.2 (Property $(P_{\text{prod}}))$.** Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\mathcal{P}$ be a measured subgroupoid of $\mathcal{G}$. We say that the pair $(\mathcal{G}, \mathcal{P})$ satisfies *Property* $(P_{\text{prod}})$ if it verifies the following assertions.

$(P_{\text{prod}})_1$ There exist measured subgroupoids $\mathcal{A}, \mathcal{N} \subseteq \mathcal{P}$ such that

(a) $\mathcal{A}$ is amenable and of infinite type, and is stably normalized by $\mathcal{N}$;

(b) $\mathcal{N}$ is everywhere non-amenable, and is stably normalized by $\mathcal{P}$;

$(P_{\text{prod}})_2$ Whenever $\mathcal{P}'$ is a measured subgroupoid of $\mathcal{G}$ that satisfies Property $(P_{\text{prod}})_1$, and such that $\mathcal{P}$ is stably contained in $\mathcal{P}'$, then $\mathcal{P}$ and $\mathcal{P}'$ are stably equal.

**Remark 7.3.** We mention the following crucial stabilities for Property $(P_{\text{prod}})$.
1. If $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$, then for every positive measure Borel subset $U \subseteq X$, the pair $(\mathcal{G}|_U, \mathcal{P}|_U)$ also satisfies Property $(\mathrm{P}_{\mathrm{prod}})$.

2. If there exists a Borel partition $X^* = \sqcup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets, such that for every $i \in I$, the pair $(\mathcal{G}|_{X_i}, \mathcal{P}|_{X_i})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$, then $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$.

These stabilities will allow us to take restrictions and work up to a countable Borel partition in the proofs below. All other properties that will appear later in this section (Properties $(\mathrm{P}_{\mathrm{spec}})$, $(\mathrm{P}_{\mathrm{cofact}})$, $(\mathrm{P}_{\mathrm{fact}})$, $(\mathrm{P}_{\mathrm{vert}})$) and the next two also satisfy the same stabilities.

Property $(\mathrm{P}_{\mathrm{prod}})$ is the groupoid-theoretic translation of the following idea: if $P = F_1 \times \cdots \times F_k$ is a maximal direct product in $G$, then (unless possibly if this product comes from a clique and no factor contains an infinite amenable subgroup) there exists an infinite amenable subgroup $A$ in some $F_i$, normalized by $N = F_1 \times \cdots \times \hat{F_j} \times \cdots \times F_k$, which is itself normalized by $P$. It turns out that conversely, maximal subgroups of $G$ with this property are either product parabolic subgroups, or else they could be contained in a parabolic subgroup whose type is an isolated clique. The following lemma, which is inspired from [HHI23, Lemma 5.4], establishes this fact at the groupoid-theoretic level.

**Lemma 7.4.** 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 $\rho : \mathcal{G} \to G$. Let $\mathcal{P} \subseteq \mathcal{G}$ be a measured subgroupoid. Then

1. If $(\mathcal{P}, \rho)$ is of maximal product type and nowhere of isolated clique type, then $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$.

2. If $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$, then there exists a Borel partition $X = X^1 \sqcup X^2$ such that $(\mathcal{P}|_{X^1}, \rho)$ is of maximal product type, and $\mathcal{P}|_{X^2}$ is stably contained in a subgroupoid $Q$ of $\mathcal{G}|_{X^2}$ such that $(Q, \rho)$ is of isolated clique type.

**Remark 7.5.** When $\Gamma$ does not contain any isolated clique, then (as in [HHI23]) the above lemma gives a characterization of pairs $(\mathcal{P}, \rho)$ of maximal product type that is independent of the choice of an action-like cocycle $\rho : \mathcal{G} \to G$. But when $\Gamma$ has isolated cliques, whether or not subgroupoids of isolated clique type satisfy $(\mathrm{P}_{\mathrm{prod}})$ depends on the vertex groups. This is why there is an asymmetry in the statement of the lemma.

In particular, we mention that with our definitions, it is unclear to us whether or not a subgroupoid of the form $\rho^{-1}(C)$, where $C$ is a parabolic subgroup associated to an isolated clique on several vertices, with nonamenable vertex groups, always satisfies Property $(\mathrm{P}_{\mathrm{prod}})$. This is the case as soon as one of the vertex groups contains an infinite amenable subgroup $A$, by taking $\mathcal{A} = \rho^{-1}(A)$ and $\mathcal{N} = \rho^{-1}(N)$, where $N$ is the direct product of the other vertex groups of the clique. The difficulty comes from the case where the vertex groups are torsion groups, e.g. free Burnside groups. This uncertainty required us to make a few adjustments in the statements and in the arguments.

*Proof. Step 1* Let us first prove that if $(\mathcal{P}, \rho)$ is of maximal product type and nowhere of isolated clique type, then $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})_1$.

Up to passing to a conull Borel subset of $X$ and performing a countable Borel partition of $X$, we can assume that $\mathcal{P} = \rho^{-1}(P)$ for some maximal product parabolic subgroup $P$ not of isolated clique type. Write $P = P_1 \times P_2$ as a product of two infinite parabolic
subgroups. Up to changing the roles of $P_1$ and $P_2$, Lemma 6.5, page 64 lets us assume
that $P_2$ contains a non-abelian free subgroup. If $P_1$ is amenable, then the conclusion
follows by letting $\mathcal{A} = \rho^{-1}(P_1)$ and $\mathcal{N} = \rho^{-1}(P_2)$: indeed

• $\mathcal{A}$ is amenable because $A$ is and $\rho$ has trivial kernel (Lemma 4.7, page 41), of infinite type because $A$ is infinite and $\rho$ is action-like, and (stably) normalized by $\mathcal{N}$ because $A$ is normalized by $P_2$ (Example 4.6, page 41).

• $\mathcal{N}$ is everywhere non-amenable because $P_2$ is non-amenable and $\rho$ is action-like, and $\mathcal{N}$ is (stably) normalized by $\mathcal{P}$ because $N$ is normalized by $P$ (Example 4.6).

Otherwise, if $P_1$ is non-amenable, let $A \subseteq P_2$ be an infinite cyclic subgroup; then the
conclusion follows by letting $\mathcal{A} = \rho^{-1}(A)$ and $\mathcal{N} = \rho^{-1}(P_1)$.

**Step 2** Let us prove that if $\mathcal{P}$ satisfies $(P_{\text{prod}})_1$, then there exist a countable partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets $X_i$, and for every $i \in I$, a parabolic subgroup $Q_i$ which is either a product or of isolated clique type, such that $\mathcal{P}|_{X_i} \subseteq \rho^{-1}(Q_i)|_{X_i}$.

Let $\mathcal{A}, \mathcal{N} \subseteq \mathcal{P}$ be measured subgroupoids given by $(P_{\text{prod}})_1$. Up to replacing $X$ by a
conull Borel subset and taking a countable Borel partition, we will assume that

• $\mathcal{A}$ is normalized by $\mathcal{N}$, and $\mathcal{N}$ is normalized by $\mathcal{P}$;

• $(\mathcal{A}, \rho)$ is tightly $\mathcal{A}$-supported for some parabolic subgroup $A \subseteq G$ (Lemma 4.16, page 44) – notice that $A \neq \{1\}$ because $\mathcal{A}$ is of infinite type and $\rho$ has trivial kernel;

• $(\mathcal{N}, \rho)$ is tightly $\mathcal{N}$-supported for some parabolic subgroup $N \subseteq G$ (Lemma 4.16) – notice that $N$ is non-amenable because $\mathcal{N}$ is everywhere non-amenable and $\rho$ has trivial kernel, see Lemma 4.7, page 41;

• $\mathcal{N} \subseteq \rho^{-1}(A \times A^{\perp})$ and $\mathcal{P} \subseteq \rho^{-1}(N \times N^{\perp})$ using the invariance of the support by the normalizer (Lemma 4.17).

Notice in particular that $N \subseteq A \times A^{\perp}$, by definition of the support. Now there are two cases.

• Case 1: Either $N$ splits as a direct product of two infinite parabolic subgroups, or $N^{\perp} \neq \{1\}$. In this case $Q := N \times N^{\perp}$ is a product parabolic subgroup, we are done since $\mathcal{P} \subseteq \rho^{-1}(Q)$.

• Case 2: $N$ does not split as a direct product of two infinite parabolic subgroups, and $N^{\perp} = \{1\}$. Then $\mathcal{P} \subseteq \rho^{-1}(N)$, and we will now show that in this case $N$ is of isolated clique type, which will conclude. As $N \subseteq A \times A^{\perp}$, we deduce using Lemma 2.7, page 22 that $N = (N \cap A) \times (N \cap A^{\perp})$. But we assumed that $N$ does not split as a direct product, so either $N \subseteq A$ or $N \subseteq A^{\perp}$. If the latter holds we have $\{1\} \subsetneq A \subseteq N^{\perp}$ which contradicts our assumption that $N^{\perp}$ is trivial. Hence $N \subseteq A$, and $A^{\perp} = \{1\}$ because $N^{\perp} = \{1\}$. Lemma 6.15, page 72 (applied to the amenable groupoid $\mathcal{A}$, which is normalized by the non-amenable groupoid $\mathcal{N}$) thus ensures that the clique factor $A_0$ of $A$ is non-amenable, and $N \cap A_0 \neq \{1\}$. As $N \subseteq A$, as $N \cap A_0 \neq \{1\}$, as above using again Lemma 2.7, page 22, and that $N$ does not split as a direct product we obtain that $N$ is equal to a conjugate of a vertex group. Since $N^{\perp} = \{1\}$, this vertex group is isolated. This completes the proof in Case 2.

**Step 3** We now prove that if $(\mathcal{P}, \rho)$ is of maximal product type, then $(\mathcal{G}, \mathcal{P})$ satisfies Property $(P_{\text{prod}})_2$: this will complete the proof of the first part of the lemma.
Let $\mathcal{P}'$ be a measured subgroupoid of $\mathcal{G}$ that satisfies Property $(\mathrm{P}_{\mathrm{prod}})_1$, such that $\mathcal{P}$ is stably contained in $\mathcal{P}'$. By Step 2, we can find a countable partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets $X_i$, and for every $i \in I$, a parabolic subgroup $Q_i$ which is either a product or of isolated clique type, such that $\mathcal{P}'|_{X_i} \subseteq \rho^{-1}(Q_i)|_{X_i}$. Without loss of generality, we will assume that all subsets $X_i$ have positive measure. Since $\mathcal{P}$ is stably contained in $\mathcal{P}'$, up to passing again to a conull Borel subset and performing a countable partition, we will assume that $\mathcal{P}|_{X_i} \subseteq \rho^{-1}(Q_i)|_{X_i}$. Since $(\mathcal{P}, \rho)$ is of maximal product type, up to a further partition again, we have $\mathcal{P}|_{X_i} = \rho^{-1}(P_i)|_{X_i}$ for some maximal product parabolic subgroup $P_i$. It thus follows from Lemma 4.14, page 43 that $P_i \subseteq Q_i$. We now observe that in fact $P_i = Q_i$. Indeed, either $Q_i$ is a product and then this follows from the maximality of $P_i$. Or $Q_i$ is of isolated clique type and contains the product parabolic subgroup $P_i$, then the corresponding clique is not reduced to a point; so $Q_i$ is a product and as above $P_i = Q_i$. It follows from the above that $\mathcal{P}'$ is stably contained in $\mathcal{P}$, showing that $\mathcal{P}$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})_2$.

**Step 4** We finally prove the second assertion of the lemma.

So assume that $(\mathcal{G}, \mathcal{P})$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$. By Step 2, there exist a countable partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets $X_i$, and for every $i \in I$, a parabolic subgroup $Q_i$ which is either of product type or of isolated clique type, such that $\mathcal{P}|_{X_i} \subseteq \rho^{-1}(Q_i)|_{X_i}$. Up to increasing $Q_i$, we will assume that $Q_i$ is either a maximal product or of isolated clique type (in particular $Q_i$ is not a proper subgroup of a parabolic subgroup of isolated clique type). Let $I_2 \subseteq I$ be the subset consisting of all $i$ such that $Q_i$ is of isolated clique type, and let $I_1 = I \setminus I_2$. Let $X^1$ (resp. $X^2$) be the union of all $X_i$ with $i \in I_1$ (resp. with $i \in I_2$). Let us now prove that $\mathcal{P}|_{X_i}$ is actually (stably) equal to $\rho^{-1}(Q_i)|_{X_i}$ for all $i \in I_1$. So let $\mathcal{P}'$ be a subgroupoid of $\mathcal{G}$ such that for every $i \in I_1$, one has $\mathcal{P}'|_{X_i} = \rho^{-1}(Q_i)|_{X_i}$, with $\mathcal{P}'|_{X^2} = \mathcal{P}|_{X^2}$. Then $\mathcal{P}$ is stably contained in $\mathcal{P}'$. In addition $\mathcal{P}'$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})_1$: this uses Step 1 of this proof for $\mathcal{P}'|_{X^1}$, and the fact that $\mathcal{P}$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$ for $\mathcal{P}'|_{X^2}$. So $(\mathrm{P}_{\mathrm{prod}})_2$ for $(\mathcal{G}, \mathcal{P})$ implies that $\mathcal{P}$ and $\mathcal{P}'$ are stably equal. In particular the second assertion of the lemma follows with $\mathcal{Q}$ a groupoid verifying $\mathcal{Q}|_{X_i} := \rho^{-1}(Q_i)|_{X_i}$ for every $i \in I_2$. $\square$

## 7.3 Recognizing factors in products

**Special subproducts.** Let $P = F_1 \times \cdots \times F_n$ be a product parabolic subgroup, where the subgroups $F_j$ with $j \ge 1$ are parabolic subgroups that do not further split as direct products of parabolic subgroups (in particular the clique factor of $P$ is fully decomposed). A *subproduct* is a subgroup of $P$ equal to the product of finitely many (possibly only one) factors $F_j$. A subproduct $S$ of $P$ is *special within* $G$ if $N_G(S) \supsetneq N_G(P)$, that is to say $S \times S^\perp \supsetneq P \times P^\perp$.

Let now $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to G$ be an action-like cocycle. Let $\mathcal{P}$ be a measured subgroupoid such that $(\mathcal{P}, \rho)$ is of product type. Let $X^* = \cup_{i \in I} X_i$ be a partition of a conull Borel subset $X^* \subseteq X$, and for every $i \in I$, $P_i$ be a parabolic subgroup of $G$ of product type, such that $\mathcal{P}|_{X_i} = \rho^{-1}(P_i)|_{X_i}$. Let $\mathcal{S}$ be a measured subgroupoid of $\mathcal{P}$. We say that $(\mathcal{S}, \rho)$ is of special subproduct type within $(\mathcal{P}, \rho)$ if, up to passing to a further conull Borel subset
and refining the above partition, for every $i \in I$, there exists a special subproduct $S_i$ of $P_i$ such that $S_{|X_i} = \rho^{-1}(S_i)_{|X_i}$. We observe that this definition does not depend on the decomposition of $X$ we chose.

We highlight the following property, which translates the notion of a special subproduct at the groupoid-theoretic level.

**Definition 7.6 (Property (P<sub>spec</sub>))**. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\mathcal{S} \subseteq \mathcal{P}$ be two measured subgroupoids of $\mathcal{G}$. We say that the triple $(\mathcal{G}, \mathcal{P}, \mathcal{S})$ satisfies *Property (P<sub>spec</sub>)* if the following two properties hold.
(P<sub>spec</sub>)_1 The subgroupoid $\mathcal{S}$ is stably normal in $\mathcal{P}$.

(P<sub>spec</sub>)_2 There exists a measured subgroupoid $\mathcal{N} \subseteq \mathcal{G}$ that stably normalizes $\mathcal{S}$, such that for every measured subgroupoid $\mathcal{N}'$ of $\mathcal{G}$ that stably normalizes $\mathcal{P}$, and every Borel subset $U \subseteq X$ of positive measure, one has $\mathcal{N}_{|U} \not\subseteq \mathcal{N}'_{|U}$.

**Lemma 7.7.** 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 $\rho : \mathcal{G} \to G$. Let $\mathcal{S} \subseteq \mathcal{P} \subseteq \mathcal{G}$ be measured subgroupoids, with $(\mathcal{P}, \rho)$ of product type.

1. If $(S, \rho)$ is of special subproduct type within $(\mathcal{P}, \rho)$, then $(\mathcal{G}, \mathcal{P}, S)$ satisfies Property (P<sub>spec</sub>).

2. If $(\mathcal{G}, \mathcal{P}, S)$ satisfies Property (P<sub>spec</sub>), then there exist 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 product parabolic subgroup $P_i$, and a special subproduct $S_i \subseteq P_i$, such that $\mathcal{P}_{|X_i} = \rho^{-1}(P_i)_{|X_i}$ and $(S_{|X_i}, \rho)$ is tightly $S_i$-supported.

Remark that in our definition of (P<sub>spec</sub>) – in contrast with (P<sub>prod</sub>) – we do not have any maximality condition. This explains the asymmetry in the above lemma.

*Proof.* Without loss of generality, we can assume that $\mathcal{P} = \rho^{-1}(P)$ for some product parabolic subgroup $P \subseteq G$.

We first prove Assertion 1, so assume that $(S, \rho)$ is of special subproduct type within $(\mathcal{P}, \rho)$. That $(\mathcal{G}, \mathcal{P}, S)$ satisfies (P<sub>spec</sub>)_1 follows from the fact that subproducts are normal in the ambient product, and Example 4.6, page 41. We now prove that it satisfies (P<sub>spec</sub>)_2. Up to passing to a conull Borel subset and taking a countable partition of $X$, we will assume that $S = \rho^{-1}(S)$ for some special subproduct $S \subseteq P$. Let $\mathcal{N} = \rho^{-1}(S \times S^{\perp})$, which normalizes $S$. Let $\mathcal{N}'$ be a measured subgroupoid of $\mathcal{G}$ that stably normalizes $\mathcal{P}$. Let $U \subseteq X$ be a positive measure Borel subset. Since $(\mathcal{P}, \rho)$ is tightly $P$-supported (Lemma 4.15, page 43) and stably normalized by $\mathcal{N}'$, Lemma 4.17, page 44 ensures that there exists a positive measure Borel subset $V \subseteq U$ such that $\mathcal{N}'_{|V} \subseteq \rho^{-1}(P \times P^{\perp})_{|V}$. Since $S$ is a special subproduct, we have $S \times S^{\perp} \not\subseteq P \times P^{\perp}$. It thus follows from Lemma 4.14, page 43 that $\mathcal{N}_{|V} \not\subseteq \mathcal{N}'_{|V}$; in particular $\mathcal{N}_{|U} \not\subseteq \mathcal{N}'_{|U}$, so (P<sub>spec</sub>)_2 holds.

We now prove the second assertion, so assume that $(\mathcal{G}, \mathcal{P}, S)$ satisfies Property (P<sub>spec</sub>). Without loss of generality, we can assume that $(S, \rho)$ is tightly $S$-supported for some parabolic subgroup $S \subseteq P$. Since $S$ is stably normal in $\mathcal{P}$ (by (P<sub>spec</sub>)_1), Lemma 4.17, page 44, ensures that $\mathcal{P} = \rho^{-1}(P)$ is stably contained in $\rho^{-1}(S \times S^{\perp})$, and Lemma 4.14, page 43 ensures that $P \subseteq S \times S^{\perp}$. Therefore $S$ is a subproduct of $P$ (see Lemma 2.7). Let $\mathcal{N}$ be a measured groupoid as in (P<sub>spec</sub>)_2. Let $\mathcal{N}' = \rho^{-1}(P \times P^{\perp})$, which normalizes
P. Since $\mathcal{N}$ stably normalizes $\mathcal{S}$, by Lemma 4.17, there exists a positive measure Borel subset $U \subseteq X$ such that $\mathcal{N}_{|U} \subseteq \rho^{-1}(S \times S^{\perp})|_{U}$. It follows from $(P_{\text{spec}})_{2}$ that $S \times S^{\perp}$ is not contained in $P \times P^{\perp}$. So $\mathcal{S}$ is a special subproduct, which completes the proof. $\square$

**Clique-inclusive co-factors.** Let $P$ be a parabolic subgroup of product type. Write $P = C_0 \times F_1 \times \cdots \times F_k$ as a direct product of parabolic subgroups, where the $F_j$ do not further split as a direct product of proper parabolic subgroups and $C_0$ denotes the clique factor of $P$. A *clique-inclusive co-factor* of $P$ is a subgroup of the form $C_0 \times F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$ for some $j \in \{1, \dots, k\}$. By convention, if $P = C_0$, there is no clique-inclusive co-factor.

