###### Proof.

By definition, there exists a fundamental domain $X_{\hat{G}}$ for the action
of $\hat{G}$ on $\hat{\Omega}$ such that the cocycle $\hat{c}:H\times
X_{\hat{G}}\to\hat{G}$ is $L^{1}$-integrable.

Let $S=\\{g_{1},\dots,g_{k}\\}$ be a (finite) set of representatives of the
right cosets of $G$ in $\hat{G}$. Then $X_{G}=g_{1}X_{\hat{G}}\cup\dots\cup
g_{k}X_{\hat{G}}$ is a fundamental domain for the $G$-action on
$\hat{\Omega}$.

We claim that the associated cocycle $c:H\times X_{G}\to G$ is
$L^{1}$-integrable. Indeed, let $h\in H$. For $x\in X_{G}$, there exists a
unique $\hat{x}\in X_{\hat{G}}$ and $j\in\\{1,\dots,k\\}$ such that
$x=g_{j}\hat{x}$. Also by definition of $X_{\hat{G}}$, there exists a unique
element $\hat{g}\in\hat{G}$ such that $\hat{g}hx\in X_{\hat{G}}$. Using the
fact that the actions of $\hat{G}$ and $H$ on $\hat{\Omega}$ commute, we see
that $(\hat{g}g_{j})h\hat{x}\in X_{\hat{G}}$, showing that
$\hat{c}(h,\hat{x})=\hat{g}g_{j}$. On the other hand, the set
$S^{-1}=\\{g_{1}^{-1},\dots,g_{k}^{-1}\\}$ is a set of representatives of the
left cosets of $G$ in $\hat{G}$, so there exist $i\in\\{1,\dots,k\\}$ and
$g\in G$ such that $\hat{g}=g_{i}^{-1}g$. We then have $ghx\in
g_{i}X_{\hat{G}}$, showing that $c(h,x)=g$. It follows that $c(h,x)\in
S\hat{c}(h,\hat{x})S^{-1}$. As this is true for every $h\in H$ and $x\in
X_{G}$ and $S$ is a finite set, the lemma follows. ∎

Let now $G,H$ be as in the statement of Proposition 7.1. Let $\Omega$ be an
$(L^{1},L^{0})$-measure equivalence coupling from $H$ to $G$, and let $X_{G}$
be a Borel fundamental domain for the $G$-action on $\Omega$ such that the
measure equivalence cocycle $c:H\times X_{G}\to G$ is $L^{1}$-integrable.

Let $\hat{G}$ be the finite-index extension of $G$ introduced in Section 6.1.
Let $\hat{\Omega}$ be the induced coupling between $\hat{G}$ and $H$, namely
$\hat{\Omega}=G\backslash(\hat{G}\times\Omega)$ – here $\hat{G}\times G$ acts
on $\hat{G}$ via $(\hat{g},g)\cdot h=\hat{g}hg^{-1}$, and $\hat{G}$ acts
trivially on $\Omega$ while $H$ acts trivially on $\hat{G}$, and we are taking
the quotient by the diagonal action of $G$ on $\hat{G}\times\Omega$. Notice
that $X_{\hat{G}}=\\{\mathrm{id}\\}\times X_{G}$, identified to a subset of
$\hat{\Omega}$, is a Borel fundamental domain for the action of $\hat{G}$ on
$\hat{\Omega}$. The associated measure equivalence cocycle $H\times
X_{\hat{G}}\to\hat{G}$ takes its values in $G$ and coincides with $c$. In
particular $\hat{\Omega}$ is an $(L^{1},L^{0})$-measure equivalence coupling
from $H$ to $\hat{G}$. We can therefore apply Lemma 7.2 and obtain a Borel
fundamental domain $\hat{X}_{G}$ for the $G$-action on $\hat{\Omega}$, such
that the measure equivalence cocycle $\hat{c}:H\times\hat{X}_{G}\to G$ is
$L^{1}$-integrable.