Let now $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\rho: \mathcal{G} \to \mathcal{G}$ be an action-like cocycle. Let $\mathcal{P}$ be a measured subgroupoid such that $(\mathcal{P}, \rho)$ is of product type. Let $X^* = \bigsqcup_{i \in I} X_i$ be a partition of a conull Borel subset $X^* \subseteq X$, and for every $i \in I$, $P_i$ be a product parabolic subgroup of $\mathcal{G}$, such that $P_{|X_i} = \rho^{-1}(P_i)|_{X_i}$. Let $\mathcal{Q}$ be a measured subgroupoid of $\mathcal{P}$. We say that $(\mathcal{Q}, \rho)$ is of *clique-inclusive co-factor type within* $(\mathcal{P}, \rho)$ if, up to passing to a further conull Borel subset and refining the above partition, for every $i \in I$, there exists a clique-inclusive co-factor $Q_i$ of $P_i$ such that $Q_{|X_i} = \rho^{-1}(Q_i)|_{X_i}$.

We now aim to give a characterization of subgroupoids of clique-inclusive co-factor type. For this, we highlight the following property, an informal explanation is given below it.

**Definition 7.8 (Property $(P_{\text{cofact}}))$.** Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$. Let $\mathcal{Q} \subseteq \mathcal{P}$ be measured subgroupoids of $\mathcal{G}$. We say that the triple $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies *Property $(P_{\text{cofact}})$* if the following three properties hold.

(P_{\text{cofact}})_1 \text{ The groupoid } \mathcal{Q} \text{ is stably normal in } \mathcal{P}.

(P_{\text{cofact}})_2 \text{ There exists a measured subgroupoid } \mathcal{B} \subseteq \mathcal{P} \text{ of infinite type such that}

(a) the groupoid $\mathcal{B}$ is stably normalized by a measured subgroupoid $\mathcal{N}$ of $\mathcal{G}$ such that

i. whenever $\mathcal{N}'$ is a measured subgroupoid of $\mathcal{G}$ that stably normalizes $\mathcal{P}$, for every positive measure Borel subset $U \subseteq X$, one has $\mathcal{N}_{|U} \not\subseteq \mathcal{N}'_{|U}$;

ii. for every positive measure Borel subset $U \subseteq X$, if $\mathcal{B}_{|U}$ is contained in a measured subgroupoid $\mathcal{S}$ of $P_{|U}$ such that $(\mathcal{G}_{|U}, P_{|U}, \mathcal{S})$ satisfies Property $(P_{\text{spec}})$, and if $\mathcal{N}'$ is a measured subgroupoid of $\mathcal{G}_{|U}$ that stably normalizes $\mathcal{S}$, then for every positive measure Borel subset $V \subseteq U$, one has $\mathcal{N}_{|V} \not\subseteq \mathcal{N}'_{|V}$;

(b) the groupoid $\mathcal{Q}$ stably normalizes $\mathcal{B}$.

(P_{\text{cofact}})_3 \text{ Whenever } \mathcal{Q}' \text{ is a measured subgroupoid of } \mathcal{P} \text{ such that } (\mathcal{G}, \mathcal{P}, \mathcal{Q}') \text{ satisfies Properties } (P_{\text{cofact}})_1 \text{ and } (P_{\text{cofact}})_2, \text{ and such that } \mathcal{Q} \text{ is stably contained in } \mathcal{Q}', \text{ then } \mathcal{Q} \text{ and } \mathcal{Q}' \text{ are stably equal.}

The idea behind Property $(P_{\text{cofact}})$ is the following. Let $P = F_1 \times \cdots \times F_n$ be a product parabolic subgroup, and for simplicity let us assume that its clique factor is trivial. Let $j \in \{1, \dots, n\}$. The group $Q = F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_n$ is normal in $P$, as in $(P_{\text{cofact}})_1$. Under our assumption that the defining graph $\Gamma$ is strongly reduced, the subgraph $\Lambda_j$ that gives the type of $F_j$ is not collapsible. We can therefore find $B \subseteq F_j$, conjugate to a vertex
<figure><img src="image_20.png" /><figcaption>Figure 17: Objects appearing in Step 1 of the proof of Lemma 7.9</figcaption></figure>

group, whose normalizer is not contained in the normalizer of $F_j$. In particular $N_G(B)$ is not contained in $N_G(P)$ (as expressed in its groupoid-theoretic form by $(P_{\text{cofact}})_2(a.i))$. And if $F_j$ is contained in a special subproduct $S$, then $N_G(B)$ is also not contained in $N_G(S)$ (as expressed in its groupoid-theoretic form by $(P_{\text{cofact}})_2(a.ii)$). Finally notice that $B$ is normalized by $Q$, as in $(P_{\text{cofact}})_2(b)$. Conversely, we will prove (in a groupoid-theoretic context) that these properties are enough to ensure that a subgroup(oid) is contained in one of co-factor type. And the maximality property $(P_{\text{cofact}})_3$ is here to ensure that we get the full co-factor, not just a proper subgroup(oid). (In the case where the clique factor of $P$ is non-trivial, the above remains true, except that it now only provides a characterization of the co-factors which are clique-inclusive.) This is the contents of our next lemma, which as explained above crucially uses the assumption that the underlying graph $\Gamma$ is strongly reduced.

**Lemma 7.9.** Let $G$ be a graph product of countably infinite groups over a strongly reduced finite simple graph $\Gamma$. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$. Let $\mathcal{Q} \subseteq \mathcal{P} \subseteq \mathcal{G}$ be measured subgroupoids, with $(\mathcal{P}, \rho)$ of product type and nowhere of clique type.

Then $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$ if and only if $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies Property $(P_{\text{cofact}})$.

We summarize in Figure 17 the objects appearing in the proof and the link between them.

**Proof. Step 1** Let us prove that if $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$, then $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies $(P_{\text{cofact}})_1$ and $(P_{\text{cofact}})_2$.

Notice that if $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$, then $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies $(P_{\text{cofact}})_1$, because any co-factor is normal in the ambient product (see Example 4.6). We now prove that $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ also satisfies Property $(P_{\text{cofact}})_2$.

Up to replacing $X$ by a conull Borel subset and partitioning it into countably many Borel subsets, we can (and will) assume that $\mathcal{P} = \rho^{-1}(P)$ and $\mathcal{Q} = \rho^{-1}(Q)$, where
* $P = C_0 \times F_1 \times \dots \times F_k$ is a product parabolic subgroup, with (possibly trivial) clique factor $C_0$, with $k \ge 1$, and no $F_i$ decomposes further as a direct product;

* And $Q = C_0 \times F_1 \times \dots \times \hat{F}_j \times \dots \times F_k$ for some $j \in \{1, \dots, k\}$.

We now fix $j \in \{1, \dots, k\}$ given by the description of $Q$. Up to a global conjugation, we will also assume that $P = G_\Lambda$ and $Q = G_\Upsilon$ for some induced subgraphs $\Upsilon \subseteq \Lambda \subseteq \Gamma$. We let $\Lambda_j \subseteq \Lambda$ be the subgraph such that $F_j = G_{\Lambda_j}$ – notice that $\Lambda_j$ is not reduced to a point. We recall from Lemma 4.15, page 43 that $(\mathcal{P}, \rho)$ is tightly $P$-supported.

Since $\Gamma$ is strongly reduced, the subgraph $\Lambda_j$ (which is not reduced to a point) is not collapsible. Therefore, there exists a vertex $v \in V\Lambda_j$ which is joined to a vertex $w \notin V\Lambda_j$, such that $w$ is not joined to all vertices of $\Lambda_j$. Notice in particular that

$$G_w \subseteq G_v^{\perp} \quad \text{and} \quad G_w \not\subseteq F_j \times F_j^{\perp} \quad \text{and} \quad G_w \not\subseteq P \times P^{\perp}. \qquad (7.1)$$

Let $\mathcal{B} = \rho^{-1}(G_v)$. Then $\mathcal{B}$ is contained in $\mathcal{P}$, is of infinite type (because $G_v$ is infinite and $\rho$ is action-like) and is normalized by $\mathcal{Q}$ (because $G_v$ is normalized by $\mathcal{Q}$, and using Example 4.6). This proves $(\mathrm{P}_{\mathrm{cofact}})_2(\mathcal{b})$.

We now prove $(\mathrm{P}_{\mathrm{cofact}})_2(\mathcal{a})$. Let $\mathcal{N} = \rho^{-1}(G_v \times G_v^{\perp})$, which normalizes $\mathcal{B}$.

To check $(\mathrm{P}_{\mathrm{cofact}})_2(\mathcal{a})$, let $\mathcal{N}'$ be a measured subgroupoid of $\mathcal{G}$ that stably normalizes $\mathcal{P}$. Without loss of generality, we will assume that $\mathcal{N}'$ normalizes $\mathcal{P}$. Since $(\mathcal{P}, \rho)$ is tightly $P$-supported, Lemma 4.17, page 44 ensures that $\mathcal{N}' \subseteq \rho^{-1}(P \times P^{\perp})$ (up to restricting to a conull Borel subset). And eq. (7.1) implies that $G_v \times G_v^{\perp} \not\subseteq P \times P^{\perp}$. Lemma 4.14, page 43, thus ensures that for every positive measure Borel subset $U \subseteq X$, one has $\mathcal{N}_{|U} \not\subseteq \mathcal{N}'_{|U}$, as desired.

To check $(\mathrm{P}_{\mathrm{cofact}})_2(\mathcal{a})$, let $U, S, \mathcal{N}', V$ be as in the statement. Let $W \subseteq V$ be a positive measure Borel subset such that $\mathcal{N}'_{|W}$ normalizes $S_{|W}$. Since $(G_{|W}, P_{|W}, S_{|W})$ satisfies Property $(\mathrm{P}_{\mathrm{spec}})$, the second point of Lemma 7.7 ensures that, up to replacing $W$ by a positive measure subset, $(S_{|W}, \rho)$ is tightly $S$-supported for some special subproduct $S$ of $P$. Since by assumption $B_{|W} = \rho^{-1}(G_v)_{|W}$ is contained in $S_{|W}$, it follows from Lemma 4.14, page 43 that $G_v \subseteq S$. So $F_j \subseteq S$ since $S$ is a subproduct. By Lemma 4.17, up to replacing $W$ by a conull Borel subset, we have $\mathcal{N}'_{|W} \subseteq \rho^{-1}(S \times S^{\perp})_{|W}$, which in turn is contained in $\rho^{-1}(F_j \times F_j^{\perp})_{|W}$. The first two inclusions of eq. (7.1) give that $G_v \times G_v^{\perp} \not\subseteq F_j \times F_j^{\perp}$, so Lemma 4.14 implies that $\mathcal{N}_{|W} \not\subseteq \mathcal{N}'_{|W}$. And therefore $\mathcal{N}_{|V} \not\subseteq \mathcal{N}'_{|V}$, as desired.

**Step 2** We now assume that $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies Properties $(\mathrm{P}_{\mathrm{cofact}})_1$ and $(\mathrm{P}_{\mathrm{cofact}})_2$, and we prove that there exists a measured subgroupoid $\mathcal{Q}'$ of $\mathcal{G}$ such that $(\mathcal{Q}', \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$, and $\mathcal{Q}$ is stably contained in $\mathcal{Q}'$.

Let $\mathcal{B}, \mathcal{N}$ be as in $(\mathrm{P}_{\mathrm{cofact}})_2$. Again, up to passing to a conull Borel subset of $X$ and partitioning it into at most countably many Borel subsets, we will assume that $\mathcal{P} = \rho^{-1}(P)$ where $P = F_1 \times \dots \times F_n$ is a product parabolic subgroup of $\mathcal{G}$, where the $F_i$ are chosen not to split further (in particular the clique subgroup of $P$ is decomposed as the product of its vertex groups). We will further assume that $\mathcal{B}$ is normalized by $\mathcal{N}$ and $\mathcal{Q}$, and that $\mathcal{Q}$ is normal in $\mathcal{P}$. Using Lemma 4.16, we can (and will) also assume that $(\mathcal{B}, \rho)$ is tightly $\mathcal{B}$-supported for some parabolic subgroup $\mathcal{B} \subseteq P$, and that $(\mathcal{Q}, \rho)$ is tightly $\mathcal{Q}$-supported for some parabolic subgroup $\mathcal{Q} \subseteq P$.
Since $\mathcal{B}$ is normalized by $\mathcal{N}$, Lemma 4.17 ensures (up to restricting to a conull Borel subset) that $\mathcal{N} \subseteq \rho^{-1}(B \times B^{\perp})$. For $j \in \{1, \dots, n\}$, let $B_j = B \cap F_j$.

We claim that there exists $j \in \{1, \dots, n\}$ such that $B_j \neq \{1\}$ and $B_j \neq F_j$. Indeed, otherwise $B$ would be a subproduct. But in this case, if $B$ was a nonspecial subproduct, then $B \times B^{\perp} \subseteq P \times P^{\perp}$, so $\mathcal{N} \subseteq \rho^{-1}(B \times B^{\perp}) \subseteq \rho^{-1}(P \times P^{\perp})$. Since $\mathcal{N}' := \rho^{-1}(P \times P^{\perp})$ normalizes $\mathcal{P}$, this gives a contradiction to Property $(\mathrm{P}_{\mathrm{cofact}})_2(a.i)$. If $B$ was special, then $(\mathcal{G}, \mathcal{P}, \rho^{-1}(B))$ would satisfy Property $(\mathrm{P}_{\mathrm{spec}})$ (Lemma 7.7), and the inclusion $\mathcal{N} \subseteq \rho^{-1}(B \times B^{\perp})$ would give a contradiction to Property $(\mathrm{P}_{\mathrm{cofact}})_2(a.ii)$, applied this time to $\mathcal{N}' := \rho^{-1}(B \times B^{\perp})$. Hence our claim.

Let $j \in \{1, \dots, n\}$ be as above. The above claim shows that $F_j$ contains $B_j$ as a proper parabolic subgroup, so necessarily $F_j$ is not conjugate to a vertex group. Since $\mathcal{B}$ is normalized by $\mathcal{Q}$, we have $\mathcal{Q} \subseteq \rho^{-1}(B \times B^{\perp})$ (Lemma 4.17), so $\mathcal{Q} \subseteq B \times B^{\perp}$. In particular $\mathcal{Q} \cap F_j$ is contained in $B_j \times B_j^{\perp}$. So $\mathcal{Q} \subseteq F_1 \times \dots \times F_{j-1} \times ((B_j \times B_j^{\perp}) \cap F_j) \times F_{j+1} \times \dots \times F_n$. Since $\mathcal{Q}$ is normal in $\mathcal{P} = \rho^{-1}(P)$ and $(\mathcal{Q}, \rho)$ is tightly $\mathcal{Q}$-supported, we have $P \subseteq \mathcal{Q} \times \mathcal{Q}^{\perp}$, and we deduce that $\mathcal{Q} \subseteq F_1 \times \dots \times \hat{F}_j \times \dots \times F_n$. Let $\mathcal{Q}' = \rho^{-1}(F_1 \times \dots \times \hat{F}_j \times \dots \times F_n)$. Since $F_j$ is not conjugate to a vertex group, the subgroup $F_1 \times \dots \times \hat{F}_j \times \dots \times F_n$ is a clique-inclusive co-factor, and therefore $(\mathcal{Q}', \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$. So Step 1 of this proof shows that $\mathcal{Q}'$ satisfies Properties $(\mathrm{P}_{\mathrm{cofact}})_1$ and $(\mathrm{P}_{\mathrm{cofact}})_2$. Moreover $\mathcal{Q} \subseteq \mathcal{Q}'$ (up to a conull Borel subset). Hence the wanted assertion.

**Step 3** We now assume that $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies Property $(\mathrm{P}_{\mathrm{cofact}})$, and prove that $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$.

Step 2 gives a measured subgroupoid $\mathcal{Q}'$ of $\mathcal{G}$ such that $(\mathcal{Q}', \rho)$ is of clique-inclusive type within $(\mathcal{P}, \rho)$, and $\mathcal{Q}$ is stably contained in $\mathcal{Q}'$. Step 1 then ensures that $(\mathcal{G}, \mathcal{P}, \mathcal{Q}')$ satisfies $(\mathrm{P}_{\mathrm{cofact}})_1$ and $(\mathrm{P}_{\mathrm{cofact}})_2$. And Assertion $(\mathrm{P}_{\mathrm{cofact}})_3$ then implies that $\mathcal{Q}$ is stably equal to $\mathcal{Q}'$, showing that $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$.

**Step 4** We are left with showing that if $(\mathcal{Q}, \rho)$ is of clique-inclusive co-factor type within $(\mathcal{P}, \rho)$, then $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies $(\mathrm{P}_{\mathrm{cofact}})_3$.

So assume that $\mathcal{Q} = \rho^{-1}(\mathcal{Q})$ for some clique-inclusive co-factor $\mathcal{Q}$ of $P$, and let $\mathcal{Q}'$ be a measured subgroupoid of $\mathcal{P}$ such that $(\mathcal{G}, \mathcal{P}, \mathcal{Q}')$ satisfies $(\mathrm{P}_{\mathrm{cofact}})_1$ and $(\mathrm{P}_{\mathrm{cofact}})_2$, such that $\mathcal{Q}$ is stably contained in $\mathcal{Q}'$. By Step 2 applied to the triple $(\mathcal{G}, \mathcal{P}, \mathcal{Q}')$, up to a conull Borel subset and a countable partition of the base space, $\mathcal{Q}'$ is contained in $\rho^{-1}(\mathcal{Q}')$ for some clique-inclusive co-factor $\mathcal{Q}'$ of $P$. Lemma 4.14 implies that $\mathcal{Q} \subseteq \mathcal{Q}'$, and since $\mathcal{Q}$ and $\mathcal{Q}'$ are two co-factors of $P$ it follows that $\mathcal{Q} = \mathcal{Q}'$, which concludes our proof. $\square$

**Factors.** 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 $\mathcal{P}$ be a measured subgroupoid of $\mathcal{G}$ such that $(\mathcal{P}, \rho)$ is of parabolic type. Let $X^* = \bigsqcup_{i \in I} X_i$ be a partition of a conull Borel subset $X^* \subseteq X$ into at most countably many Borel subsets, and for every $i \in I$, let $P_i$ be a parabolic subgroup of $G$, such that $\mathcal{P}|_{X_i} = \rho^{-1}(P_i)|_{X_i}$. Let $\mathcal{F}$ be a measured subgroupoid of $\mathcal{P}$. We say that $(\mathcal{F}, \rho)$ is of factor type within $(\mathcal{P}, \rho)$ if, up to passing to a further conull Borel subset and refining the above partition, for every $i \in I$, there exists a factor $F_i$ of $P_i$ such that $\mathcal{F}|_{X_i} = \rho^{-1}(F_i)|_{X_i}$. We say that $(\mathcal{F}, \rho)$ is of clique factor type within $(\mathcal{P}, \rho)$ if additionally, for every $i \in I$ such that $X_i$ has positive measure, $F_i$ is the clique factor of $P_i$.
We now aim to characterize these notions using a purely groupoid-theoretic property, which is as follows.

**Definition 7.10 (Property ($P_{\text{fact}$}))**. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$. Let $\mathcal{F} \subseteq \mathcal{P}$ be measured subgroupoids of $\mathcal{G}$. We say that the triple $(\mathcal{G}, \mathcal{P}, \mathcal{F})$ satisfies *Property* ($P_{\text{fact}}$) if the following three properties hold.
($P_{\text{fact}}$)$_1$ $\mathcal{F}$ is of infinite type.