Lemma 6.6 gives us a homomorphism $\iota:H\to\mathrm{Aut}(\mathbb{B})$ and an
almost equivariant map $\theta:\hat{\Omega}\to\mathrm{Aut}(\mathbb{B})$. In
particular we have an action of $H$ on $\mathbb{B}$. Given $v\in V\mathbb{B}$,
we denote by $G_{v}$ and $H_{v}$ its stabilizers for the actions of $G$ and
$H$, respectively.

######  Lemma 7.3.

For every $v\in V\mathbb{B}$, there exists an $(L^{1},L^{0})$-measure
equivalence coupling from $H_{v}$ to $G_{v}$.

Our proof of Lemma 7.3 is inspired by an argument coming from ongoing work of
Escalier and the first-named author [EH].

###### Proof.

As above, let $\hat{X}_{G}$ be a fundamental domain for the action of $G$ on
$\hat{\Omega}$, such that the cocycle $\hat{c}:H\times\hat{X}_{G}\to G$ is
$L^{1}$-integrable.

By Lemma 6.7, the space
$\hat{\Omega}_{v}=\theta^{-1}(\mathrm{Stab}_{\mathrm{Aut}(\mathbb{B})}(v))$ is
a measure equivalence coupling between $G_{v}$ and $H_{v}$. Let
$Y_{G_{v}},Y_{H_{v}}$ be respective Borel fundamental domains for these
actions on $\hat{\Omega}_{v}$. Lemma 6.7 also shows that $Y_{G_{v}},Y_{H_{v}}$
extend to Borel fundamental domains $Y_{G},Y_{H}$ for the actions of $G$ and
$H$ on $\hat{\Omega}$.

Let $c_{v}:H_{v}\times Y_{G_{v}}\to G_{v}$ be the measure equivalence cocycle.
Then $c_{v}$ extends to a measure equivalence cocycle $c^{\prime}:H\times
Y_{G}\to G$. The cocycle $c^{\prime}$ is cohomologous to $\hat{c}$, i.e. there
exists $\theta:Y_{G}\to G$ such that $c^{\prime}(h,x)=\theta(h\cdot
x)^{-1}\hat{c}(h,\tilde{x})\theta(x)$, where $\tilde{x}$ is the unique element
of $\hat{X}_{G}$ in the same $G$-orbit as $x$.

As usual $G$ is equipped with its standard generating set. Let $\psi:G\to G$
be defined by letting $\psi(g)$ be the unique element with smallest word
length such that $\psi(g)G_{v}=gG_{v}$ (uniqueness comes from the normal form
in the right-angled Artin group, in the sense of graph products [Gre90]).

Let $\theta^{\prime}=\psi\circ\theta$, and let $c^{\prime\prime}:H\times
Y_{G}\to G$ be defined by $c^{\prime\prime}(h,x)=\theta^{\prime}(h\cdot
x)^{-1}\hat{c}(h,\tilde{x})\theta^{\prime}(x)$. By definition of $\psi$, for
every $x\in X$, there exists $g_{x}\in G_{v}$ such that
$\theta^{\prime}(x)=\theta(x)g_{x}$. Using this and the fact that
$c^{\prime}(H_{v}\times Y_{G_{v}})\subseteq G_{v}$, we deduce that
$c^{\prime\prime}(H_{v}\times Y_{G_{v}})\subseteq G_{v}$. In addition, a
normal form argument shows that
$|c^{\prime\prime}(h,x)|_{G_{v}}\leq|\hat{c}(h,\tilde{x})|_{G}$ for every
$h\in H_{v}$ and $x\in Y_{G_{v}}$ (indeed, one writes
$\theta^{\prime}(hx)c^{\prime\prime}(h,x)\theta^{\prime}(x)^{-1}=\hat{c}(h,\tilde{x})$,
and observes that the subword $c^{\prime\prime}(h,x)$ on the left cannot be
shortened by cancellations with $\theta^{\prime}(x)$ and $\theta^{\prime}(hx)$
in view of the choice of $\theta^{\prime}$).

Therefore the cocycle $c^{\prime\prime}:H_{v}\times Y_{G_{v}}\to G_{v}$ is
$L^{1}$-integrable. It is $G_{v}$-cohomologous to $c^{\prime}$, and therefore
it is a measure equivalence cocycle from $H_{v}$ to $G_{v}$, for the coupling
$\Omega_{v}$. This completes our proof. ∎

###  7.2 Integrable embeddings

Let $G,H$ be countable groups, with $G$ finitely generated. Let $(X,\mu)$ be a
standard probability space equipped with a measure-preserving action of $H$ by
Borel automorphisms. We say that a cocycle $c:H\times X\to G$ is _$L^{1}$
-integrable_ if for every $h\in H$, one has

$\int_{X}|c(h,x)|_{G}\;d\mu(x)<+\infty.$

In the sequel of this section, we will work with the notion of integrable
embeddings, as defined by Bowen in the appendix of [Aus16b].

######  Definition 7.4 (Integrable embedding (Bowen [Aus16b, Definition
B.8])).

Let $G,H$ be countable groups, with $G$ finitely generated. Let $(X,\mu)$ be a
standard probability space equipped with a measure-preserving action of $H$ by
Borel automorphisms.

A cocycle $c:H\times X\to G$ is an _$L^{1}$ -integrable embedding_ if for
every $\varepsilon>0$, there exist an $L^{1}$-integrable cocycle
$c^{\prime}:H\times X\to G$ which is cohomologous to $c$, a Borel subset
$X_{0}\subseteq X$ with $\mu(X_{0})\geq 1-\varepsilon$, and a constant
$C=C(\varepsilon)$, such that for every $h\in H$ and almost every $x\in
X_{0}$, one has $|\\{g\in G\mid gx\in X_{0}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }c^{\prime}(g,x)=h\\}|<C.$

The following elementary lemma, analogous to Bowen’s [Aus16b, Theorem B.9],
enables to check the second property from the above definition for measure
equivalence cocycles.

######  Lemma 7.5.

Let $\Omega$ be a measure equivalence coupling between two countable groups
$G$ and $H$, let $X_{G}\subseteq\Omega$ be a fundamental domain for the
$H$-action, and let $c:H\times X_{G}\to G$ be the associated measure
equivalence cocycle. Let $\varepsilon>0$.

Then there exist $C>0$ and a measurable subset $X_{\varepsilon}$ of $X_{G}$
with $\mu(X_{\varepsilon})\geq\mu(X_{G})-\varepsilon$ such that for every
$x\in X_{\varepsilon}$ and every $g\in G$, one has

$|\\{h\in H\mid h\cdot x\in X_{\varepsilon}\text{\leavevmode\nobreak\
and\leavevmode\nobreak\ }c(h,x)=g\\}|<C.$

###### Proof.

Let $X_{H}$ be a fundamental domain for the $H$-action on $\Omega$. Then
$\Omega$ is covered by countably many pairwise disjoint $H$-translates of
$X_{H}$. For every $h\in H$, let $X_{G,h}=X_{G}\cap hX_{H}$. Then the subsets
$X_{G,h}$ form a countable partition of $X_{G}$. Choose a finite subset
$F\subseteq H$ such that $\mu(\cup_{h\in F}X_{G,h})>\mu(X_{G})-\varepsilon$.
Let $C=|F|$, and let $X_{\varepsilon}=\cup_{g\in F}X_{G,h}$.

Let now $g\in G$ and $x\in X_{\varepsilon}$. We claim that if two distinct
elements $h_{1},h_{2}\in H$ satisfy $c(h_{1},x)=c(h_{2},x)=g$, then $h_{1}x$
and $h_{2}x$ belong to different subsets of the partition $g^{-1}X_{G}=\sqcup
g^{-1}X_{G,h}$. Indeed, notice first that as $c(h_{1},x)=c(h_{2},x)=g$, we
have $h_{1}x,h_{2}x\in g^{-1}X_{G}$. Arguing by contradiction that the claim
fails, there exists $h\in H$ such that $h_{1}x,h_{2}x\in g^{-1}(X_{G}\cap
hX_{H})$. In particular $gh_{1}x,gh_{2}x\in hX_{H}$. As the actions of $G$ and
$H$ on $\Omega$ commute, it follows that $h_{1}$ and $h_{2}$ both send $gx$ to
a common fundamental domain (namely $hX_{H}$) of the $H$-action on $\Omega$.
This is a contradiction, which proves the claim.