($P_{\text{fact}}$)$_2$ There exist $n \ge 1$ and measured subgroupoids $Q_1, \dots, Q_n$ of $\mathcal{P}$ such that $(\mathcal{G}, \mathcal{P}, Q_j)$ satisfies Property $(P_{\text{cofact}})$ for every $j \in \{1, \dots, n\}$, and a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ such that for every $i \in I$, one has $\mathcal{F}_{|X_i} = (Q_1 \cap \dots \cap Q_n)|_{X_i}$.

($P_{\text{fact}}$)$_3$ For every measured subgroupoid $\mathcal{Q}$ of $\mathcal{P}$ such that $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies Property $(P_{\text{cofact}})$, and every positive measure Borel subset $U \subseteq X$, there exists a positive measure Borel subset $V \subseteq U$ such that either $\mathcal{F}_{|V} \subseteq \mathcal{Q}_{|V}$ or $(\mathcal{F} \cap \mathcal{Q})_{|V}$ is trivial.

The idea behind Property (P<sub>fact</sub>) is the following. If $P = F_1 \times \dots \times F_n$ is a product parabolic subgroup with trivial clique factor, then the factors $F_j$ are characterized as the intersections of co-factors (translated in (P<sub>fact</sub>)<sub>2</sub>) which are infinite (as in (P<sub>fact</sub>)<sub>1</sub>) and minimal (minimality is the aim of (P<sub>fact</sub>)<sub>3</sub>). And if P has a non-trivial clique factor, then all clique-inclusive co-factors contain it, and therefore the clique factor is characterized as being the intersection of all clique-inclusive co-factors. The following lemma expresses this idea at the groupoid-theoretic level.

**Lemma 7.11.** Let $G$ be a graph product of countably infinite groups over a strongly reduced finite simple graph $\Gamma$. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$. Let $P \subseteq G$ be a product parabolic subgroup with clique factor $C_0$, with $P \neq C_0$, and let $\mathcal{P} = \rho^{-1}(P)$. Let $\mathcal{F} \subseteq \mathcal{P}$ be a measured subgroupoid.

1. If $C_0 = \{1\}$, then $(\mathcal{F}, \rho)$ is of factor type within $(\mathcal{P}, \rho)$ if and only if $(\mathcal{G}, \mathcal{P}, \mathcal{F})$ satisfies Property $(P_{\text{fact}})$.

2. If $C_0 \neq \{1\}$, then $(\mathcal{F}, \rho)$ is of clique factor type within $(\mathcal{P}, \rho)$ if and only if $(\mathcal{G}, \mathcal{P}, \mathcal{F})$ satisfies Property $(P_{\text{fact}})$.

*Proof.* We first assume that $C_0 = \{1\}$ and that $(\mathcal{F}, \rho)$ is of factor type within $(\mathcal{P}, \rho)$. Write $P = F_1 \times \dots \times F_n$, where the factors $F_j$ do not further decompose as products. Up to a conull Borel subset and a countable partition, and a permutation of the factors $F_j$, we will assume that $\mathcal{F} = \rho^{-1}(F_1)$. In particular $\mathcal{F}$ is of infinite type because $F_1$ is infinite and $\rho$ is action-like, and hence $(P_{\text{fact}})_1$ is verified. Now, for every $j \in \{2, \dots, k\}$, let $Q_j = F_1 \times \dots \times \hat{F}_j \times \dots \times F_k$, and let $Q_j = \rho^{-1}(Q_j)$. Lemma 7.9, page 82 ensures that $(\mathcal{G}, \mathcal{P}, Q_j)$ satisfies Property $(P_{\text{cofact}})$. In addition $\mathcal{F} = Q_2 \cap \dots \cap Q_k$, showing that $(P_{\text{fact}})_2$ holds. Finally, let $\mathcal{Q}$ be a measured subgroupoid of $\mathcal{P}$ such that $(\mathcal{G}, \mathcal{P}, \mathcal{Q})$ satisfies $(P_{\text{cofact}})$. By Lemma 7.9, there exist a positive measure Borel subset $U \subseteq X$ and a clique-inclusive co-factor $\mathcal{Q} \subseteq \mathcal{P}$ such that $\mathcal{Q}|_U = \rho^{-1}(\mathcal{Q})|_U$. Either $\mathcal{Q} = F_2 \times \dots \times F_n$, in which case $(\mathcal{F} \cap \mathcal{Q})|_U$ is trivial, or else $F_1 \subseteq \mathcal{Q}$, in which case $\mathcal{F}|_U \subseteq \mathcal{Q}|_U$. This shows that $(P_{\text{fact}})_3$ holds.
We now assume that $C_0 \neq \{1\}$ and that $(\mathcal{F}, \rho)$ is of clique factor type within $(\mathcal{P}, \rho)$. Write $P = C_0 \times F_1 \times \cdots \times F_k$, where the $F_j$ do not further decompose as direct products. Up to a conull Borel subset and a countable partition, we will assume that $\mathcal{F} = \rho^{-1}(C_0)$. In particular $\mathcal{F}$ is of infinite type because $C_0$ is infinite and $\rho$ is action-like, showing that $(\mathrm{P}_{\mathrm{fact}})_1$ holds. For every $j \in \{1, \dots, k\}$, let $Q_j = C_0 \times F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$, and let $Q = \rho^{-1}(Q_j)$. Lemma 7.9 ensures that $(\mathcal{G}, \mathcal{P}, Q)$ satisfies Property $(\mathrm{P}_{\mathrm{cofact}})$. In addition $\mathcal{F} = Q_1 \cap \cdots \cap Q_k$, showing that $(\mathrm{P}_{\mathrm{fact}})_2$ holds. Finally, let $Q$ be a measured subgroupoid of $\mathcal{P}$ such that $(\mathcal{G}, \mathcal{P}, Q)$ satisfies $(\mathrm{P}_{\mathrm{cofact}})$. By Lemma 7.9, for every positive measure Borel subset $U \subseteq X$ there exist a positive measure Borel subset $V \subseteq U$ and a clique-inclusive co-factor $Q \subseteq P$ such that $Q|_V = \rho^{-1}(Q)|_V$. Then $C_0 \subseteq Q$, so $\mathcal{F}|_V \subseteq Q|_V$, showing that $(\mathrm{P}_{\mathrm{fact}})_3$ holds.

Conversely, let us now assume that $(\mathcal{G}, \mathcal{P}, \mathcal{F})$ satisfies Property $(\mathrm{P}_{\mathrm{fact}})$. Let $Q_1, \dots, Q_n$ be measured subgroupoids of $\mathcal{P}$ as in $(\mathrm{P}_{\mathrm{fact}})_2$. Up to a conull Borel subset and a countable partition, we will assume that $\mathcal{F} = Q_1 \cap \cdots \cap Q_n$, and, using Lemma 7.9, that $Q_j = \rho^{-1}(Q_j)$ for some clique-inclusive co-factor $Q_j$. Thus $\mathcal{F} = \rho^{-1}(F)$, where $F = Q_1 \cap \cdots \cap Q_n$ is a subproduct of $P$. Then $F$ is non-trivial as $\mathcal{F}$ is of infinite type by $(\mathrm{P}_{\mathrm{fact}})_1$ while $\rho$ has trivial kernel, and $F$ contains $C_0$. To conclude the proof, there only remains to show that

* if $C_0 = \{1\}$, then $F$ does not split as a product of at least two factors;

* if $C_0 \neq \{1\}$, then $F$ does not split as a product of $C_0$ and at least another factor.

But in both cases, if it did, then we could find a clique-inclusive co-factor $Q$ such that $F \cap Q$ is an infinite proper parabolic subgroup of $F$ (in particular $F \not\subseteq Q$). We now reach a contradiction to $(\mathrm{P}_{\mathrm{fact}})_3$ by letting $Q = \rho^{-1}(Q)$: indeed for every positive measure Borel subset $U \subseteq X$, Lemma 4.14 shows that $\mathcal{F}|_U \not\subseteq Q|_U$, while the fact that $F \cap Q$ is infinite and $\rho$ is action-like ensures that $(\mathcal{F} \cap Q)|_U$ is non-trivial. This completes our proof. □

## 7.4 Proof of the Vertex Recognition Property

We finally introduce the property that will enable us to characterize subgroupoids of vertex type<fnref n="4" /> with no reference to an action-like cocycle, under the condition that the defining graph of $G$ is transvection-free and strongly reduced. We first state the property, and then comment it below – it is inspired from Lemma 6.6.

**Definition 7.12 (Property $(\mathrm{P}_{\mathrm{vert}}))$.** Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, and let $\mathcal{V}$ be a measured subgroupoid of $\mathcal{G}$. We say that the pair $(\mathcal{G}, \mathcal{V})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$ if $\mathcal{V}$ is of infinite type and the following four properties hold.

$(\mathrm{P}_{\mathrm{vert}})_1$ There exists a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ such that for every $i \in I$, there exist measured subgroupoids

$$ \mathcal{G}|_{X_i} = \mathcal{F}_0 \supseteq \mathcal{P}_1 \supseteq \mathcal{F}_1 \supseteq \mathcal{P}_2 \supseteq \mathcal{F}_2 \supseteq \cdots \supseteq \mathcal{P}_n \supseteq \mathcal{F}_n = \mathcal{V}|_{X_i} $$

of $\mathcal{G}|_{X_i}$ (with $n \geq 1$) such that

(a) for every $j \in \{1, \dots, n\}$, the pair $(\mathcal{F}_{j-1}, \mathcal{P}_j)$ satisfies Property $(\mathrm{P}_{\mathrm{prod}})$;

(b) for every $j \in \{1, \dots, n\}$, the triple $(\mathcal{G}, \mathcal{P}_j, \mathcal{F}_j)$ satisfies Property $(\mathrm{P}_{\mathrm{fact}})$.

<footnote n="4">⁴Recall that groupoids of vertex type are defined in Section 5.1, page 53</footnote>
(P<sub>vert</sub>)<sub>2</sub> For every measured subgroupoid <i>W</i> of <i>G</i> of infinite type which is stably contained in <i>V</i>, and every measured subgroupoid <i>H</i> ⊆ <i>G</i>, if <i>W</i> is stably normalized by <i>H</i>, then <i>V</i> is stably normalized by <i>H</i>.

(P<sub>vert</sub>)<sub>3</sub> There exists a measured subgroupoid <i>N</i> of <i>G</i> that stably normalizes <i>V</i>, such that for every measured subgroupoid <i>V</i>' of <i>G</i> such that (<i>G</i>, <i>V</i}') satisfies Properties (P<sub>vert</sub>)<sub>1</sub> and (P<sub>vert</sub>)<sub>2</sub>, every measured subgroupoid <i>N</i>' of <i>G</i> that stably normalizes <i>V</i>', and every positive measure Borel subset <i>U</i> ⊆ <i>X</i>, there exists a positive measure Borel subset <i>V</i> ⊆ <i>U</i> such that either <i>N</i><sub>|<i>V</i></sub> = <i>N</i>'<sub>|<i>V</i></sub> or <i>N</i><sub>|<i>V</i></sub> &notin; <i>N</i>'<sub>|<i>V</i></sub>.

(P<sub>vert</sub>)<sub>4</sub> Whenever <i>V</i>' is a measured subgroupoid of <i>G</i> such that (<i>G</i>, <i>V</i}') satisfies Properties (P<sub>vert</sub>)<sub>1</sub>, (P<sub>vert</sub>)<sub>2</sub> and (P<sub>vert</sub>)<sub>3</sub>, if <i>V</i> is stably contained in <i>V</i>', then they are stably equal.

A few comments regarding the meaning of this property are in order. Property (P<sub>vert</sub>)<sub>1</sub> is a groupoid translation of the contents of Lemma 6.6, page 65, saying informally that vertex groups can be obtained by successively passing to maximal product parabolic subgroups and their factors. However, if we only considered Property (P<sub>vert</sub>)<sub>1</sub> and end this process too early, we could end up with a subgroupoid “of parabolic type”, i.e. of the form ρ<sup>-1</sup>(<i>B</i>) for some parabolic subgroup <i>B</i> ⊆ <i>G</i> (possibly after taking a conull subset and a countable Borel partition), where <i>B</i> could be larger than a vertex group. Also, because of the slight asymmetry in our lemma that deals with subgroupoids of maximal product type (Lemma 7.4, page 77), we could possibly end up with a subgroupoid contained in one “of clique type” (possibly, a subgroupoid strictly contained in one of vertex type), with no precise control. The role of Properties (P<sub>vert</sub>)<sub>2</sub>, (P<sub>vert</sub>)<sub>3</sub> and (P<sub>vert</sub>)<sub>4</sub> is to exclude these extra possibilities, as we now explain.

Under our assumption that the defining graph Γ is strongly reduced, a parabolic
subgroup <i>B</i> that is not conjugate to a vertex group, always contains a proper (parabolic)
subgroup <i>A</i> ⊆ <i>B</i> with N<sub><i>G</i></sub>(<i>A</i>) ⊈ N<sub><i>G</i></sub>(<i>B</i>). On the other hand a vertex group does not.
Property (P<sub>vert</sub>)<sub>2</sub> is the groupoid translation of this fact, which will allow us to exclude
subgroupoids of parabolic type but not of vertex type.

Assuming that Γ is transvection-free, if *B* is a proper subgroup of a clique subgroup *C*,
then the normalizer of *B* is strictly contained in the normalizer of any vertex subgroup
of *C*. On the other hand, the normalizer of a vertex group is not strictly contained in the
normalizer of any other vertex group. A groupoid version of this distinction is exploited
in Property (P<sub>vert</sub>)<sub>3</sub>, to exclude subgroupoids contained in one “of clique type” but not
contained in one of vertex type.

Finally, a maximality assumption is exploited in Property (P<sub>vert</sub>)<sub>4</sub> to exclude proper
subgroupoids of subgroupoids of vertex type.

Having explained the general idea behind it, we can now turn to the proof of our
main lemma, from which the Vertex Recognition Property (see Definition 5.1) follows.

**Lemma 7.13.** Let *G* be a graph product of countably infinite groups over a strongly reduced transvection-free finite simple graph Γ not reduced to a vertex. Let *G* be a measured groupoid over a standard probability space *X*, equipped with an action-like cocycle ρ : *G* → *G*. Let *V* ⊆ *G* be a measured subgroupoid.

Then ($\mathcal{V}$, $\rho$) is of vertex type if and only if ($\mathcal{G}$, $\mathcal{V}$) satisfies Property ($\mathbf{P}_{\text{vert}}$).

*Proof.* The proof has four steps.
**Step 1.** Let us show that if $(\mathcal{V}, \rho)$ is of vertex type, then $\mathcal{V}$ is of infinite type and $(\mathcal{G}, \mathcal{V})$ satisfies $(P_{\text{vert}})_1$ and $(P_{\text{vert}})_2$.

Since $\rho$ is action-like and $(\mathcal{V}, \rho)$ is of vertex type, and since vertex groups are infinite, $\mathcal{V}$ is of infinite type. Up to passing to a conull Borel subset, taking a countable partition, and conjugating the cocycle, we can assume that $\mathcal{V} = \rho^{-1}(G_v)$ for some vertex $v \in V\Gamma$. The vertex $v$ is untransvectable because $\Gamma$ is transvection-free.

For $(P_{\text{vert}})_1$, by Lemma 6.6, page 65, there exists a chain of parabolic subgroups

$$G = F_0 \supseteq P_1 \supseteq F_1 \supseteq P_2 \supseteq F_2 \supseteq \dots \supseteq P_n \supseteq F_n = G_v$$

such that

1. for every $j \in \{1, \dots, n\}$, $P_j$ is a maximal product parabolic subgroup of $F_{j-1}$ which is not of isolated clique type, and

2. for every $j \in \{1, \dots, n-1\}$, the clique factor of $P_j$ is trivial, and $F_j$ is a factor of $P_j$, and

3. $F_n$ is the clique factor of $P_n$.

For every $j \in \{1, \dots, n\}$, let $\mathcal{P}_j = \rho^{-1}(P_j)$, and let $\mathcal{F}_j = \rho^{-1}(F_j)$. These come equipped with action-like cocycles to $P_j, F_j$, respectively. Notice that for every $j \in \{1, \dots, n\}$, the pair $(\mathcal{P}_j, \rho)$ is nowhere of isolated clique type. By the first assertion of Lemma 7.4, page 77, for every $j \in \{1, \dots, n\}$, the pair $(\mathcal{F}_{j-1}, \mathcal{P}_j)$ satisfies Property $(P_{\text{prod}})$. By Lemma 7.11, using that $\Gamma$ is strongly reduced, for every $j \in \{1, \dots, n\}$, the triple $(\mathcal{G}, \mathcal{P}_j, \mathcal{F}_j)$ satisfies Property $(P_{\text{fact}})$.

For $(P_{\text{vert}})_2$, let $\mathcal{W}$ be a measured subgroupoid of infinite type which is stably contained in $\mathcal{V}$, and let $\mathcal{H} \subseteq \mathcal{G}$ be a measured subgroupoid that stably normalizes $\mathcal{W}$. Up to a countable partition of the base space, we will assume that $\mathcal{W} \subseteq \mathcal{V}$ and that $\mathcal{H}$ normalizes $\mathcal{W}$. Then $(\mathcal{W}, \rho)$ is tightly $G_v$-supported, so Lemma 4.17, page 44, ensures (up to restricting to a conull Borel subset) that $\mathcal{H} \subseteq \rho^{-1}(G_v \times G_v^{\perp})$. Therefore $\mathcal{H}$ normalizes $\mathcal{V} = \rho^{-1}(G_v)$, as desired.

**Step 2.** Let us show that if $(\mathcal{G}, \mathcal{V})$ satisfies Properties $(P_{\text{vert}})_1$ and $(P_{\text{vert}})_2$, then there exist 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 $C_i$ that is conjugate to $G_{\Upsilon_i}$, for some non-empty complete subgraph $\Upsilon_i \subseteq \Gamma$ (possibly reduced to one vertex), such that $\mathcal{V}|_{X_i} \subseteq \rho^{-1}(C_i)|_{X_i}$.

Up to a countable partition, we can assume that there exist subgroupoids

$$\mathcal{G} = \mathcal{F}_0 \supseteq \mathcal{P}_1 \supseteq \mathcal{F}_1 \supseteq \mathcal{P}_2 \supseteq \mathcal{F}_2 \supseteq \dots \supseteq \mathcal{P}_n \supseteq \mathcal{F}_n = \mathcal{V}$$

as in $(P_{\text{vert}})_1$. By inductively applying Lemmas 7.4 and 7.11, we show that for every $i \in \{1, \dots, n\}$, up to passing to a conull Borel subset and taking a countable partition,

* Either $\mathcal{P}_i = \rho^{-1}(P_i)$ for some maximal product parabolic subgroup $P_i$ whose type is not a clique, and $\mathcal{F}_i = \rho^{-1}(F_i)$ for some factor $F_i$ of $P_i$;

* Or $\mathcal{P}_i \subseteq \rho^{-1}(C)$ for some parabolic subgroup $C$ that is conjugate to $G_{\Upsilon}$ for some non-empty complete subgraph $\Upsilon \subseteq \Gamma$.

In particular, either $\mathcal{V} = \rho^{-1}(B)$ for some proper parabolic subgroup $B$, or else $\mathcal{V} \subseteq \rho^{-1}(C)$ for some parabolic group $C$ that is conjugate to $G_{\Upsilon}$ for some non-empty complete
subgraph $\Upsilon \subseteq \Gamma$. To show the wanted assertion, it suffices to exclude the case where $\mathcal{V} = \rho^{-1}(B)$ for a parabolic subgroup $B$ whose type is an induced subgraph $\Lambda \subseteq \Gamma$ that contains at least 2 vertices. Up to conjugating the cocycle we will assume that $\mathcal{V} = \rho^{-1}(G_{\Lambda})$ and aim for a contradiction. Since $\Gamma$ is strongly reduced, there exists a vertex $w \in V\Lambda$ such that $G_w \times G_w^{\perp} \not\subseteq G_{\Lambda} \times G_{\Lambda}^{\perp}$, as otherwise the subgraph $\Lambda$ would be collapsible. We will now use Property $(P_{\text{vert}})_2$ to reach a contradiction. For this, let $\mathcal{W} = \rho^{-1}(G_w)$. Then $\mathcal{W}$ is contained in $\mathcal{V}$, and $\mathcal{W}$ is of infinite type because $G_w$ is infinite and $\rho$ is action-like. Let $\mathcal{H} = \rho^{-1}(G_w \times G_w^{\perp})$, which normalizes $\mathcal{W}$ (Example 4.6, page 41). So Property $(P_{\text{vert}})_2$ implies that $\mathcal{V}$ is stably normalized by $\mathcal{H}$. As $(\mathcal{V}, \rho)$ is tightly $G_{\Lambda}$-supported (Lemma 4.15, page 43), it follows from Lemma 4.17, page 44 that $\mathcal{H} \subseteq \rho^{-1}(G_{\Lambda} \times G_{\Lambda}^{\perp})$. But since $G_w \times G_w^{\perp} \not\subseteq G_{\Lambda} \times G_{\Lambda}^{\perp}$, this is a contradiction to Lemma 4.14, page 43. This contradiction completes our proof of Step 2.

**Step 3.** Let us show that if $(\mathcal{V}, \rho)$ is of vertex type, then $(\mathcal{G}, \mathcal{V})$ satisfies Property $(P_{\text{vert}})_3$.

Again we assume that $\mathcal{V} = \rho^{-1}(G_v)$ for some $v \in V\Gamma$. Let $\mathcal{N} = \rho^{-1}(G_v \times G_v^{\perp})$, which normalizes $\mathcal{V}$ (Example 4.6, page 41). Let $\mathcal{V}'$ be a measured subgroupoid of $\mathcal{G}$ that satisfies $(P_{\text{vert}})_1$ and $(P_{\text{vert}})_2$, let $\mathcal{N}'$ be a measured subgroupoid of $\mathcal{G}$ that stably normalizes $\mathcal{V}'$ and let $U \subseteq X$ be a positive measure Borel subset. By Lemma 4.16, page 44, we can find a positive measure Borel subset $V \subseteq U$ such that $(\mathcal{V}'|_V, \rho)$ is tightly $C$-supported for some parabolic subgroup $C \subseteq G$. Up to partitioning, we can (and will) also assume that $\mathcal{N}'|_V$ normalizes $\mathcal{V}'|_V$. By the previous step, $C$ is conjugate to $G_{\Upsilon}$ for some non-empty complete subgraph $\Upsilon \subseteq \Gamma$. Since $v$ is untransvable, it follows from Lemma 2.9, page 23, that either $C = G_v$, or else $G_v \times G_v^{\perp} \not\subseteq C \times C^{\perp}$. If $C = G_v$, then Lemma 4.17, page 44 ensures that $\mathcal{N}'|_V \subseteq \mathcal{N}|_V$ (up to a null set), and we are done. If $G_v \times G_v^{\perp} \not\subseteq C \times C^{\perp}$, then Lemma 4.14, page 43 ensures that $\mathcal{N}|_V \not\subseteq \rho^{-1}(C \times C^{\perp})|_V$. As $\mathcal{N}'|_V$ normalizes $\mathcal{V}'|_V$, we also have $\mathcal{N}'|_V \subseteq \rho^{-1}(C \times C^{\perp})|_V$ (Lemma 4.17, page 44). So we deduce that $\mathcal{N}|_V \not\subseteq \mathcal{N}'|_V$, and again Property $(P_{\text{vert}})_3$ holds.

**Step 4.** Let us finally show that if $(\mathcal{G}, \mathcal{V})$ satisfies Properties $(P_{\text{vert}})_1$, $(P_{\text{vert}})_2$ and $(P_{\text{vert}})_3$, then there exist 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 $A_i$ that is conjugate to a vertex subgroup, such that $\mathcal{V}|_{X_i} \subseteq \rho^{-1}(A_i)|_{X_i}$.

Otherwise, in view of Step 2 (and Lemma 4.16, page 44), there would exist a positive measure Borel subset $U \subseteq X$ such that $(\mathcal{V}|_U, \rho)$ is tightly $C$-supported for some parabolic subgroup $C$ conjugate to $G_{\Upsilon}$, where $\Upsilon \subseteq \Gamma$ is a complete subgraph of size at least 2. Let $\mathcal{N}$ be a measured subgroupoid given by $(P_{\text{vert}})_3$. Since $\mathcal{N}$ stably normalizes $\mathcal{V}$, Lemma 4.17, page 44, ensures that, up to replacing $U$ by a positive measure subset, we have $\mathcal{N}|_U \subseteq \rho^{-1}(C \times C^{\perp})|_U$. Let $A \subseteq C$ be a parabolic subgroup that is equal to a conjugate of a vertex group. Let $\mathcal{V}'$ be a measured subgroupoid of $\mathcal{G}$ such that $\mathcal{V}'|_U = \rho^{-1}(A)|_U$, and $\mathcal{V}'|_{X \setminus U} = \mathcal{V}|_{X \setminus U}$. Using Step 1, we see that $(\mathcal{G}, \mathcal{V}')$ satisfies Properties $(P_{\text{vert}})_1$ and $(P_{\text{vert}})_2$. Let $\mathcal{N}'$ be such that $\mathcal{N}'|_U = \rho^{-1}(A \times A^{\perp})|_U$, and $\mathcal{N}'|_{X \setminus U} = \mathcal{N}|_{X \setminus U}$. Then $\mathcal{N}'$ stably normalizes $\mathcal{V}'$. As every vertex of $\Gamma$ is untransvable, the inclusion $C \times C^{\perp} \subseteq A \times A^{\perp}$ is strict. Lemma 4.14, page 43 thus implies that $\mathcal{N}|_V \not\subseteq \mathcal{N}'|_V$ for every positive measure Borel subset $V \subseteq U$, contradicting $(P_{\text{vert}})_3$.
**Conclusion.** If ($\mathcal{V}$, $\rho$) is of vertex type, then $(\mathrm{P}_{\mathrm{vert}})_1$, $(\mathrm{P}_{\mathrm{vert}})_2$ and $(\mathrm{P}_{\mathrm{vert}})_3$ are verified by Steps 1 and 3. Finally, take $\mathcal{V}'$ verifying the hypothesis of $(\mathrm{P}_{\mathrm{vert}})_4$. By Step 4, up to taking a conull Borel subset and a countable partition, there exists a subgroup $A \subseteq \mathcal{G}$ that is conjugate to a vertex group, such that $\mathcal{V}' \subseteq \rho^{-1}(A)$. Up to refining the partition, we also assume that $\mathcal{V} = \rho^{-1}(B)$ for some subgroup $B \subseteq \mathcal{G}$ that is conjugate to a vertex group. Since $\mathcal{V}$ is stably contained in $\mathcal{V}'$, we deduce that $\rho^{-1}(B)$ is stably contained in $\rho^{-1}(A)$. By Lemma 4.14, page 43, this implies that $B \subseteq A$, and in fact $A = B$ since they are both conjugate to vertex groups. Hence $(\mathcal{G}, \mathcal{V})$ satisfies $(\mathrm{P}_{\mathrm{vert}})$.

Conversely, if $(\mathcal{G}, \mathcal{V})$ verifies $(\mathrm{P}_{\mathrm{vert}})$ then using the maximality assumption provided by Property $(\mathrm{P}_{\mathrm{vert}})_4$ and the above Step 4, we get that $(\mathcal{V}, \rho)$ is of vertex type. $\square$

We are now in position to complete our proof of Proposition 7.1.

*Proof*. Let $G, H$ be two graph products of countably infinite groups over transvection-free strongly reduced finite simple graphs. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, equipped with action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$. Let $\mathcal{V} \subseteq \mathcal{G}$ be a measured subgroupoid, and assume that the pair $(\mathcal{V}, \rho_G)$ is of vertex type; we aim to prove that $(\mathcal{V}, \rho_H)$ is of vertex type.

By Lemma 7.13, applied to the cocycle $\rho_G$, the pair $(\mathcal{G}, \mathcal{V})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$. And by Lemma 7.13, applied to the cocycle $\rho_H$, the pair $(\mathcal{V}, \rho_H)$ is of vertex type, as desired. $\square$

# 8 The reducible case and proof of Proposition 5.3

In this section, we complete our proof of Proposition 5.3, by treating the case of reducible graph products.

## 8.1 Goal of the section

The main result of this section is the following proposition, which extends Proposition 7.1 to the case of direct products.

**Proposition 8.1.** The class of graph products of countably infinite groups over finite simple graphs $\Gamma$ that splits as a join $\Gamma = \Gamma_1 \circ \cdots \circ \Gamma_k$ (with $k \ge 1$) of transvection-free strongly reduced graphs $\Gamma_j$ on at least 2 vertices, satisfies the Vertex Recognition Property.

In view of Corollary 6.4, this covers all graph products over transvection-free graphs with no partial conjugation, and therefore Proposition 5.3 follows from Proposition 8.1. In the previous section, we have treated the case where $k = 1$, we now need to tackle the reducible case.

## 8.2 Proof of the Vertex Recognition Property

Given a graph product $G$ that splits as a direct product with trivial clique factor, the following property will enable us to recognize the co-factors of the product, at a groupoid-theoretic level.
**Definition 8.2 (Property ($P'_{\text{cofact}}$))**. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\mathcal{Q}$ be a measured subgroupoid of $\mathcal{G}$. We say that $(\mathcal{G}, \mathcal{Q})$ satisfies *Property* ($P'_{\text{cofact}}$) if the following two properties hold.

($P'_{\text{cofact}}$)$_1$ The groupoid $\mathcal{Q}$ is of infinite type and stably normal in $\mathcal{G}$, and it stably normalizes an amenable measured subgroupoid $\mathcal{A}$ of $\mathcal{G}$ of infinite type.

($P'_{\text{cofact}}$)$_2$ Whenever $\mathcal{Q}'$ is a measured subgroupoid of $\mathcal{G}$ such that $(\mathcal{G}, \mathcal{Q}')$ satisfies Property ($P'_{\text{cofact}}$)$_1$, and such that $\mathcal{Q}$ is stably contained in $\mathcal{Q}'$, then $\mathcal{Q}$ and $\mathcal{Q}'$ are stably equal.

The idea behind Property ($P'_{\text{cofact}}$) is the following. Assume that a graph product $G$ splits non-trivially as a product $G = F_1 \times \cdots \times F_k$, with trivial clique factor. Then each $Q_j = F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$ is normal in $G$ and normalizes an infinite amenable subgroup $A \subseteq F_j$. Conversely, any maximal subgroup of $G$ with these properties is a co-factor. The following lemma is the groupoid-theoretic version of this fact (since all clique factors are trivial in this section, for simplicity, we will talk about “subgroupoids of co-factor type” as a shortcut for “subgroupoids of clique-inclusive co-factor type”).

**Lemma 8.3.** Let $G$ be a graph product over a finite simple graph $\Gamma$ with countably infinite vertex groups and trivial clique factor. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$. Let $\mathcal{Q} \subseteq \mathcal{G}$ be a measured subgroupoid.

Then $(\mathcal{G}, \mathcal{Q})$ satisfies Property ($P'_{\text{cofact}}$) if and only if $G$ is reducible and $(\mathcal{Q}, \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$.

*Proof.* Let $G = F_1 \times \cdots \times F_k$ be the decomposition of $G$ into its irreducible components (possibly with $k=1$).

**Step 1** We first assume that $G$ is reducible (i.e. $k \ge 2$) and that $(\mathcal{Q}, \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$, and prove that $(\mathcal{G}, \mathcal{Q})$ satisfies $(P'_{\text{cofact}})_1$.

Up to a conull Borel subset and a countable partition, we will assume that $\mathcal{Q} = \rho^{-1}(\mathcal{Q})$, where $\mathcal{Q} = F_1 \times \cdots \times \hat{F}_j \times \cdots \times F_k$ for some $j \in \{1, \dots, k\}$. Then $\mathcal{Q}$ is of infinite type (because $\mathcal{Q}$ is infinite and $\rho$ is action-like) and normalized by $\mathcal{G}$ (because $\mathcal{Q}$ is normal in $G$). Let $A \subseteq F_j$ be an infinite amenable subgroup: this always exists, because the type of $F_j$ (i.e. the underlying subgraph of $\Gamma$) contains at least 2 vertices. The conclusion follows by letting $\mathcal{A} = \rho^{-1}(A)$.

**Step 2** Let us prove that if $(\mathcal{G}, \mathcal{Q})$ satisfies $(P'_{\text{cofact}})_1$ then $G$ is reducible (i.e. $k \ge 2$) and $\mathcal{Q}$ is stably contained in a subgroupoid $\mathcal{Q}'$ of $\mathcal{G}$ such that $(\mathcal{Q}', \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$.

Let $\mathcal{A}$ be given by $(P'_{\text{cofact}})_1$. Without loss of generality, we will assume that $\mathcal{A}$ is normalized by $\mathcal{Q}$. For every $j \in \{1, \dots, k\}$, let $\rho_j : \mathcal{G} \to F_j$ be the cocycle obtained by post-composing $\rho$ by the projection $G \to F_j$. Up to a countable Borel partition of $X$, we will assume that $(\mathcal{A}, \rho)$ is tightly $\mathcal{A}$-supported for some parabolic subgroup $A \subseteq G$ (and $A \ne \{1\}$ because $\mathcal{A}$ is of infinite type).

• Let $j \in \{1, \dots, k\}$ be such that $A \cap F_j \ne \{1\}$ (such a $j$ exists by Lemma 2.7, page 22). In particular $(\rho_j)_{|_A}$ is nowhere trivial (by the second point of the definition of tight support, page 43).
• Let us prove that $(\rho_j)|_Q$ is stably trivial.

So assume towards a contradiction that $(\rho_j)|_Q$ is not stably trivial.

• We first show that there exists a Borel subset $V$ of positive measure such that $\rho_j$ is nowhere trivial on $Q|_V$.

Let $U \subseteq X$ be a Borel subset of maximal measure such that $\rho_j$ is stably trivial on $Q|_U$. Our assumption ensures that $V = X\setminus U$ has positive measure. And $\rho_j$ is nowhere trivial on $Q|_V$.

• We now show that $(\rho^{-1}(F_j)|_V, \rho_j)$ is $F_j$-elementary (in the sense of Definition 6.8). Lemma 6.11, page 69 and Remark 6.12, page 69 ensure that $(\mathcal{A}, \rho_j)$ is $F_j$-elementary. Applying Lemma 6.14, page 72 twice then shows that $(\mathcal{Q}, \rho_j)$ and $(\mathcal{G}|_V, \rho_j)$ are $F_j$-elementary. In particular $(\rho^{-1}(F_j)|_V, \rho_j)$ is $F_j$-elementary.

• Since $\rho_j : \rho^{-1}(F_j)|_V \to F_j$ is action-like and the type of $F_j$ is not a clique, it contradicts Lemma 6.10, page 68.

This contradiction shows that $(\rho_j)|_Q$ is stably trivial.

• Let us show that $G$ is reducible.

Assume that $G$ is irreducible, namely that $k = 1$. Then $\rho_1 = \rho$ and since $\rho$ has trivial kernel, the previous point implies that $Q$ is stably trivial. This contradicts the fact that $Q$ is of infinite type (by $(\mathrm{P}'_{\mathrm{cofact}})_1$).

• Taking $Q := F_1 \times \dots \times \hat{F}_j \times \dots F_k$ and $Q' := \rho^{-1}(Q)$ gives the wanted conclusion for Step 2.

**Conclusion** If $G$ is reducible and $(Q, \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$, then it verifies $(\mathrm{P}'_{\mathrm{cofact}})_1$ by Step 1. To check $(\mathrm{P}'_{\mathrm{cofact}})_2$ take $Q'$ such that $(\mathcal{G}, Q')$ satisfies $(\mathrm{P}'_{\mathrm{cofact}})_1$ with $Q$ stably contained in $Q'$. By Step 2, there exists $Q''$ such that $Q'$ is also stably contained in $Q''$ and $(Q'', \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$. Since an inclusion of co-factors is always an equality, we deduce that $Q$ and $Q''$ are stably equal, in particular $Q$ and $Q'$ are stably equal.

Now assume that $(\mathcal{G}, Q)$ satisfies $(\mathrm{P}'_{\mathrm{cofact}})$. By Step 2, $G$ is reducible, and there exists $Q'$ such that $Q$ is stably contained in $Q'$ and $(Q', \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$. By Step 1, $(\mathcal{G}, Q')$ verifies $(\mathrm{P}'_{\mathrm{cofact}})_1$ and thus $(\mathrm{P}'_{\mathrm{cofact}})_2$ for $(\mathcal{G}, Q)$ gives that $Q$ and $Q'$ are stably equal. In particular $(Q, \rho)$ is of co-factor type within $(\mathcal{G}, \rho)$. $\square$

We now consider the following property which is a very slight variation over Property $(\mathrm{P}_{\mathrm{fact}})$ from the previous section (Definition 7.10), with $(\mathrm{P}'_{\mathrm{cofact}})$ instead of $(\mathrm{P}_{\mathrm{cofact}})$.

**Definition 8.4 (Property $(\mathrm{P}'_{\mathrm{fact}}))$.** Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$. Let $\mathcal{F} \subseteq \mathcal{G}$ be a measured subgroupoid. We say that the pair $(\mathcal{G}, \mathcal{F})$ satisfies *Property* $(\mathrm{P}'_{\mathrm{fact}})$ if the following three properties hold.

($\mathrm{P}'_{\mathrm{fact}}$)$_1$ The subgroupoid $\mathcal{F}$ is of infinite type.

($\mathrm{P}'_{\mathrm{fact}}$)$_2$ There exist measured subgroupoids $Q_1, \dots, Q_n$ of $\mathcal{G}$, with $n \ge 1$, and such that $(\mathcal{G}, Q_j)$ satisfies Property $(\mathrm{P}'_{\mathrm{cofact}})$ for every $j \in \{1, \dots, n\}$, and there exists a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ such that for every $i \in I$, one has $\mathcal{F}|_{X_i} = (Q_1 \cap \dots \cap Q_n)|_{X_i}$.

($\mathrm{P}'_{\mathrm{fact}}$)$_3$ For every subgroupoid $\mathcal{Q}$ of $\mathcal{G}$ such that $(\mathcal{G}, \mathcal{Q})$ satisfies Property $(\mathrm{P}'_{\mathrm{cofact}})$, and every positive measure Borel subset $U \subseteq X$, there exists a positive measure Borel subset $V \subseteq U$ such that either $\mathcal{F}|_V \subseteq Q|_V$ or $(\mathcal{F} \cap \mathcal{Q})|_V$ is trivial.
Replacing the use of Lemma 7.9, page 82 by Lemma 8.3 in the proof of the first point of Lemma 7.11, page 85 we obtain the following statement.

**Lemma 8.5.** Let $G$ be a reducible graph product of countably infinite groups over a finite simple graph $\Gamma$ with trivial clique factor. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho : \mathcal{G} \to G$. Let $\mathcal{F} \subseteq \mathcal{G}$ be a measured subgroupoid.

Then $(\mathcal{F}, \rho)$ is of factor type within $(\mathcal{G}, \rho)$ if and only if $(\mathcal{G}, \mathcal{F})$ satisfies Property $(\mathrm{P}'_{\mathrm{fact}})$. $\square$

For the following statement, we refer to Definition 7.12, page 86 from the previous section for the definition of Property $(\mathrm{P}_{\mathrm{vert}})$.

**Lemma 8.6.** Let $G$ be a reducible graph product of countably infinite groups over a finite simple graph $\Gamma$ with no transvection and no partial conjugation. 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 $\mathcal{V} \subseteq \mathcal{G}$ be a measured subgroupoid.

Then $(\mathcal{V}, \rho)$ is of vertex type if and only if there exist a measured subgroupoid $\mathcal{F} \subseteq \mathcal{G}$ such that $(\mathcal{G}, \mathcal{F})$ satisfies Property $(\mathrm{P}'_{\mathrm{fact}})$, and a countable Borel partition $X^* = \cup_{i \in I} X_i$ of a conull Borel subset $X^* \subseteq X$ such that for every $i \in I$, one has $\mathcal{V}|_{X_i} \subseteq \mathcal{F}|_{X_i}$, and the pair $(\mathcal{F}|_{X_i}, \mathcal{V}|_{X_i})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$.