When considering the $H$-action on $X_{G}$, the above claim translates as
follows: if two distinct elements $h_{1},h_{2}\in H$ are such that
$c(h_{1},x)=c(h_{2},x)=g$, then $h_{1}\cdot x$ and $h_{2}\cdot x$ belong to
different subsets of the partition $X_{G}=\sqcup_{h\in H}X_{G,h}$. Since only
$C$ of these subsets are contained in $X_{\varepsilon}$, the lemma follows. ∎

An immediate corollary of Lemma 7.5 is the following, see also [Aus16b,
Theorem B.9].

######  Corollary 7.6.

Let $G$ and $H$ be countable groups, with $G$ finitely generated. If there is
an $(L^{1},L^{0})$-measure equivalence coupling from $H$ to $G$, then there is
an $L^{1}$-integrable embedding from $H$ to $G$. ∎

###  7.3 Stabilizers of rank $1$ vertices are virtually cyclic

In this section, we prove Proposition 7.1. We already know that there exists
an $(L^{1},L^{0})$-measure equivalence coupling from $H_{v}$ to $G_{v}$ (Lemma
7.3). If we additionally knew that $H_{v}$ were finitely generated, then we
could use a theorem of Bowen [Aus16b, Theorem B.10] to deduce that the growth
of $H_{v}$ is at most equal to the growth of $G_{v}$, whence at most linear.
Being infinite, $H_{v}$ is therefore virtually isomorphic to $\mathbb{Z}$.
However, we do not know _a priori_ that $H_{v}$ is finitely generated. In this
section, we extend Bowen’s theorem in order to deal with this issue, see
Theorem 7.10.

Given a finitely generated group $G$, with a finite generating set $S$, we let
$B_{G,S}(e,n)=\\{g\in G\mid|g|_{S}\leq n\\}$. The _growth function_
$\mathrm{gr}_{G,S}:\mathbb{N}\to\mathbb{N}$ is defined as

$\mathrm{gr}_{G,S}(n)=|B_{G,S}(e,n)|.$

Given two functions $f,g:\mathbb{N}\to\mathbb{N}$, write $f\preceq g$ if there
exists $A$ such that $f(n)\leq Ag(An+A)+A$. Write $f\sim g$ if $f\preceq g$
and $g\preceq f$: this is an equivalence relation on maps
$\mathbb{N}\to\mathbb{N}$. The growth functions associated to two finite
generating sets of the same group are always equivalent; we denote by
$\mathrm{gr}_{G}$ their equivalence class. When $S$ is clear from context, we
will also simply write $B_{G}(e,n)$ instead of $B_{G,S}(e,n)$.

######  Lemma 7.7.

Let $L$ be a countable group, let $G$ be a finitely generated group, and
assume that there exists an $(L^{1},L^{0})$-measure equivalence coupling from
$L$ to $G$.

Then for every finitely generated subgroup $L^{\prime}$ of $L$, one has
$\mathrm{gr}_{L^{\prime}}\preceq\mathrm{gr}_{G}$.

###### Proof.

Lemma 7.5 ensures that any $L^{1}$-integrable measure equivalence cocycle
$L\times X\to G$ is an $L^{1}$-integrable embedding. In particular, the same
holds true for the restriction $L^{\prime}\times X\to G$. The corollary is
therefore a consequence of Bowen’s theorem regarding the behavior of growth
with respect to $L^{1}$-integrable embeddings [Aus16b, Theorem B.10]. ∎

Given a probability measure-preserving action $G\curvearrowright(X,\mu)$, a
Borel subset $Z\subset X$ and $x\in X$, the _return time set_ of $x$ with
respect to $Z$ is $R_{Z}(x)=\\{g\in G\mid gx\in Z\\}$.