*Proof*. Let $\Gamma = \Gamma_1 \circ \cdots \circ \Gamma_k$ be the decomposition of $\Gamma$ as a join of irreducible subgraphs (well-defined up to permutation of the factors), and let $G = F_1 \times \cdots \times F_k$ be the corresponding decomposition of $G$ as a direct product (since $G$ is reducible, $k \ge 2$). Notice that as $\Gamma$ is transvection-free, no $\Gamma_i$ is reduced to a point, in other words $G$ has trivial clique factor.

We first assume that $(\mathcal{V}, \rho)$ is of vertex type. Up to a countable partition of the base space $X$, and up to conjugating the cocycle, we will assume that $\mathcal{V} = \rho^{-1}(G_v)$ for some $v \in V\Gamma$. Let $j \in \{1, \dots, k\}$ be such that $v \in V\Gamma_j$. Let $\mathcal{F} = \rho^{-1}(F_j)$. By Lemma 8.5, page 93 the pair $(\mathcal{G}, \mathcal{F})$ satisfies Property $(\mathrm{P}'_{\mathrm{fact}})$. In addition, $\Gamma_j$ has no transvection and no partial conjugation (Remark 2.1, page 18), so it is strongly reduced by Lemma 6.3, page 63. It thus follows from Lemma 7.13, page 87 (applied to the groupoid $\mathcal{F}$ and to the cocycle $\rho$, viewed as an action-like cocycle $\mathcal{F} \to F_j$), that $(\mathcal{F}, \mathcal{V})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$.

Conversely, assume that there exist a subgroupoid $\mathcal{F}$ as in the lemma – up to reasoning on each subset of the partition, we will assume that $(\mathcal{F}, \mathcal{V})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$. Since $(\mathcal{G}, \mathcal{F})$ satisfies Property $(\mathrm{P}'_{\mathrm{fact}})$, Lemma 8.5 ensures that, up to a countable Borel partition of $X$ and passing to a conull Borel subset, we have $\mathcal{F} = \rho^{-1}(F_j)$ for some $j \in \{1, \dots, k\}$. Since $\Gamma_j$ is transvection-free and strongly reduced, and $(\mathcal{F}, \mathcal{V})$ satisfies Property $(\mathrm{P}_{\mathrm{vert}})$, it follows from Lemma 7.13, page 87 that $(\mathcal{V}, \rho)$ is of vertex type, as desired. $\square$

We are now in position to complete the proof of the main proposition of the section.

*Proof of Proposition 8.1.* Let $G$ and $H$ be two graph products of countably infinite groups over finite simple graphs that split (possibly trivially) as joins of strongly reduced transvection-free finite simple graphs on at least 2 vertices.
Let $\mathcal{G}$ be a measured groupoid equipped with two action-like cocycles $\rho_G : \mathcal{G} \to \mathcal{G}$ and $\rho_H : \mathcal{G} \to \mathcal{H}$.

Let us first show that $\mathcal{G}$ is reducible if and only if $\mathcal{H}$ is.

If $\mathcal{G}$ is reducible there exists a subgroupoid $\mathcal{Q}$ in $\mathcal{G}$ such that $(\mathcal{Q}, \rho_G)$ is of co-factor type within $(\mathcal{G}, \rho_G)$. By Lemma 8.3, page 91, the pair $(\mathcal{G}, \mathcal{Q})$ verifies $(\mathrm{P}'_{\mathrm{cofact}})$ and therefore $\mathcal{H}$ is also reducible.

If $\mathcal{G}$ is reducible, the conclusion of Proposition 8.1, page 90 is now an immediate consequence of Lemma 8.6, page 93, as we have characterized subgroupoids of $\mathcal{G}$ of vertex type with respect to some action-like cocycle $\rho$, using a groupoid-theoretic property that does not refer to the cocycle $\rho$.

And if $\mathcal{G}$ is irreducible, the conclusion of Proposition 8.1, page 90 follows from Lemma 7.13, page 87. $\square$

*Proof of Proposition 5.3.* By Corollary 6.4, if a finite simple graph $\Gamma$ is transvection-free and has no partial conjugation, then $\Gamma$ splits as $\Gamma = \Gamma_1 \circ \cdots \circ \Gamma_k$ (possibly with $k = 1$), in such a way that every $\Gamma_j$ is transvection-free, strongly reduced, and has at least two vertices. Proposition 5.3 thus follows from Proposition 8.1. $\square$

This completes the proof of the main theorem of Part II (Theorem 1.2, page 7).

# 9 Amenable vertex groups and untransvable vertices

## 9.1 Main result

The contents of this section is not needed for the main theorems of Part II, but will be used in Section 13. In this section specifically, we will consider graph products where all vertex groups are amenable. In this case, more can be said without assuming that the graph has no transvection and no partial conjugation. Namely, given a subgroupoid $\mathcal{G}$ over a standard probability space $X$ equipped with an action-like cocycle $\rho : \mathcal{G} \to \mathcal{G}$, we can still recognize subgroupoids that correspond to parabolic subgroups of untransvable vertex type, in the following sense.

Let $\Gamma$ be a finite simple graph. Recall that a vertex $v \in V\Gamma$ is *untransvable* if there does not exist any vertex $w \in V\Gamma$ distinct from $v$ and such that $\mathrm{lk}(v) \subseteq \mathrm{star}(w)$. A parabolic subgroup $P \subseteq \mathcal{G}$ is of untransvable vertex type if it is conjugate to $G_v$ for some untransvable vertex $v$.

Given $\mathcal{G}$ and $\rho$ as above, and a measured subgroupoid $\mathcal{H} \subseteq \mathcal{G}$, we say that $(\mathcal{H}, \rho)$ is of untransvable vertex type if there exist a conull Borel subset $X^* \subseteq X$, a partition $X^* = \cup_{i \in I} X_i$ into at most countably many Borel subsets, and for every $i \in I$, a parabolic subgroup $P_i$ of untransvable vertex type such that for every $i \in I$, one has $\mathcal{H}|_{X_i} = \rho^{-1}(P_i)|_{X_i}$.

We say that a finite simple graph $\Gamma$ is *clique-reduced* if there do not exist distinct vertices $v, w \in V\Gamma$ with $\mathrm{star}(v) = \mathrm{star}(w)$. Equivalently $\Gamma$ is clique-reduced if and only if there does not exist any collapsible clique (in the sense of Definition 6.1, page 62) on at least 2 vertices in $\Gamma$.

The main result of this section is the following proposition.
**Proposition 9.1.** Let $G, H$ be two graph products over clique-reduced finite simple graphs with countably infinite amenable vertex groups. Let $\mathcal{G}$ be a measured groupoid over a standard probability space, equipped with two action-like cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$. Let $\mathcal{V} \subseteq \mathcal{G}$ be a measured subgroupoid.

Then, the pair $(\mathcal{V}, \rho_G)$ is of untransvable vertex type if and only if the pair $(\mathcal{V}, \rho_H)$ is of untransvable vertex type.

**Remark 9.2.** The assumption that the two graphs are clique-reduced is necessary. Otherwise, here is a counterexample. Let $G$ be the graph product over the graph $\bar{\Gamma}$ from Figure 11, page 62, with vertex groups $\{G_v\}_v$ isomorphic to $\mathbb{Z}^2$. Then $G$ is also isomorphic to the graph product $H$ over the graph $\Gamma$ (from the same figure) with vertex groups isomorphic to $\mathbb{Z}$. We fix an isomorphism $\theta : H \to G$ sending vertex groups inside vertex groups. Let $\mathcal{G}$ be a measured groupoid with an action-like cocycle $\rho_G : \mathcal{G} \to G$, and let $\rho_H = \rho_G$ (through the chosen isomorphism $H \to G$). If $v \in V\bar{\Gamma}$, then $(\rho_G^{-1}(G_v), \rho_G)$ is of untransvable vertex type, while $(\rho_G^{-1}(G_v), \rho_H)$ is not.

Our motivation behind proving Proposition 9.1 is that it is a useful in the measure equivalence classification of right-angled Artin groups. In fact, as recorded in Corollary 9.4 below, it provides finer information than [HH22a], which only dealt with right-angled Artin groups with finite outer automorphism groups. Among concrete applications, let us mention the following.

* Corollary 9.4 below will be used in Section 10 to provide new concrete examples of right-angled Artin groups that can be distinguished from the viewpoint of measure equivalence.

* In Part III of the present work, Proposition 9.1 will enable us to obtain quantitative estimates and solve the inverse problem in quantitative orbit equivalence for a large family of right-angled Artin groups.

* In ongoing work of Huang and the second-named author (which has largely motivated the present section), the authors expect to use the results obtained here towards finer results in measure equivalence classification of the right-angled Artin groups.

To state our main corollary, we make the following definition.

**Definition 9.3 (Untransvable extension graph).** Let $G$ be a graph product over a finite simple graph $\Gamma$. The untransvable extension graph is the subgraph $\Gamma_G^{ue} \subseteq \Gamma_G^e$ spanned by all vertices corresponding to parabolic subgroups of untransvable vertex type.

Notice that $\Gamma_G^{ue}$ is a $G$-invariant subgraph of $\Gamma_G^e$, for the $G$-action on $\Gamma_G^e$ coming from the $G$-action by conjugation on itself. We warn the reader that $\Gamma_G^{ue}$ might be disconnected, and might be empty.

We also refer to Definition 4.22, page 47 for the definition of an ME-witness.

**Corollary 9.4.** Let $G$ and $H$ be two graph products over finite simple graphs $\Gamma_G, \Gamma_H$, with countably infinite amenable vertex groups.

Then $(\Gamma_G^{ue}, \Gamma_H^{ue})$ is an ME-witness for $G, H$. In particular, if $G$ and $H$ are measure equivalent, then $\Gamma_G^{ue}$ and $\Gamma_H^{ue}$ are isomorphic.
*Proof.* Let $\mathcal{G}$ be a measured groupoid over a standard probability space, and let $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$ be two action-like cocycles. Notice that $(\mathcal{G}, \rho_G)$ is stably $\Gamma_G^{ue}$ in the sense of Section 4.4 if and only if it is of untransvable vertex type. Therefore, the first point of Definition 4.22 is exactly the contents of Proposition 9.1, page 95. And the second point of Definition 4.22 is a consequence of Lemma 5.4, page 54, which characterizes adjacency in the (untransvable) extension graph with no reference to the action-like cocycle. Finally, the additional part of the corollary follows from Proposition 4.24, page 47. $\square$

In particular, Corollary 9.4 establishes the untransvable extension graph as a new invariant in the measure equivalence classification of right-angled Artin groups. Notice that when $\Gamma_G$ and $\Gamma_H$ are transvection-free, then $\Gamma_G^{ue} = \Gamma_G^e$ and $\Gamma_H^{ue} = \Gamma_H^e$, and Corollary 9.4 recovers [HH22a, Theorem 2]. Let us mention that, although we believe that the techniques from [HH22a] could have been extended to prove the results from the present section, we decided to use a different path, more in the spirit of the previous sections of the present work.

## 9.2 Detecting clique subgroups

We make the following definition.

**Definition 9.5 (Property ($P_{clique}$))**. Let $\mathcal{P}$ be a measured groupoid over a standard probability space. Let $C \subseteq \mathcal{P}$ be a measured subgroupoid. We say that $(\mathcal{P}, C)$ satisfies Property $(P_{clique})$ if the following two properties hold.

(P_{clique})_1 The subgroupoid C is amenable, of infinite type, and stably normal in P.

(P_{clique})_2 If $C'$ is a measured subgroupoid of $\mathcal{P}$ such that $(\mathcal{P}, C')$ satisfies Property $(P_{clique})_1$, then $C'$ is stably contained in $C$.

**Lemma 9.6.** Let $G$ be a graph product of countably infinite amenable groups over a finite simple graph $\Gamma$. 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 product parabolic subgroup, and let $\mathcal{P} = \rho^{-1}(P)$.

Let $C \subseteq P$ be the clique factor of $P$.

* If $C = \{1\}$ then no measured subgroupoid $C$ of $\mathcal{P}$ is such that $(\mathcal{P}, C)$ satisfies Property $(P_{clique})$.

* If $C$ is non-trivial then a measured subgroupoid $C \subseteq \mathcal{P}$ is such that $(\mathcal{P}, C)$ satisfies Property $(P_{clique})$ if and only if $C$ is stably equal to $\rho^{-1}(C)$.

*Proof.* We first assume that $C = \{1\}$, and assume towards a contradiction that there exists a measured subgroupoid $C \subseteq \mathcal{P}$ such that $(\mathcal{P}, C)$ satisfies Property $(P_{clique})$. Then $(\mathcal{P}, \mathcal{P})$ satisfies Property $(P'_{cofact})$ (recall Definition 8.2, page 91). So Lemma 8.3, page 91 applies and shows that $\mathcal{P}$ is stably equal to $\rho^{-1}(Q)$ for some cofactor $Q \subseteq P$. This is a contradiction to Lemma 4.14, page 43 using that $\rho$ is action-like.

We now assume that $C \neq \{1\}$.

* We first show that $\rho^{-1}(C)$ satisfies Property $(P_{clique})$.
  The subgroup $C$ is infinite and amenable because vertex groups are infinite and amenable. Let $C = \rho^{-1}(C)$. Then $C$ is amenable because $C$ is amenable and $\rho$ has
trivial kernel (Lemma 4.7, page 41), of infinite type because C is infinite and ρ is action-like, and normal in P because C is normal in P (Example 4.6, page 41). So (P, C) satisfies Property (P<sub>clique</sub>)<sub>1</sub>.

To check Property (P<sub>clique</sub>)<sub>2</sub>, let C' ⊆ P be an amenable subgroupoid of infinite type which is stably normal in P. We aim to prove that C' is stably contained in C. Without loss of generality, we will assume that C' is normalized by P. We write P = C × F<sub>1</sub> × ... × F<sub>k</sub>, where the subgroups F<sub>j</sub> do not further split as a direct product. For every j ∈ {1, ..., k}, let ρ<sub>j</sub> : P → F<sub>j</sub> be the cocycle obtained by post-composing ρ by the projection G → F<sub>j</sub>. Up to a countable Borel partition of X, we will assume using Lemma 4.16, page 44 that (C', ρ) is tightly A-supported for some parabolic subgroup A ⊆ P. We aim to prove that A ⊆ C. So assume by contradiction that this is not the case. Using Lemma 2.7, page 22 we can find j ∈ {1, ..., k} such that A ∩ F<sub>j</sub> ≠ {1}. In particular (ρ<sub>j</sub>)<sub>|C'</sub> is nowhere trivial. Since C' is amenable, Lemma 6.11, page 69 and Remark 6.12, page 69 ensure that (C', ρ<sub>j</sub>) is F<sub>j</sub>-elementary. Lemma 6.14, page 72 therefore implies that (P, ρ<sub>j</sub>) is also F<sub>j</sub>-elementary. In particular (ρ<sup>-1</sup>(F<sub>j</sub>), ρ<sub>j</sub>) is F<sub>j</sub>-elementary, which contradicts Lemma 6.10, page 68 (since F<sub>j</sub> is not a clique). Therefore A ∩ F<sub>j</sub> = {1} for all j and thus A ⊆ C as wanted.

* The stability of Property (P<sub>clique</sub>) under restriction and stabilization implies that any subgroupoid stably equal to ρ<sup>-1</sup>(C), satisfies itself Property (P<sub>clique</sub>).

* Now if C satisfies Property (P<sub>clique</sub>), we showed in the first point that it is stably contained in ρ<sup>-1</sup>(C). By the maximality property (P<sub>clique</sub>)<sub>2</sub> it is therefore stably equal to ρ<sup>-1</sup>(C). □

## 9.3 Proof of Proposition 9.1

When all vertex groups are amenable, subgroupoids of isolated clique type are amenable, and therefore they do not satisfy Property (P<sub>prod</sub>) from Definition 7.2, page 76. Lemma 7.4, page 77 thus reformulates as follows in this case.

**Lemma 9.7.** Let G be a graph product of countably infinite amenable groups over a finite simple graph Γ. Let G be a measured groupoid over a standard probability space X, equipped with an action-like cocycle ρ : G → G. Let P ⊆ G be a measured subgroupoid.

Then (P, ρ) is of maximal product type and nowhere of isolated clique type if and only if (G, P) satisfies Property (P<sub>prod</sub>). □

We now introduce the following variation over Property (P<sub>vert</sub>) (Definition 7.12).

**Definition 9.8 (Property (P'<sub>vert</sub>))**. Let G be a measured groupoid over a standard probability space X, and let V be a measured subgroupoid of G. We say that the pair (G, V) satisfies Property (P'<sub>vert</sub>) if the following two properties hold.

(P'<sub>vert</sub>)<sub>1</sub> There exists a countable Borel partition X* = ⋃<sub>i∈I</sub> X<sub>i</sub> of a conull Borel subset X* ⊆ X such that for every i ∈ I, there exist measured subgroupoids

$$G|_{X_i} = F_0 \supseteq P_1 \supseteq F_1 \supseteq P_2 \supseteq F_2 \supseteq \dots \supseteq P_n \supseteq C_n = V|_{X_i}$$

of G|<sub>X<sub>i</sub></sub> such that
1. for every $j \in \{1, \dots, n\}$, the pair $(\mathcal{F}_{j-1}, \mathcal{P}_j)$ satisfies Property $(P_{\text{prod}})$ (see Definition 7.2, page 76);

2. for every $j \in \{1, \dots, n-1\}$, the pair $(\mathcal{P}_j, \mathcal{F}_j)$ satisfies Property $(P'_{\text{fact}})$ (see Definition 8.4, page 92);

3. for every $j \in \{1, \dots, n-1\}$ and every positive measure Borel subset $U \subseteq X_i$, there does not exist any subgroupoid $\mathcal{C}$ of $(\mathcal{P}_j)_{|U}$ such that $((\mathcal{P}_j)_{|U}, \mathcal{C})$ satisfies Property $(P_{\text{clique}})$ (see Definition 9.5, page 96);

4. the pair $(\mathcal{P}_n, \mathcal{C}_n)$ satisfies Property $(P_{\text{clique}})$.

$(P'_{\text{vert}})_2$ For every measured subgroupoid $\mathcal{W}$ of $\mathcal{G}$ of infinite type which is stably contained in $\mathcal{V}$, and every measured subgroupoid $\mathcal{H} \subseteq \mathcal{G}$, if $\mathcal{W}$ is stably normalized by $\mathcal{H}$, then $\mathcal{V}$ is stably normalized by $\mathcal{H}$.

**Lemma 9.9.** Let $\mathcal{G}$ be a graph product of countably infinite amenable groups over a clique-reduced finite simple graph $\Gamma$. Let $\mathcal{G}$ be a measured groupoid over a standard probability space $X$, equipped with an action-like cocycle $\rho: \mathcal{G} \to \mathcal{G}$. Let $\mathcal{V} \subseteq \mathcal{G}$ be a measured subgroupoid.

Then $(\mathcal{V}, \rho)$ is of untransvable vertex type if and only if $(\mathcal{G}, \mathcal{V})$ satisfies Property $(P'_{\text{vert}})$.

*Proof.* We first assume that $(\mathcal{V}, \rho)$ is of untransvable vertex type, and prove that $(\mathcal{G}, \mathcal{V})$ satisfies Property $(P'_{\text{vert}})$.