######  Lemma 7.8.

Let $L$ and $J$ be finitely generated groups, with $L$ infinite, equipped with
word metrics $|\cdot|_{L}$ and $|\cdot|_{J}$ (associated to finite generating
sets). Let $L\curvearrowright(X,\mu)$ be a measure-preserving action on a
standard probability space, and $c:L\times X\to J$ an $L^{1}$-integrable
cocycle. Let $M>0$. Assume that there exists $M^{\prime}>0$ such that for
every $g\in L$ satisfying $|g|_{L}\geq M^{\prime}$, one has
$\displaystyle\int_{X}|c(g,x)|_{J}\;d\mu(x)\leq M|g|_{L}$. Let $X_{0}\subseteq
X$ be a measurable subset with $\mu(X_{0})\geq 0.9$.

Then there exists $n_{0}>0$ such that for every $n\geq n_{0}$, there exists a
measurable subset $Y_{n}\subset X_{0}$ with $\mu(Y_{n})\geq 0.3$, such that
for every $x\in Y_{n}$, one has

$\frac{|\\{g\in B_{L}(e,n):g\in R_{X_{0}}(x)\ \mathrm{and}\ |c(g,x)|_{J}\leq
120M|g|_{L}\\}|}{|B_{L}(e,n)|}\geq 0.1.$

###### Proof.

We follow Bowen’s proof of [Aus16b, Theorem B.10] very closely. For $g\in L$,
we write $\kappa(g)=\displaystyle\int_{X}|c(g,x)|_{J}\;d\mu(x)$. As $L$ is
infinite and there are only finitely many elements in $L$ such that
$\kappa(g)>M|g|_{L}$, there exists $n_{0}$ such that for every $n\geq n_{0}$,
one has

$\frac{1}{|B_{L}(e,n)|}\sum_{g\in B_{L}(e,n)}\frac{\kappa(g)}{|g|_{L}}\leq
2M.$

We now fix $n\geq n_{0}$, and find $Y_{n}$ as in the lemma. As

$\int_{X}\left(\frac{1}{|B_{L}(e,n)|}\sum_{g\in
B_{L}(e,n)}\frac{|c(g,x)|_{J}}{|g|_{L}}\right)d\mu(x)\leq 2M,$

we deduce from the Markov inequality that there exists a measurable subset
$X_{1}\subset X$ with $\mu(X_{1})\geq 0.9$ such that

$\frac{1}{|B_{L}(e,n)|}\sum_{g\in B_{L}(e,n)}\frac{|c(g,x)|_{J}}{|g|_{L}}\leq
20M$

for any $x\in X_{1}$. For every $x\in X_{1}$, let

$L_{x}=\\{g\in B_{L}(e,n)\mid|c(g,x)|_{J}\leq 120M|g|_{L}\\},$

then $|L_{x}|\geq(5/6)|B_{L}(e,n)|$. Let $X_{2}=X_{0}\cap X_{1}$. Then
$\mu(X_{2})\geq 0.8$. By [Aus16b, Lemma B.11],

$\int_{X_{2}}\frac{|R_{X_{2}}(x)\cap B_{L}(e,n)|}{|B_{L}(e,n)|}d\mu(x)\geq
2\mu(X_{2})-\mu(X)\geq 0.6.$

Thus

$\displaystyle\int_{X_{2}}\frac{|R_{X_{0}}(x)\cap
L_{x}|}{|B_{L}(e,n)|}d\mu(x)\geq\int_{X_{2}}\frac{|R_{X_{2}}(x)\cap
L_{x}|}{|B_{L}(e,n)|}d\mu(x)$
$\displaystyle\geq\int_{X_{2}}\frac{|R_{X_{2}}(x)\cap
B_{L}(e,n)|+|L_{x}|-|B_{L}(e,n)|}{|B_{L}(e,n)|}d\mu(x)\geq
0.6+\mu(X_{2})(5/6-1)\geq 0.4.$

Then the measurable subset

$X_{3}:=\left\\{x\in X_{2}\Bigm{|}\frac{|R_{X_{0}}(x)\cap
L_{x}|}{|B_{L}(e,n)|}\geq 0.1\right\\}$

satisfies $\mu(X_{3})\geq 0.3$, otherwise

$\displaystyle\int_{X_{2}}\frac{|R_{X_{0}}(x)\cap
L_{x}|}{|B_{L}(e,n)|}d\mu(x)=\int_{X_{3}}\frac{|R_{X_{0}}(x)\cap
L_{x}|}{|B_{L}(e,n)|}d\mu(x)+\int_{X_{2}\setminus
X_{3}}\frac{|R_{X_{0}}(x)\cap L_{x}|}{|B_{L}(e,n)|}d\mu(x)$
$\displaystyle\leq\mu(X_{3})+0.1\mu(X_{2}\setminus
X_{3})\leq\mu(X_{3})+0.1\mu(X_{2})<0.4,$

which is a contradiction. Now the lemma follows with $Y_{n}=X_{3}$. ∎

Recall that a group $L$ is _locally virtually cyclic_ if every finitely
generated subgroup of $L$ is virtually cyclic (possibly finite).

######  Lemma 7.9.

Let $L$ be a locally virtually cyclic countable group with bounded torsion.
Then either $L$ is virtually cyclic, or else there exists an infinite cyclic
subgroup $L_{1}\subseteq L$ such that for every $k\in\mathbb{N}$, there exists
an infinite cyclic subgroup $L_{k}\subseteq L$ that contains $L_{1}$ as a
subgroup of index at least $k$.

###### Proof.

Suppose first that $L$ is locally finite, i.e. every finitely generated
subgroup is finite. Then $L$ must be finite because of the bounded torsion
assumption.

Now assume $L$ is not locally finite. The conclusion is obvious if $L$ is
finitely generated, so we assume otherwise. The group $L$ has an infinite
finitely generated subgroup, which is virtually cyclic by our assumption. Thus
there exists an infinite cyclic subgroup $L^{\prime}_{1}\subseteq L$. By
adding extra elements successively, we can find an infinite increasing
sequence of finitely generated, whence virtually cyclic, subgroups
$L^{\prime}_{1}\subsetneq L^{\prime}_{2}\subsetneq\dots$ In particular, as $k$
goes to $+\infty$, the index $[L^{\prime}_{k}:L^{\prime}_{1}]$ goes to
$+\infty$.

Let $K$ be a bound on the cardinality of a finite subgroup of $L$. Observe
that every infinite virtually cyclic group with no finite subgroup of
cardinality larger than $K$, contains an infinite cyclic subgroup of index at
most $2K$: this follows, for instance, from the fact that every infinite
virtually cyclic group is either finite-by-$\mathbb{Z}$ or finite-by-dihedral.

For every $k\in\mathbb{N}$, let $L^{\prime\prime}_{k}\subseteq L^{\prime}_{k}$
be an infinite cyclic subgroup of order at most $2K$. Letting
$L_{1}(k)=L^{\prime}_{1}\cap L^{\prime\prime}_{k}$, there is a bound on the
index $[L^{\prime}_{1}:L_{1}(k)]$. Therefore, up to a subsequence, we can
assume that all groups $L_{1}(k)$ are equal to the same group $L_{1}$. As
$L_{1}\subseteq L^{\prime\prime}_{k}$ and the index
$[L^{\prime\prime}_{k}:L_{1}]$ goes to $+\infty$, the lemma follows (by
letting $L_{k}=L^{\prime\prime}_{\sigma(k)}$ for some increasing map
$\sigma:\mathbb{N}\to\mathbb{N}$ chosen so that
$[L^{\prime\prime}_{\sigma(k)}:L_{1}]\geq k$). ∎

######  Theorem 7.10.

Let $L$ be a countable group with bounded torsion, and assume that there
exists an $L^{1}$-integrable embedding from $L$ to $\mathbb{Z}$.