Up to passing to a conull Borel subset of $X$, taking a countable partition, and conjugating the cocycle, we will assume that $\mathcal{V} = \rho^{-1}(G_v)$, where $v \in V\Gamma$ is untransvable. We start with $(P'_{\text{vert}})_1$. Let

$$G = F_0 \supseteq P_1 \supseteq F_1 \supseteq \dots \supseteq P_n \supseteq C_n = G_v$$

be parabolic subgroups provided by Lemma 6.6, page 65 (here we write $F_n = C_n$). For every $j \in \{1, \dots, n\}$, let $\mathcal{P}_j = \rho^{-1}(P_j)$ and $\mathcal{F}_{j-1} = \rho^{-1}(F_{j-1})$, and let $C_n = \rho^{-1}(C_n)$. Since $P_j$ is a maximal product parabolic subgroup of $F_{j-1}$ and is not of isolated clique type, Lemma 7.4, page 77 ensures that $(\mathcal{F}_{j-1}, \mathcal{P}_j)$ satisfies Property $(P_{\text{prod}})$ for every $j \in \{1, \dots, n\}$. Since $P_j$ has trivial clique factor for every $j \in \{1, \dots, n-1\}$, and $F_j$ is a factor of $P_j$, Lemma 8.5, page 93 applies and shows that $(\mathcal{P}_j, \mathcal{F}_j)$ satisfies Property $(P'_{\text{fact}})$. Since $C_n$ is the clique factor of $P_n$, Lemma 9.6, page 96 ensures that $(\mathcal{P}_n, \mathcal{C}_n)$ satisfies Property $(P_{\text{clique}})$. On the other hand, for every $j \in \{1, \dots, n-1\}$, since $P_j$ has trivial clique factor, Lemma 9.6 ensures that for every positive measure Borel subset $U \subseteq X$, there does not exist any measured subgroupoid $\mathcal{C}$ of $(\mathcal{P}_j)_{|U}$ such that $((\mathcal{P}_j)_{|U}, \mathcal{C})$ satisfies Property $(P_{\text{clique}})$. This proves Property $(P'_{\text{vert}})_1$.

We now prove that $(\mathcal{G}, \mathcal{V})$ satisfies Property $(P'_{\text{vert}})_2$. Let $\mathcal{W}$ and $\mathcal{H}$ be as in $(P'_{\text{vert}})_2$. Since $\mathcal{W}$ is of infinite type and stably contained in $\mathcal{V}$, up to a countable Borel partition of a conull Borel subset of $X$, we will assume that $(\mathcal{W}, \rho)$ is tightly $G_v$-supported. As $\mathcal{W}$ is stably normalized by $\mathcal{H}$, Lemma 4.17, page 44 allows us to further assume that $\mathcal{H} \subseteq \rho^{-1}(G_v \times G_v^{\perp})$. But since $G_v$ is normal in $G_v \times G_v^{\perp}$, it follows that $\mathcal{V}$ is (stably) normalized by $\mathcal{H}$ (Example 4.6, page 41), as desired.

Conversely, let us assume that $(\mathcal{G}, \mathcal{V})$ satisfies Property $(P'_{\text{vert}})$, and prove that $(\mathcal{V}, \rho)$ is of untransvable vertex type.
Up to a countable partition, we can assume that there exist measured subgroupoids

$$G = F_0 \supseteq P_1 \supseteq F_1 \supseteq P_2 \supseteq F_2 \supseteq \dots \supseteq P_n \supseteq C_n = V$$

as in $(P'_{\text{vert}})_1$. Using Lemmas 8.5, 9.6 and 9.7, up to a further partition, we can assume that there exists a chain of parabolic subgroups

$$G = F_0 \supseteq P_1 \supseteq F_1 \supseteq \dots \supseteq P_n \supseteq C_n$$

such that

1. for every $j \in \{1, \dots, n\}$, one has $P_j = \rho^{-1}(P_j)$, and $F_{j-1} = \rho^{-1}(F_{j-1})$, and $C_n = \rho^{-1}(C_n)$, and

2. for every $j \in \{1, \dots, n\}$, $P_j$ is a maximal product parabolic subgroup of $F_{j-1}$ and not of isolated clique type (Lemma 9.7, page 97 using the first assertion of $(P'_{\text{vert}})_1$),

3. for every $j \in \{1, \dots, n-1\}$, the clique factor of $P_j$ is trivial (Lemma 9.6, using the third assertion of $(P'_{\text{vert}})_1$), and $F_j$ is a factor of $P_j$ (Lemma 8.5, page 93 using the second assertion of $(P'_{\text{vert}})_1$), and

4. $C_n$ is the clique factor of $P_n$ (Lemma 9.6, page 96 using the fourth assertion of $(P'_{\text{vert}})_1$).

We now claim that $C_n$ is conjugate to $G_v$ for some $v \in V\Gamma$. This will be enough to conclude our proof, as Lemma 6.6, page 65 then implies that $v$ is untransvable.

We are thus left with proving the above claim. Assume that $C_n$ is conjugate to $G_{\Upsilon}$, for a clique subgraph $\Upsilon \subseteq \Gamma$ which contains at least two vertices. Up to conjugating the cocycle, we will assume without loss of generality that $C_n = G_{\Upsilon}$. Since $\Gamma$ is clique-reduced, we can find a vertex $w \in V\Upsilon$ such that $lk(w) \notin \Upsilon \circ \Upsilon^{\perp}$. We now aim for a contradiction using Property $(P'_{\text{vert}})_2$. So let $W = \rho^{-1}(G_w)$, and let $H = \rho^{-1}(G_w \times G_w^{\perp})$. Then $W \subseteq V$, and $W$ is normalized by $H$ (Example 4.6, page 41). Property $(P'_{\text{vert}})_2$ implies that $V$ is stably normalized by $H$. Since $(V, \rho)$ is tightly $G_{\Upsilon}$-supported (Lemma 4.15, page 43), Lemma 4.17 enables us to find a positive measure Borel subset $U \subseteq X$ such that $H|_U \subseteq \rho^{-1}(G_{\Upsilon} \times G_{\Upsilon}^{\perp})$. It follows from Lemma 4.14, page 43 that $G_w \times G_w^{\perp} \subseteq G_{\Upsilon} \times G_{\Upsilon}^{\perp}$, which contradicts our choice of $w$. This contradiction completes our proof. $\square$

We are now in position to complete the proof of the main proposition of this section.

*Proof of Proposition 9.1.* Assume that $(V, \rho_G)$ is of untransvable vertex type. By Lemma 9.9 applied to the cocycle $\rho_G$, the pair $(G, V)$ satisfies Property $(P'_{\text{vert}})$. And Lemma 9.9 applied to the cocycle $\rho_H$ then proves that $(V, \rho_H)$ is of untransvable vertex type. By symmetry the other implication also holds. $\square$

# 10 Examples

To conclude this part, we provide some examples illustrating the different cases treated in the past sections. All the graph products presented in this section are in fact right-angled Artin groups (though they are sometimes viewed as graph products with non-cyclic vertex groups). So our work provides new cases of the measure equivalence classification of right-
angled Artin groups, beyond those treated in [HH22a]. On the other hand, some cases remain open, such as right-angled Artin groups defined over trees.

**Example 10.1.** The following example is an illustration of Theorem 1.2.

The two graph products defined in Figure 18 are not measure equivalent.

Indeed, the graph (namely the pentagon) is transvection-free and has no partial conjugation. Therefore Theorem 1.2 applies: if $G$ and $H$ were measure equivalent, then we would have a graph isomorphism sending the vertex group $F_2$ to an orbit equivalent vertex group in $H$. Such a vertex group does not exist in $H$. Therefore $G$ and $H$ cannot be measure equivalent.

<figure><img src="image_21.png" /><figcaption>Figure 18: Groups from Example 10.1</figcaption></figure>

**Example 10.2.** The following example is an illustration of Corollary 5.7.

Let $G$ and $H$ be as in Figure 19 (the two leftmost graphs). If $G$ and $H$ were measure equivalent, then we would have a graph homomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that $G_v$ is measure equivalent to $H_{\sigma(v)}$ for all $v \in V\Gamma_G$. In particular $\sigma$ must send blue vertices to blue vertices. In particular, by looking at the restriction of $\sigma$ to the leftmost pentagon in $\Gamma_G$ (all of whose vertices are blue), there would exist a graph homomorphism from a pentagon to a segment of length 3, which is not the case. Therefore $G$ and $H$ are not measure equivalent.

<figure><img src="image_22.png" /><figcaption>Figure 19: Graphs and groups from Example 10.2 and Remark 10.3</figcaption></figure>

**Remark 10.3.** We do not know however whether or not the groups $H$ and $G'$ are measure equivalent. There do exist graph homomorphisms $\Gamma_G \to \Gamma_H$ and $\Gamma_H \to \Gamma_G$, such that a vertex and its image always carry measure equivalent vertex groups, so Corollary 5.7 does not provide any obstruction. In fact, if the brown vertex group was isomorphic to $F_2$,
then $G'$ would be isomorphic to an index 2 subgroup of $H$ (whence measure equivalent to $H$). However, with the brown vertex group isomorphic to $F_3$, we have no reason to expect that the groups $G'$ and $H$ should be measure equivalent.

**Example 10.4.** The following example is an illustration of Proposition 9.1.

Let $G$ and $H$ be the two groups defined in the left half of Figure 20, page 101. Then $G$ and $H$ are not measure equivalent.

Indeed, first remark that in Figure 20 all untrans vectorable vertices are drawn in orange. Second, note that $H$ can be rewritten as a right-angled Artin group over the graph drawn at the top of the right half of Figure 20 (in particular, with this decomposition all vertex groups are now amenable). In the latter graph, all vertices are trans vectorable. If $G$ and $H$ were measure equivalent then we could apply Proposition 9.1, page 95 and get the existence of an untrans vectorable vertex in the rightmost graph defining $H$. Hence $G$ and $H$ can not be measure equivalent. For the same reason, the groups $H$ and $H'$ are not measure equivalent.

In fact, as a consequence of Corollary 9.4, we can also prove that $G$ and $H'$ are not measure equivalent. Indeed, the untrans vectorable extension graph $\Gamma_{H'}^e$ (see Definition 9.3) is totally disconnected: its vertices represent the conjugates of the leftmost orange vertex group and of the rightmost orange vertex group, and no two distinct conjugates of these vertex groups commute. On the other hand $\Gamma_G^e$ contains an edge, because all orange vertex groups are untrans vectorable.

One could provide many variations over the above example. But many cases remain open: the most basic ones is that our methods do not allow to decide whether or not right-angled Artin groups defined over trees of diameter at least 3 are measure equivalent or not.

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

also decomposes
as:

<figure><img src="image_26.png" /></figure>

also decomposes
as:

<figure><img src="image_27.png" /><figcaption>Figure 20: Groups from Example 10.4</figcaption></figure>
## Part III

# Quantitative aspects and the inverse problem for RAAGs

The goal of the present part is to obtain a quantitative version of the previous classification theorem (Theorem 1.2, page 7) and solve the inverse problem in quantitative measure equivalence for a large family of right-angled Artin groups. We refer to Section 3.1.3, page 33 for the definition of quantitative orbit equivalence.

In Section 11 we prove that when two graph products are defined over the same graph, the integrability at the level of the vertex groups can be lifted at the level of the graph products themselves (Theorem 1.10). This actually does not need any assumption on the defining graph and relies on a previous result of Huang and the second named author [HH22a], generalizing an argument of Gaboriau [Gab05, P$_{ME}$6]. In Section 12 we show a generalized version of Theorem 1.11, proving for a large family of graph products that the integrability at the level of the graph products passes at the level of the vertex groups.

The inverse problem is finally solved in Section 13.

## 11 From the factors to the graph product

We prove here Theorem 1.10 which we recall below – in the introduction, we used the language of integrable orbit equivalence couplings instead of pairings, but these are equivalent, see Section 3.1.3.

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

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

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

This theorem is a quantitative version of [HH22a, Proposition 4.2]. Our proof is actually based on the pairing built in [HH22a] to prove the latter proposition. In order to keep this article self contained we recall the construction below.

**Construction of the pairing** From now on we fix an enumeration $\{v_1, \dots, v_n\}$ of the vertices of $\Gamma$.

For all $i \in \{1, \dots, n\}$, let $(Y_i, \nu_i)$ be an orbit equivalence pairing between $G_{v_i}$ and $H_{v_i}$. Consider also a free measure-preserving action of $G$ on a standard probability space $Z$. Let $X = Z \times Y_1 \times \dots \times Y_n$ and endow $X$ with the product probability measure denoted by $\mu$. An element $x$ of $X$ is denoted by $x = (z, y_1, \dots, y_n)$ where $z \in Z$ and $y_i \in Y_i$ for all $i \in \{1, \dots, n\}$. We also denote by $r_i: G \to G_{v_i}$ the retraction which is the identity on $G_{v_i}$ and sends all other factors to $\{1\}$. We can thus define a free measure-preserving $G$-action on $X$, where the action on each $Y_i$ is through $r_i$, that is to say:

$$g(z, y_1, \dots, y_n) = (gz, r_1(g)y_1, \dots, r_n(g)y_n). \qquad (11.1)$$
To define the action of $H$ on $X$ we first define the action of $H_{v_i}$ on $X$ for $i \in \{1, \dots, n\}$. So let $i \in \{1, \dots, n\}$ and denote by $c_i : G_{v_i} \times Y_i \to H_{v_i}$ and $c'_i : H_{v_i} \times Y_i \to G_{v_i}$ the two orbit equivalence cocycles associated to the pairing between $G_{v_i}$ and $H_{v_i}$. Recall that $c_i(g, y_i)$ is defined for a.e. $y_i \in Y_i$, as the unique $h \in H_{v_i}$ such that $gy_i = hy_i$. Similarly $c'_i(h, y_i)$ is the unique $g \in G_{v_i}$ such that $gy_i = hy_i$. Hence, for $h \in H_{v_i}$ we define for a.e. $(z, y_1, \dots, y_n) \in X$,

$$h(z, y_1, \dots, y_n) := c'_i(h, y_i)(z, y_1, \dots, y_n).$$

This leads to a well defined free action of $H_{v_i}$ on $X$.

To obtain an action of the entire group $H$, we now show that if $h_i, h_j$ belong to adjacent vertex groups $H_{v_i}, H_{v_j}$ respectively, then $h_i(h_j(z, y_1, \dots, y_n)) = h_j(h_i(z, y_1, \dots, y_n))$. First remark that, by definition of the retractions, when $g$ belongs to some vertex group $G_{v_l}$ the above eq. (11.1) becomes

$$g(z, y_1, \dots, y_n) = (gz, y_1, \dots, y_{l-1}, gy_l, y_{l+1}, \dots, y_n). \quad (11.2)$$

Finally, since for all $l$ the cocycle $c'_l$ goes from $H_{v_l} \times Y_l$ to $G_{v_l}$ and since $v_i$ and $v_j$ are adjacent in $\Gamma$, we thus get that $c'_i(h_i, y_i)$ and $c'_j(h_j, y_j)$ belong to adjacent vertex groups. Therefore, using eq. (11.2) for the second and the fourth equalities, we get

$$ \begin{aligned} h_i(h_j(z, y_1, \dots, y_n)) &= h_i(c'_j(h_j, y_j)(z, y_1, \dots, y_n)), \\ &= c'_i(h_i, y_i)(c'_j(h_j, y_j)(z, y_1, \dots, y_n)), \\ &= c'_j(h_j, y_j)(c'_i(h_i, y_i)(z, y_1, \dots, y_n)), \\ &= h_j(c'_i(h_i, y_i)(z, y_1, \dots, y_n)), \\ &= h_j(h_i(z, y_1, \dots, y_n)). \end{aligned} $$

The action defined at the level of the vertex groups thus extends at the level of the entire graph product $H$. The fact that this $H$-action is free follows from the freeness of the $G$-action on $Z$ and a normal form argument.

We can now use this construction to prove our result.

*Proof of Theorem 1.10.* Keeping all notations from above, let $X = Z \times Y_1 \times \dots \times Y_n$ be the pairing defined in the last paragraph, where for all $i \in \{1, \dots, n\}$ we choose $Y_i$ to be a $(\varphi, \psi)$-integrable orbit equivalence pairing from $G_{v_i}$ to $H_{v_i}$. Denote by $c : G \times X \to H$ and $c' : H \times X \to G$ the two cocycles associated to the pairing $(X, \mu)$. Finally recall that an element $x$ of $X$ is denoted by $x = (z, y_1, \dots, y_n)$ where $z \in Z$ and $y_i \in Y_i$ for all $i \in \{1, \dots, n\}$.

For any $i \in \{1, \dots, n\}$ denote by $S_G^i$ and $S_H^i$ finite generating sets of $G_{v_i}$ and $H_{v_i}$ respectively. Then $S_G := \bigsqcup_{i=1}^n S_G^i$ generates $G$ and $S_H := \bigsqcup_{i=1}^n S_H^i$ generates $H$. Therefore, to prove that this pairing is $(\varphi, \psi)$-integrable we only need to show that for all $s \in S_G$
and all $s' \in S_H$ we have

$$ \int_{x \in X} \varphi(|c(s, x)|_{S_H}) d\mu(x) < +\infty, \qquad (11.3) $$

$$ \int_{x \in X} \psi(|c'(s', x)|_{S_G}) d\mu(x) < +\infty. \qquad (11.4) $$

**$\varphi$-integrability** Consider $s \in S_G$ and let $i \in \{1, \dots, n\}$ be such that $s \in S_G^i$. For a.e. $x \in X$, using first the definition of the action of $H$ on $X$ and then using that $c'_i(c_i(s, y_i), y_i) = s$, we obtain

$$ c_i(s, y_i)x = c'_i(c_i(s, y_i), y_i)x = sx, $$

and thus $c(s, x) = c_i(s, y_i)$. Therefore, by choice of $S_H$ we get that

$$ |c(s, x)|_{S_H} = |c_i(s, y_i)|_{S_H} = |c_i(s, y_i)|_{S_H^i}. $$

Finally, using that all spaces $Z$ and $Y_j$ have measure 1 and that $c_i$ is $\varphi$-integrable on $Y_i$, we thus obtain that eq. (11.3) is verified.

**$\psi$-integrability** Now consider $s' \in S_H$ and let $i \in \{1, \dots, n\}$ be such that $s' \in S_H^i$. Then by definition of the $H$-action on $X$, we have $s'x = c'_i(s', y_i)x$ for a.e. $x \in X$. Therefore $c'(s', x) = c'_i(s', y_i)$ for a.e. $x \in X$, and since the latter element belongs to $G_{v_i}$ we thus obtain

$$ |c'(s', x)|_{S_G} = |c'_i(s', y_i)|_{S_G} = |c'_i(s', y_i)|_{S_G^i}. $$

But by assumption, $c'_i$ is $\psi$-integrable, thus eq. (11.4) is verified and hence the pairing constructed above is $(\varphi, \psi)$-integrable. $\square$

# 12 From the graph product to the factors

The goal of this section is to prove Theorem 1.11, page 12, which states that the quantification at the level of the graph products passes at the level of the vertex groups.

## 12.1 The general statement and applications

### 12.1.1 Main result

What we actually show here is the following more general version of the latter theorem. We refer to Definition 5.1, page 53 for the definition of the Vertex Recognition Property and to Section 2.2.3, page 22 for the definition of an untransvable vertex.

**Theorem 12.1.** Let $\varphi, \psi: [1, +\infty] \to [1, +\infty[$ be two increasing unbounded functions and let $G$ and $H$ be two graph products defined over $\Gamma_G, \Gamma_H$ respectively, with infinite finitely generated vertex groups. Assume that there exists a $(\varphi, \psi)$-integrable measure equivalence coupling from $G$ to $H$.

1. If for all $v \in V\Gamma_G$ and all $w \in V\Gamma_H$ the vertex groups $G_v$ and $H_w$ are amenable then for all untransvable vertices $v \in V\Gamma_G$, their exist $\sigma(v) \in V\Gamma_H$ and a $(\varphi, \psi)$-integrable measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$.
2. If $G$ and $H$ both belong to a same class of groups $C$ satisfying the Vertex Recognition Property, then there exist a graph homomorphism $\sigma: \Gamma_G \to \Gamma_H$ and for all $v \in V\Gamma_G$ a $(\varphi, \psi)$-integrable measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$.

3. If $\Gamma_G$ and $\Gamma_H$ are transvection-free with no partial conjugation, then there exists a graph isomorphism $\sigma: \Gamma_G \to \Gamma_H$ and for all $v \in V\Gamma_G$ a $(\varphi, \psi)$-integrable measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$.