Then $L$ is virtually cyclic (possibly finite).

###### Proof.

We write $J=\mathbb{Z}$. We choose once and for all a generator for $J$, and
consider the associated word length on $J$ for the rest of the argument. For
$C>0$, we say that a map between two sets $f:E_{1}\to E_{2}$ is _at most
$C$-to-1_ if the cardinality of each point inverse of $f$ is at most $C$.

Without loss of generality, we can assume that $L$ is infinite. By Lemma 7.7,
every finitely generated subgroup of $L$ has growth at most linear, and
therefore is virtually cyclic (possibly finite), see e.g. [Jus71]. Assuming
towards a contradiction that $L$ is not virtually cyclic, let $L_{1}\subseteq
L$ be an infinite cyclic subgroup of $L$ given by Lemma 7.9. We will prove
that there exists $K>0$ such that for every other infinite cyclic subgroup
$L_{2}$ containing $L_{1}$, one has $[L_{2}:L_{1}]\leq K$. This will
contradict Lemma 7.9 and conclude our proof.

By assumption, there exists a measure-preserving action of $L$ on a standard
probability space $(X,\mu)$, a cocycle $c:L\times X\to J$ which is
$L^{1}$-integrable, a Borel subset $X_{0}\subseteq X$ with $\mu(X_{0})\geq
0.9$, and a constant $C>0$, such that for every $x\in X_{0}$, the restriction
$c(\cdot,x)_{|R_{X_{0}}(x)}$ is at most $C$-to-$1$.

Let $L_{2}$ be an infinite cyclic subgroup of $L$ that contains $L_{1}$, and
let $k=[L_{2}:L_{1}]$. For every $i\in\\{1,2\\}$, we choose a generator
$g_{i}$ of $L_{i}$, and we endow $L_{i}$ with the word metric with respect to
this choice of generator. For $g\in L_{2}$, let

$\kappa(g)=\int_{X}|c(g,x)|_{J}\;d\mu(x).$

Let $M_{1}=\kappa(g_{1})$ and $D=\max\\{\kappa(g_{2}^{i})\\}_{0\leq i\leq
k-1}$. Let $g\in L_{2}$, and write it as $g=\ell g_{1}+ig_{2}$ with
$\ell\in\mathbb{Z}$ and $0\leq i\leq k-1$. Using the cocycle relation, we see
that $\kappa(g)\leq|\ell|M_{1}+D$. On the other hand $|g|_{L_{2}}=k|\ell|\pm
i$. Therefore, there exists $M^{\prime}\in\mathbb{N}$ such that
$\kappa(g)\leq\frac{2M_{1}}{k}|g|_{L_{2}}$ whenever $|g|_{L_{2}}\geq
M^{\prime}$.

Recall that $\mu(X_{0})\geq 0.9$. We can therefore apply Lemma 7.8 with
$M=\frac{2M_{1}}{k}$, and with $L_{2}$ in place of $L$, and deduce that for
$n$ large enough, there exists a Borel subset $Y_{n}\subseteq X_{0}$ with
$\mu(Y_{n})\geq 0.3$ such that for every $x\in Y_{n}$, one has

$\frac{|\\{g\in B_{L_{2}}(e,n):g\in R_{X_{0}}(x)\ \mathrm{and}\
|c(g,x)|_{J}\leq 240\frac{M_{1}}{k}|g|_{L_{2}}\\}|}{|B_{L_{2}}(e,n)|}\geq
0.1.$

Let $B_{L_{2},x}^{\mathrm{good}}(e,n)$ be the set defined as in the numerator
of the above expression. By definition of $X_{0}$, for $x\in Y_{n}$, the map
$c(\cdot,x)$ restricted to $B_{L_{2},x}^{\mathrm{good}}(e,n)$ is at most
$C$-to-1. On the other hand,