The second assertion applies in particular whenever $\Gamma_G$ and $\Gamma_H$ are transvection-free and strongly reduced (Proposition 7.1, page 74).

The three conclusions of the above theorem provide quantitative versions of three situations already encountered in Part II of this work. Namely,

* the first part is a quantitative version of the work from Section 9;

* the second part is a version of Corollary 5.7, page 55;

* the third part (which is exactly Theorem 1.11, page 12 from the introduction), is a quantitative version of Theorem 1.2.

We also refer to Section 10 for illustrations of these various situations.

Recall also that when $G$ is a right-angled Artin group, asking $\Gamma_G$ to be transvection-free with no partial conjugation is equivalent to requiring $\text{Out}(G)$ to be finite. So in particular the first and second assertions of our theorem apply to a larger family of right-angled Artin groups than the ones with finite outer automorphism group.

Finally, we refer to Remark 1.12 for a discussion on the asymmetry between Theorem 1.10 (stated in terms of orbit equivalence) and Theorem 12.1 (stated in terms of measure equivalence).

### 12.1.2 Obstructions in the non-amenable world

As explained in the introduction, quantitative measure equivalence responds well to geometry, which sometimes provides us with upper bounds to the possible quantifications between two given groups. So far, most of the known obstructions were in the amenable world, our Theorem 12.1 above allows to extend them to a large family of non-amenable groups. We recall below some of the known geometric obstructions and detail the case of the isoperimetric profile that will be used in the next section.

Among the previously known obstructions were the results of Bowen and Austin in [Aus16]. Indeed, Bowen proved that growth is stable under integrable measure equivalence. Austin moreover showed that two $L^1$-measure equivalent groups of polynomial growth have bi-Lipschitz asymptotic cones. This implies for example that $\mathbb{Z}^4$ and the Heisenberg group $H_3$, although having the same growth rate, cannot be $L^1$-measure equivalent. Another obstruction relies on the notion of *isoperimetric profile*. Let $S_G$ be a finite generating set of $G$ and recall that the $S_G$-boundary of a finite subset $A$ of $G$ is defined as $\partial_{S_G} A := \{g \in A : \exists s \in S_G, gs \notin A\}$. The *isoperimetric profile* of a finitely generated group $G$, with respect to a finite generating set $S$ is the function $I_G: \mathbb{N}^* \to \mathbb{R}$ defined by

$$I_G(n) := \sup_{A \subseteq G, 0 < |A| \le n} \frac{|A|}{|\partial_{S_G} A|}.$$

Note that (by the Følner criterion) a group is amenable if and only if its isoperimetric profile is unbounded.
Delabie, Koivisto, Le Maître and Tessera [DKLMT22] showed the monotonicity under quantitative measure equivalence of the isoperimetric profile and more precisely of its asymptotic behaviour. More formally, if $f$ and $g$ are two real functions we denote $f \le g$ if there exists some constant $C > 0$ such that $f(x) = O(g(Cx))$ as $x$ tends to infinity. We write $f \approx g$ if $f \le g$ and $g \le f$.

**Theorem 12.2** ([DKLMT22, Th.1]). Let $G$ and $H$ be two finitely generated groups admitting a $(\varphi, L^0)$-integrable measure equivalence coupling from $G$ to $H$. If both $\varphi$ and $t/\varphi(t)$ are non-decreasing then $\varphi \circ I_H \le I_G$.

As mentioned above, when $G$ and $H$ are non-amenable groups, both their isoperimetric profiles are bounded. Therefore, the latter theorem does not provide any obstruction for non-amenable groups. Our Theorem 12.1, however, does: it allows us to extend the condition on the profile to the world of graph products as in the example below.

**Example 12.3.** Let $G$ and $H$ be the graph products defined in Figure 21.

By [DKLMT22, Theorem 1.9], for all $\varepsilon > 0$ there exists an $(L^{1/2-\varepsilon}, L^0)$-integrable orbit equivalence pairing from $\mathbb{Z}^2$ to $\mathbb{Z}$. Therefore by Theorem 1.10, the groups $G$ and $H$ are $(L^{1/2-\varepsilon}, L^0)$-integrably orbit equivalent for all $\varepsilon > 0$.

Now assume that we have a $(\varphi, L^0)$-integrable measure equivalence coupling from $G$ to $H$. The orange vertex on the left is untransvable, therefore Theorem 12.1 provides us with a vertex group in $G$ that is $(\varphi, L^0)$-integrably measure equivalent to $\mathbb{Z}^2$, namely a $(\varphi, L^0)$-integrable measure equivalence coupling from $\mathbb{Z}^2$ to $\mathbb{Z}$.

Recall that $I_{\mathbb{Z}}(x) \approx x$ and $I_{\mathbb{Z}^2}(x) \approx \sqrt{x}$. Applying Theorem 12.2 to the latter coupling gives that the integrability $\varphi$ cannot be better than $L^{1/2}$.

<figure><img src="image_28.png" /><figcaption>Figure 21: Graph products of Example 12.3</figcaption></figure>

The next section contains the final tools needed to prove the theorem.

## 12.2 Integrability at the vertex level

The strategy of the proof of Theorem 12.1 is the following. In the three cases we will have, inside the coupling $\Omega$, a subcoupling between a vertex group of $G$ and a vertex group of $H$. It will be obtained using the results from Part II and Lemma 12.4 below.

In the first case, we will be able to obtain such a coupling for all untransvable vertices using Section 9. When working with the second case, namely the Vertex Recognition Property, we use Corollary 5.7 instead. And the third case relies on Proposition 5.10. The final tool will be Lemma 12.5, which provides a way to obtain the wanted integrability of the subcouplings between vertex groups.

We first start by rewriting Lemma 3.3 in the graph product framework and with the groupoid theoretic vocabulary, as explained in Remark 4.4. We refer to Section 4 for background on measured groupoids.
**Lemma 12.4** (Restating Lemma 3.3). Let $G$ and $H$ be two graph products of countably infinite vertex groups, defined over finite simple graphs $\Gamma_G, \Gamma_H$. Let $(\Omega, m, X_G, X_H)$ be a measure equivalence coupling from $G$ to $H$, chosen so that $X_G \cap X_H$ has positive measure. Let $\mathcal{G}$ be the associated measured groupoid over $X_G \cap X_H$ and $\rho_G$ (resp. $\rho_H$) the corresponding cocycles. Let $v \in V\Gamma_G$ and $w \in V\Gamma_H$. Assume that there exists a positive measure Borel subset $U \subseteq X_G \cap X_H$ such that

$$ \rho_G^{-1}(G_v)|_U = \rho_H^{-1}(H_w)|_U. $$

Then there exist a $(G_v \times H_w)$-invariant Borel subspace $\Omega' \subseteq \Omega$, and Borel fundamental domains $X_v, X_w$ for the actions of $G_v, H_w$ on $\Omega'$, such that

1. $(\Omega', m, X_v, X_w)$ is a measure equivalence coupling from $G_v$ to $H_w$, and

2. $X_v, X_w$ are contained in fundamental domains for the respective actions of $G, H$ on $\Omega$.

The following lemma, contrarily to the previous one, relies on the graph product structure since it makes use of the normal form.

**Lemma 12.5.** Let $G$ and $H$ be two graph products over finite simple graphs $\Gamma_G, \Gamma_H$, and let $P_G \subseteq G$ and $P_H \subseteq H$ be two respective parabolic subgroups. Let $(\Omega, m, X_G, X_H)$ be a $(\varphi, \psi)$-integrable measure equivalence coupling from $G$ to $H$. If there exist

* a $(P_G \times P_H)$-invariant Borel subset $\Omega' \subseteq \Omega$, such that $(\Omega', m, X_{P_G}, X_{P_H})$ is a measure equivalence coupling from $P_G$ to $P_H$;

* and $X'_G, X'_H$ fundamental domains for the $G$- and $H$-actions respectively, such that $X_{P_G} \subseteq X'_G$ and $X_{P_H} \subseteq X'_H$;

then $P_G$ and $P_H$ are $(\varphi, \psi)$-integrably measure equivalent over $\Omega'$ (for suitable fundamental domains).

Recall that when $(\Omega, m, X_G, X_H)$ is a measure equivalence coupling from $G$ to $H$, we have an induced action of $G$ on $X_H$ and an induced action of $H$ on $X_G$; both actions are denoted by “.” (see page 31 and Figure 1, page 12). Recall also that given a fundamental domain $X_H$ one can identify $\Omega/H$ with $X_H$ and – through this identification – changing the fundamental domain amounts to conjugating the cocycles and vice versa (see Remark 3.1, page 32).

*Proof.* Up to replacing $\Omega'$ and the fundamental domains by translates, we can (and will) assume without loss of generality that $P_G = G\Upsilon$ and $P_H = H\Lambda$ for some induced subgraphs $\Upsilon \subseteq \Gamma_G$ and $\Lambda \subseteq \Gamma_H$.

We will prove that $\Omega'$ is $(\varphi, L^0)$-integrable (and thus exhibit an appropriate fundamental domain). By symmetry (i.e. interchanging the roles of $G$ and $H$), this will be enough to derive that $\Omega'$ is also $(L^0, \psi)$-integrable (again, for an appropriate fundamental domain), and the lemma will follow.

If $x$ belongs to $X_H$ we will denote by $x'$ the unique element of $X'_H$ in the same $H$-orbit as $x$. Denote by $c$ and $c'$ the measure equivalence cocycles associated to the fundamental domains $X_H$ and $X'_H$ respectively, namely

$$ c: G \times X_H \to H, \\ c': G \times X'_H \to H. $$
By assumption, c is $\varphi$-integrable and since $X_{PH} \subseteq X'_H$ we have $c'(P_G \times X_{PH}) \subseteq P_H$. Moreover, since $X'_H$ and $X_H$ are two fundamental domains for the same H-action, the above two cocycles are cohomologous. That is to say, there exists $\alpha : X_H \to H$ such that $c'(g, x') = \alpha(g \cdot x)c(g, x)\alpha(x)^{-1}$ for all $g \in G$ and a.e. $x \in X_H$.

Now let $\vartheta : H \to H$ be defined by letting $\vartheta(h)$ be the element with the smallest word length such that $P_H\vartheta(h) = P_Hh$. Equivalently $\vartheta(h)$ is the unique representative of the coset $P_Hh$ whose head (see page 19 for the definition) contains no element in $P_H$. By definition $\vartheta$ is constant on all right cosets of $P_H$. Now let

$$X_H'' := \{\vartheta(\alpha(x)) \cdot x : x \in X_H\}.$$

As above, if $x$ belongs to $X_H$ we denote by $x''$ the element of $X_H''$ corresponding to the same H-orbit. We denote by $X_{PH}''$ the image of $X_{PH}$ under the identification $x' \mapsto x''$. Let us prove that $X_{PH}''$ is the appropriate fundamental domain.

For all $g \in P_G$ and a.e. $x \in X_H$ we define

$$c_1''(g, x'') = \vartheta(\alpha(g \cdot x)) c(g, x) \vartheta(\alpha(x))^{-1}.$$

Remark that by definition of $\vartheta$ this cocycle $c_1''$ is $P_H$-cohomologous to $c'$ (through the identification $x' \mapsto x''$). In particular, its restriction to $P_G \times X_{PH}''$ takes its values in $P_H$ and also defines a measure equivalence cocycle on $\Omega'$ (namely, the measure equivalence cocycle associated to the fundamental domain $X_{PH}'')$.

Now, for all $v \in V\Gamma_H$ let $S_{H_v}$ be a finite generating set of $H_v$ and recall that $S_H = \cup_{v \in V\Gamma_H} S_{H_v}$ generates $H$. We also let $S_{PH} := \cup_{v \in V\Lambda} S_{H_v}$, where $\Lambda \subseteq \Gamma_H$ is the type of $P_H$ – notice that $S_{PH}$ is a finite generating set of $P_H$. Our definition of $\vartheta$, together with Lemma 2.2, page 19 imply that for a.e. $x'' \in X_{PH}''$ we have

$$|c_1''(g, x'')|_{S_{PH}} = |c_1''(g, x'')|_{S_H} \le |c(g, x)|_{S_H}.$$

Hence $c_1''$ is a $\varphi$-integrable cocycle since c is itself $\varphi$-integrable, as wanted. $\square$

## 12.3 Proof of the main theorem

We now turn to the proof of Theorem 12.1.

*Proof of Theorem 12.1.* Let $(\Omega, m, X_G, X_H)$ be a $(\varphi, \psi)$-integrable measure equivalence coupling from $G$ to $H$, and let $c_1 : G \times X_H \to H$ and $c_2 : H \times X_G \to G$ be the two associated measure equivalence cocycles. Up to replacing $X_H$ by $hX_H$ for some $h \in H$, we can (and will) assume that $\mu(X_G \cap X_H) > 0$ – notice indeed that this changes the measure equivalence cocycle from $G$ to $H$ by a conjugation by $h$, and therefore does not affect its integrability. Let $X = X_G \cap X_H$.

Let $\mathcal{G}$ be the measured groupoid over $X$ given by the restriction of the $G$- and $H$-actions on $X_H$ and $X_G$ respectively (see Example 4.2, page 38). It is naturally equipped with two cocycles $\rho_G : \mathcal{G} \to G$ and $\rho_H : \mathcal{G} \to H$, which are action-like (Lemma 4.13, page 42). We now distinguish three different cases corresponding to the three different sets of hypotheses in our theorem.
**Case 1** If for all $v \in V\Gamma_G$ and all $w \in V\Gamma_H$ the vertex groups $G_v$ and $H_w$ are amenable then Proposition 9.1, page 95 ensures that for every untransvable vertex $v \in V\Gamma_G$ there exist $\sigma(v) \in V\Gamma_H$, $h \in H$ and a Borel subset $U \subseteq X_G \cap X_H$ of positive measure such that

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

Considering now $H'$ the graph product over $\Gamma_H$ with vertex groups $\{hH_vh^{-1}\}_{v \in V\Gamma_H}$, we can apply Lemma 12.4 to $G$ and $H'$ and obtain a measure equivalence coupling $\Omega'$ from $G_v$ to $hH_{\sigma(v)}h^{-1}$ verifying the hypotheses of Lemma 12.5. The latter lemma gives the $(\varphi, \psi)$-integrability of $\Omega'$, for well-chosen fundamental domains.

**Case 2** If $G$ and $H$ both belong to a class of groups $C$ having the Vertex Recognition Property then the proof of Corollary 5.7, page 55 ensures that there exists a graph homomorphism $\sigma : \Gamma_G \to \Gamma_H$ such that for all $v \in V\Gamma_G$, the groups $G_v$ and $H_{\sigma(v)}$ verify the hypothesis of Lemma 12.4. And therefore, as above there exists a $(\varphi, \psi)$-integrable coupling from $G_v$ to $H_{\sigma(v)}$.

**Case 3** If we assume that $\Gamma_G$ and $\Gamma_H$ are transvection-free graphs with no partial conjugation, then Proposition 5.10 and Remark 5.11 ensure that there exist a graph isomorphism $\sigma : \Gamma_G \to \Gamma_H$, and for all $v \in V\Gamma_G$ a Borel subset $\Omega_v \subseteq \Omega$ which is a measure equivalence coupling from $G_v$ to $H_{\sigma(v)}$ and satisfy the hypotheses of Lemma 12.5. The latter result then gives the desired integrability. $\square$

# 13 The inverse problem for right-angled Artin groups

The goal of this section is to prove Theorem 1.13, page 13 which answers the inverse problem for right-angled Artin groups, namely: given a right-angled Artin group $H$ and a quantification $\varphi$, can we find a group $G$ and a measure equivalence coupling which is $(\varphi, L^0)$-integrable from $G$ to $H$?

As discussed in the introduction (see Section 1.2.3, page 13), we also ask this coupling to be optimal (or close to be). The optimality here comes from the geometry of the involved groups, more precisely from the obstruction given by the isoperimetric profile of the vertex groups (see Section 12.1.2).

## 13.1 Main theorem

The following restates Theorem 1.13 from the introduction.

**Theorem 1.13.** Let $\Gamma$ be a finite simple graph that contains an untransvable vertex and let $\rho$ be a non-decreasing map such that $x \mapsto x/\rho(x)$ is non-decreasing. If $H$ is a right-angled Artin group over $\Gamma$ then there exists a group $G$ and an orbit equivalence pairing from $G$ to $H$ which is a $(\varphi_\varepsilon, L^0)$-integrable for all $\varepsilon > 0$, where

$$ \varphi_\varepsilon(x) := \frac{\rho \circ \log(x)}{(\log \circ \rho \circ \log(x))^{1+\varepsilon}}. $$

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 $\varphi \le \rho \circ \log$.
**Discussion on the hypotheses** First remark that the hypotheses on $\rho$ lead us to deal with quantifications having sub-logarithmic growth. On the other hand, Huang and the second-named author showed a superrigidity phenomenon under $(L^1, L^0)$-measure equivalence [HH23, Theorem 1]. More precisely, they showed that if $G$ is a right-angled Artin group with finite outer automorphism group and $H$ is any countable group with bounded torsion such that there exists an $(L^1, L^0)$-measure equivalence coupling from $H$ to $G$, then $H$ is finitely generated and quasi-isometric to $G$. In some cases this implies that $G$ and $H$ are commensurable [Hua18]. We thus have here two completely different behaviors: a lot of flexibility for sub-logarithmic quantification versus superrigidity under $L^1$-integrability.

Second, we mention that one could extend the above theorem when $H$ is a graph product over $\Gamma$ of non-trivial finitely generated free abelian groups up to adding some assumptions on the map $\varphi_{\varepsilon}$ (concavity) as in [Esc22, Corollary 1.8].

On the other hand, we do not know whether the theorem holds when $H$ is a non-abelian free group, and more generally we cannot treat the case where $H$ splits as a graph product of non-abelian free groups.

## 13.2 Proof of Theorem 1.13

The proof relies on two main tools: the resolution of the above inverse problem for $H = \mathbb{Z}$ (or equivalently, when $\Gamma$ is reduced to one vertex) already established in [Esc22], and the results specific to graph products developed in the present article, namely Theorems 1.10 and 12.1 (pages 12 and 104).

We start by recalling the known results for the case of $\Gamma$ reduced to one vertex and then turn to our proof of the above Theorem 1.13.

**Background on the inverse problem** When $H = \mathbb{Z}$ (or equivalently, when $\Gamma$ is reduced to one vertex) building couplings with prescribed integrability with $H$ has been done by the first named author in [Esc22, Theorem 1.7], which we recall below. The optimality is obtained in this case using the obstruction given in Theorem 12.2, which relies on the isoperimetric profile (see Section 12.1.2 for the definition of the profile $I_G$).

**Theorem 13.1** (Escalier [Esc22]). For all non-decreasing maps $\rho$ such that $x \mapsto x/\rho(x)$ is non-decreasing, there exists a group $\Delta$ such that

* $I_{\Delta} \simeq \rho \circ \log$;

* There exists an orbit equivalence pairing that is $(\varphi_{\varepsilon}, \exp \circ \rho)$-integrable from $\Delta$ to $\mathbb{Z}$ for all $\varepsilon > 0$, where $\varphi_{\varepsilon}(x) := \rho \circ \log(x)/(\log \circ \rho \circ \log(x))^{1+\varepsilon}$.

We mention that the group $\Delta$ given by the above theorem, is a diagonal product of lamplighter groups as defined by Brieussel-Zheng in [BZ21].

**Claim 13.2.** The coupling given by the above theorem is optimal up to a log factor in the sense of Theorem 1.13, namely: for every non-decreasing map $\varphi$ if there exists a $(\varphi, L^0)$-integrable measure equivalence coupling from $\Delta$ to $\mathbb{Z}$ then $\varphi \ll I_{\Delta} \simeq \rho \circ \log$.

*Proof of the Claim.* Let $\Delta$ be the group given by the above theorem. Assume that there exists a $(\varphi, L^0)$-integrable measure equivalence coupling from $\Delta$ to $H$, for some non-decreasing map $\varphi$ and let us prove that $\varphi \ll I_{\Delta} \simeq \rho \circ \log$.
• Let us first prove that $x \mapsto x/\varphi(x)$ is necessarily non-decreasing.

So assume towards a contradiction that $x \mapsto x/\varphi(x)$ is decreasing. Then $\varphi \le \varphi$ and thus any $\varphi$-integrable cocycle is at least $L^1$-integrable. Therefore the coupling is also $(L^1, L^0)$-integrable from $\Delta$ to $H$. Thus, recalling that $I_H(x) \simeq x$ and $I_\Delta \simeq \rho \circ \log$ and using Theorem 12.2, we get that in this case

$$x \simeq I_H(x) \le I_\Delta(x) \simeq \rho \circ \log(x)$$

But the map $x \mapsto x/\rho(x)$ is non-decreasing, which contradicts the above inequality. Hence the map $x \mapsto x/\varphi(x)$ has to be non-decreasing.

• Since $t \mapsto t/\varphi(t)$ is non-decreasing, we can now apply Theorem 12.2 and obtain that $\varphi \circ I_H \le I_\Delta$. Since $I_H \simeq \text{id}$, the last inequality becomes $\varphi \le I_\Delta \simeq \rho \circ \log$. As wanted. □

**The case of graph products** We now turn to the proof of the general case. The existence of the group $G$ and the pairing are given by Theorems 1.10 and 13.1. The optimality is then obtained using Claim 13.2 and Theorem 12.1.

*Proof of Theorem 1.13.* Let $\Gamma$ be a finite simple graph that contains an untrans vectorable vertex, denoted by $v$, and $H$ be right-angled Artin group over $\Gamma$.

**Construction of the coupling** Let $\rho$ and for all $\varepsilon > 0$ let $\varphi_\varepsilon$ be as in the theorem. Then by Theorem 13.1 there exists a group $\Delta$ admitting a $(\varphi_\varepsilon, L^0)$-integrable orbit equivalence pairing from $\Delta$ to $\mathbb{Z}$. Denote by $G$ the graph product over $\Gamma$, with vertex groups $G_v = \Delta$ and $G_w = H_w$ for all $w \ne v$ in $V\Gamma$. Then by Theorem 1.10, page 12 there exists a $(\varphi_\varepsilon, L^0)$-integrable orbit equivalence pairing from $G$ to $H$.

**Optimality** Let $\varphi$ be a non-decreasing map and assume that we have a $(\varphi, L^0)$-integrable measure equivalence coupling from $G$ to $H$. By the first point of Theorem 12.1, page 104 there exists $\sigma(v) \in V\Gamma$ and a $(\varphi, L^0)$-integrable measure equivalence coupling from $G_v = \Delta$ to $H_{\sigma(v)} \simeq \mathbb{Z}$. So applying Claim 13.2 to the latter coupling leads $\varphi \le \rho \circ \log$. Therefore, the $(\varphi_\varepsilon, L^0)$-integrable coupling constructed in the first part of the proof is optimal up to a logarithmic error, as wanted. □
# Part IV

## Rigidity under ergodicity assumptions

The goal of this part is to prove orbit equivalence rigidity theorems for actions of graph products that satisfy certain strong ergodicity assumptions. Our main theorems are Theorem 14.2 (giving a rigidity statement among actions of graph products) and Theorem 15.1 (giving a superrigidity statement, comparing two actions $G \not\to X$ and $H \not\to Y$ where only $G$ is a graph product, while $H$ is an arbitrary countable group). This is in the vein of the work of Monod and Shalom for direct products [MS06], and generalizes the case of right-angled Artin groups [HHI23].

## 14 Rigidity among graph products

### 14.1 Some terminology

We start by recalling some terminology that will be used throughout the section.

**Ergodicity assumptions.** Recall that a measure-preserving action of a countable group $G$ on a standard probability space $(X, \mu)$ is

* weakly mixing if the diagonal action of $G$ on $X \times X$ is ergodic.

* totally ergodic if every infinite subgroup of $G$ acts ergodically on $X$.

* mildly mixing if it is ergodic and if for any measurable subset $U \subseteq X$ with $\mu(U), \mu(X\setminus U) > 0$, and any sequence of pairwise distinct elements $g_n$ in $G$, one has $\liminf \mu(g_nU \Delta U) > 0$. By [SW82], this is equivalent to the following requirement, which is in fact the original definition of mild mixing due to Furstenberg and Weiss [FW78]: for every measure-preserving ergodic $G$-action on a standard Borel space $S$ equipped with a non-atomic $\sigma$-finite Borel measure, the diagonal $G$-action on $X \times S$ is ergodic.

* strongly mixing if given any two Borel subsets $U, V \subseteq X$, and any sequence $(g_n)_{n \in \mathbb{N}} \in G^{\mathbb{N}}$ of pairwise distinct elements, one has $\lim_{n \to +\infty} \mu(g_nU \cap V) = \mu(U)\mu(V)$.

We notice that strongly mixing actions are in particular mildly mixing, that mildly mixing actions of countably infinite groups are weakly mixing, and that weakly mixing actions are ergodic (and totally ergodic actions are also ergodic).

We also mention that if $G$ has a subgroup $H$ which acts weakly mixingly on $X$, then the $G$-action on $X$ is weakly mixing. This is however not true for the total ergodicity, mild mixing and strong mixing conditions.

**(Virtual) conjugation of actions.** Two measure-preserving actions $G \not\to (X, \mu)$ and $H \not\to (Y, \nu)$ on standard probability spaces are *conjugate* if there exist a group isomorphism $\theta: G \to H$ and a measure space isomorphism $f: X \to Y$ such that $f(g \cdot x) = \theta(g)f(x)$ for every $g \in G$ and almost every $x \in X$.

There exists a weaker notion of *virtually conjugate* actions, which takes into account the possibility of passing to finite-index subgroups and taking the quotient by a finite normal subgroup. We now recall the definition, see e.g. [Kid08b, Definition 1.3].
First, given a countable group $G$, a finite-index subgroup $H \subseteq G$, and a measure-preserving action of $H$ on a standard probability space $X$, there is a notion of an action of $G$ on a standard finite measure space $\text{Ind}_H^G(X)$, induced from that of $H$, defined as follows. Let $\theta: H \to G$ be the inclusion map. The space $X \times G$ (where $G$ is equipped with the counting measure) is equipped with a measure-preserving action of $H \times G$ given by $(h,g) \cdot (x,g') = (hx, \theta(h)g'g^{-1})$. We then let $\text{Ind}_H^G(X) := H\backslash(X \times G)$: this is naturally a standard finite measure space (isomorphic to $X \times G/H$), which we renormalize to a probability space. The $G$-action on $X \times G$ descends to a probability measure-preserving $G$-action on $\text{Ind}_H^G(X)$.

Notice for future use that, following the notation from [Vae07, Notation 4.3], we can more generally define the induced action $G \to \text{Ind}_H^G(X, \theta)$ whenever $\theta: H \to G$ is an injective homomorphism with finite-index image. In the particular case where $\theta$ is an isomorphism, then the $G$-action on $X$ and the $H$-action on $\text{Ind}_H^G(X, \theta)$ are conjugate (via the isomorphism $\theta: H \to G$ and the measure space isomorphism $x \mapsto [x, e]$, where $[x, e]$ denotes the class of $(x, e)$).

Notice also that, if a countable group $G$ has a measure-preserving action on a standard probability space $(X, \mu)$, and if $F_G \subsetneq G$ is a finite normal subgroup, then there is a quotient measure-preserving action of $G/F_G \to (X, \mu)/F_G$, where $(X, \mu)/F_G$ is the space of $F_G$-orbits in $X$, renormalized to a probability measure.

Two measure-preserving actions $G \to (X, \mu)$ and $H \to (Y, \nu)$ are *virtually conjugate* if we can find

* short exact sequences

$$1 \to F_G \to G \to G_1 \to 1 \quad \text{and} \quad 1 \to F_H \to H \to H_1 \to 1,$$

where $F_G \subsetneq G$ and $F_H \subsetneq H$ are finite;

* finite-index subgroups $G_2 \subseteq G_1$ and $H_2 \subseteq H_1$;

* and conjugate measure-preserving actions $G_2 \to (X_2, \mu_2)$ and $H_2 \to (Y_2, \nu_2)$, such that

- the action $G_1 \to (X, \mu)/F_G$ is conjugate to $G_1 \to \text{Ind}_{G_2}^{G_1}(X_2)$, and

- the action $H_1 \to (X, \mu)/F_H$ is conjugate to $H_1 \to \text{Ind}_{H_2}^{H_1}(X_2)$.

## 14.2 Rigidity of actions within the class of graph products

We make the following definition.

**Definition 14.1.** Let $G$ be a graph product of countable groups over a finite simple graph. A free, probability measure preserving action of $G$ on a standard probability space $X$ is *vertexwise ergodic* (resp. *vertexwise weakly mixing*, resp. *vertexwise totally ergodic*) if for every vertex group $G_v$, the action of $G_v$ on $X$ is ergodic (resp. weakly mixing, resp. totally ergodic).

When $G$ is a graph product with countably infinite vertex groups, notice that vertexwise weak mixing and vertexwise total ergodicity are satisfied for all strongly mixing actions of $G$ on $X$. Besides Bernoulli actions, this includes for example actions coming from embedding $G$ as a closed subgroup of $\text{SL}(n, \mathbb{R})$ with non-compact image, and using
the Howe–Moore property – see [HW99, Theorem 3.2] for examples of faithful representations of graph products into $SL(n, \mathbb{Z})$. See more generally [BdlN22, Corollary B] for constructions of faithful representations of graph products into $GL(n, \mathbb{C})$, assuming the linearity over $\mathbb{C}$ of the vertex groups.

Recall that a group $G$ has the *unique root property* if for every $g_1, g_2 \in G$ and $n \ge 1$, if $g_1^n = g_2^n$, then $g_1 = g_2$. In this section, we establish the following theorem. We refer to Section 3.1.1 for the notions of stable orbit equivalence and compression constants.

**Theorem 14.2.** Let $\Gamma_G, \Gamma_H$ be two connected finite simple graphs, with no vertex joined to every other vertex. Let $G, H$ be graph products with countably infinite torsion-free vertex groups over $\Gamma_G, \Gamma_H$, respectively. Let $G \to X$ and $H \to Y$ be two free, vertexwise ergodic, measure-preserving actions on standard probability spaces. Assume that either

* $G \to X$ is vertexwise weakly mixing, or

* $G \to X$ is vertexwise totally ergodic, or

* $H$ has the unique root property.

If there is a stable orbit equivalence $f$ from $G \to X$ to $H \to Y$ with compression constant $\kappa(f) \ge 1$, then the two actions are conjugate (and every SOE cocycle $c: G \times X \to H$ associated to $f$ is cohomologous to a group isomorphism $G \to H$).

We mention that the last two cases will be useful in the next section to obtain superrigidity statements (Section 15).

**Remark 14.3** (Discussion on the connectedness hypothesis). The assumption that the defining graph $\Gamma_G$ is connected cannot be removed. In fact, Bernoulli actions of non-abelian free groups fail to satisfy the form of rigidity provided by Theorems 14.2, in view of works of Bowen [Bow11a, Bow11b].

**Remark 14.4** (Examples of groups with the unique root property). Assuming that $H$ satisfies the unique root property is equivalent to assuming that each of its vertex groups satisfies the unique root property [BF19, Theorem 6.5]. The class of groups that satisfy this property is vast. It includes, among others, all torsion-free nilpotent groups [Mal49], and more generally all bi-orderable groups (including all Baumslag–Solitar groups $BS(1, n)$ with $n \ge 1$, Thompson’s group or more generally diagram groups [GS06]), all torsion-free hyperbolic groups (using the fact that the centralizer of every non-trivial element is cyclic [BH99, Corollary III.Γ.3.10]) and more generally toral relatively hyperbolic groups (even more generally, torsion-free groups whose maximal abelian subgroups are malnormal, called *CSA groups*, which also include limit groups [MR96, Proposition 8]), and some torsion-free finite-index subgroups of surface mapping class groups [BP09] and Out($F_N$) [Gue22]. The unique root property is also stable under certain group-theoretic operations like taking wreath products [Bau60].

**Proof strategy and outline of the section.** Our proof of Theorem 14.2 follows the strategy used by Huang, Ioana and the second-named author in [HHI23]. In particular we will follow very closely the presentation of [HHI23, Propositions 6.1 and 6.2]. The idea is the following. Let $c$ be a stable orbit equivalence cocycle associated to $f$.

* We first untwist $c$ (i.e. change it within its cohomology class) to a group homomorphism on some vertex group of $G$. The main tools are our recognition statements from Section 7, ensuring in particular that the orbit equivalence relation
"remembers" the maximal product parabolic subgroups, together with the work of Monod-Shalom on direct products [MS06].

* Once this is done, a propagation argument enables to untwist the cocycle on the whole group $G$. We give three versions of this propagation argument: one uses weak mixing at the level of the vertex groups, one uses total ergodicity, and the third relies on the unique root property for $H$.

The section is organized as follows. After presenting the propagation arguments in Section 14.2.1, we review and establish a few extra tools in Sections 14.2.2 to 14.2.4, and complete our proof of Theorem 14.2 in Section 14.2.5.

### 14.2.1 Three propagation arguments

**Propagation via weak mixing.** The idea of exploiting weak mixing to propagate rigidity from a normal subgroup to an ambient group arose in the work of Popa, see [Pop07, Lemma 3.6], which finds its roots in [Pop06b, Lemma 5.7]. Popa's arguments were formulated in the language of von Neumann algebras; we will use Furman's formulation [Fur07, Lemma 3.5] whose statement is more ergodic-theoretic.

**Lemma 14.5.** Let $G, H$ be countable groups. Let $G \rightsquigarrow X$ be a measure-preserving action on a standard probability space $X$, and $c: G \times X \to H$ be a cocycle. Let $G_1, \dots, G_k$ be subgroups of $G$ satisfying the following three assumptions:

* the set $G_1 \cup \dots \cup G_k$ generates $G$;

* for every $j \in \{1, \dots, k-1\}$, the subgroup $G_{j+1}$ normalizes $G_j$;

* for every $j \in \{1, \dots, k-1\}$, the subgroup $G_j$ acts weakly mixingly on $X$.

Assume that there exists a group homomorphism $\theta_1: G_1 \to H$ such that for every $g \in G_1$ and almost every $x \in X$, one has $c(g, x) = \theta_1(g)$.

Then there exists a group homomorphism $\theta: G \to H$ such that for every $g \in G$ and almost every $x \in X$, one has $c(g, x) = \theta(g)$.

*Proof.* We first prove by induction that for every $j \in \{1, \dots, k\}$, there exists a group homomorphism $\theta_j: G_j \to H$ such that for every $g \in G_j$ and almost every $x \in X$, one has $c(g, x) = \theta_j(g)$. This holds when $j=1$ by assumption. And assuming this holds for $G_j$, then [Fur07, Lemma 3.5], applied to the subgroup $G_j$ of $G$, with $Y$ reduced to a point and $\rho = \theta_j$, ensures that it also holds for $G_{j+1}$.

Now, as $G_1 \cup \dots \cup G_k$ generates $G$, using the cocycle relation, we deduce that for every $g \in G$, the map $c(g, \cdot)$ is almost everywhere constant. Since $G$ is countable, there exists a conull Borel subset $X^* \subseteq X$ such that for every $g \in G$, the map $c(g, \cdot)|_{X^*}$ is constant. And the cocycle relation ensures that the map $\theta$ sending $g$ to the constant value of $c(g, \cdot)|_{X^*}$ is a group homomorphism, as desired. $\square$

**Propagation via ergodicity and unique roots.** The following lemma exploits the unique root property to propagate rigidity. It is a very slight variation over [HHI23, Lemma 4.2].

**Lemma 14.6.** Let $G, H$ be countable groups, and assume that $H$ satisfies the unique root property. Let $G \rightsquigarrow X$ be a measure-preserving action on a standard probability
space $X$, and $c: G \times X \to H$ be a cocycle. Let $G_1, \dots, G_k$ be subgroups of $G$ satisfying the following three assumptions:

* the set $G_1 \cup \dots \cup G_k$ generates $G$;

* for every $j \in \{1, \dots, k-1\}$, the subgroup $G_{j+1}$ centralizes $G_j$;

* for every $j \in \{1, \dots, k-1\}$, the subgroup $G_j$ acts ergodically on $X$.

Assume that there exists a group homomorphism $\theta_1: G_1 \to H$ such that for every $g \in G_1$ and almost every $x \in X$, one has $c(g, x) = \theta_1(g)$.

Then there exists a group homomorphism $\theta: G \to H$ such that for every $g \in G$ and almost every $x \in X$, one has $c(g, x) = \theta(g)$.

*Proof.* As in the previous proof, it is enough to prove by induction that for every $j \in \{1, \dots, k\}$, there exists a group homomorphism $\theta_j: G_j \to H$ such that for every $g \in G_j$ and almost every $x \in X$, one has $c(g, x) = \theta_j(g)$. In fact, it is enough to prove that for every $g \in G_j$, the map $c(g, \cdot)$ is essentially constant: indeed, the map sending $g$ to the essential value of $c(g, \cdot)$ is then automatically a group homomorphism, in view of the cocycle relation.

This holds for $j = 1$ by assumption. Assume that it holds for some $j \in \{1, \dots, k-1\}$, and let $g \in G_{j+1}$. Let $X = \cup_{i \in I} X_i$ be a countable Borel partition such that for every $i \in I$, the map $c(g, \cdot)|_{X_i}$ is constant, with value denoted by $h_i \in H$ (see also Figure 22). We aim to prove that $h_{i_1} = h_{i_2}$ for any $i_1, i_2 \in I$.

<figure><img src="image_29.png" /><figcaption>Figure 22: On $X_i$ the value of $c(g, \cdot)$ is constant, equal to $h_i$.</figcaption></figure>

So let $i_1, i_2 \in I$. Since $G_j$ acts ergodically on $X$, there exists $g' \in G_j$ such that $g'X_{i_1} \cap X_{i_2} =: V$ has positive measure. Let also $U := (g')^{-1}(V)$ (see also Figure 23). Since $g \in G_{j+1}$ and $g' \in G_j$, we have $gg' = g'g$. So for every $x \in U$, we have $gg' \cdot x = g'g \cdot x$. Applying the cocycle $c$ gives

$$c(g, g' \cdot x)c(g', x) = c(g', g \cdot x)c(g, x),$$

in other words

$$h_{i_2} \theta_j(g') = \theta_j(g') h_{i_1}. \qquad (14.1)$$

On the other hand, using Poincaré recurrence, there also exists $k > 0$ such that the subset $X_{i_1} \cap (g')^k X_{i_1} =: Z$ has positive measure. Let $W := (g')^{-k}(Z)$ (see also Figure 24). Then for every $x \in W$, we have $g(g')^k \cdot x = (g')^k g \cdot x$, so applying the cocycle $c$ gives

$$c(g, (g')^k \cdot x)c((g')^k, x) = c((g')^k, g \cdot x)c(g, x),$$

in other words

$$h_{i_1} \theta_j(g')^k = \theta_j(g')^k h_{i_1}. \qquad (14.2)$$
This rewrites as $(h_{i_1} \theta_j(g') h_{i_1}^{-1})^k = \theta_j(g')^k$. Using the unique root property for $H$, we deduce that $h_{i_1}$ and $\theta_j(g')$ commute. Therefore, Equation (14.1) yields $h_{i_2} = h_{i_1}$, as desired. $\square$

<figure><img src="image_30.png" /></figure>

We represent here how we obtain eq. (14.1). In orange are represented the actions of elements in $G$, while in blue are written the corresponding values of the cocycle $c$. The sets $U$ and $V$ are drawn in brown.

<figure><img src="image_31.png" /><figcaption>Figure 24: Illustration for eq. (14.2)</figcaption></figure>

**Propagation via total ergodicity.** The following lemma uses total ergodicity to propagate rigidity. The argument is a slight variation over the previous one, but does not necessit to assume that $H$ satisfies the unique root property.

**Lemma 14.7.** Let $G, H$ be countable groups, and assume that $G$ is torsion-free. Let $G \prec X$ be a measure-preserving action on a standard probability space $X$, and let