$c(B_{L_{2},x}^{\mathrm{good}}(e,n)\times\\{x\\})\subseteq
B_{J}\left(e,\frac{240M_{1}n}{k}\right),$

thus

$|B_{L_{2},x}^{\mathrm{good}}(e,n)|\leq\frac{480CM_{1}n}{k}+C.$

For $n$ large enough, it follows that

$\frac{481CM_{1}n/k}{|B_{L_{2}}(e,n)|}\geq 0.1.$

As $|B_{L_{2}}(e,n)|=2n+1$, we deduce that $\frac{481CM_{1}n}{2kn}\geq 0.1$.
Thus $k\leq 2405CM_{1}$. ∎

######  Remark 7.11.

Theorem 7.10 does not hold if $L$ is not assumed to have bounded torsion. The
simplest example is to take $L=\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$.
Then $L$ and $\mathbb{Z}$ have orbit equivalent actions on
$X=\\{0,1\\}^{\mathbb{N}}$ (equipped with the product measure of the uniform
measure on $\\{0,1\\}$). Here $\mathbb{Z}$ acts as the odometer, and $L$ acts
by coordinatewise addition; two elements are in the same orbit if and only if
they have the same tail. Letting $c:L\times X\to\mathbb{Z}$ be the orbit
equivalence cocycle, it is easy to see that $|c(s,\cdot)|$ is bounded for
every $s\in L$.

###### Proof of Proposition 7.1.

Let $v\in V\mathbb{B}$ be a rank $1$ vertex. By Lemma 7.3, there is an
$(L^{1},L^{0})$-measure equivalence coupling from $H_{v}$ to $G_{v}$. By
Corollary 7.6, this yields in particular an $L^{1}$-integrable embedding from
$H_{v}$ to $G_{v}$. Since $G_{v}$ is isomorphic to $\mathbb{Z}$, Theorem 7.10
implies that $H_{v}$ is virtually cyclic. And $H_{v}$ is infinite because it
is measure equivalent to the infinite group $G_{v}$, so $H_{v}$ is virtually
isomorphic to $\mathbb{Z}$. ∎

##  8 Controlling the factor actions

Throughout the section, we let $G$ be a right-angled Artin group with
$|\operatorname{Out}(G)|<\infty$, and $H$ be a countable group with bounded
torsion, such that there exists an $(L^{1},L^{0})$-measure equivalence
coupling $\Omega$ from $H$ to $G$. Let $\hat{\Omega}$ be the measure
equivalence coupling between $H$ and $\hat{G}$ defined by letting
$\hat{\Omega}=(\hat{G}\times\Omega)/G$ as in Section 7.1. Let
$\iota:H\to\mathrm{Aut}(\Gamma^{e})$ and
$\theta:\hat{\Omega}\to\mathrm{Aut}(\Gamma^{e})$ be the maps given by Lemma
6.6 through the canonical isomorphism
$\mathrm{Aut}(\mathbb{B})\simeq\mathrm{Aut}(\Gamma^{e})$ recalled in Section
2.2. In particular $H$ acts on $\Gamma^{e}$, and therefore $H$ acts on $G$ by
flat-preserving bijections, see Section 2.3.

For $\mathsf{v}\in V\Gamma^{e}$, we let
$\hat{\Omega}_{\mathsf{v}}=\theta^{-1}(\mathrm{Stab}_{\mathrm{Aut}(\Gamma^{e})}(\mathsf{v}))$
and
$H_{\mathsf{v}}=\iota^{-1}(\mathrm{Stab}_{\mathrm{Aut}(\Gamma^{e})}(\mathsf{v}))$.
As usual we denote by $G_{\mathsf{v}}$ and $\hat{G}_{\mathsf{v}}$ the
stabilizers of $\mathsf{v}$ for the actions of $G$ and $\hat{G}$,
respectively. Since the actions of $\hat{G}$ and of $\mathrm{Aut}(\Gamma^{e})$
on $\Gamma^{e}$ have the same orbits of vertices (Lemma 6.4), it follows from
Corollary 4.4 that $\hat{\Omega}_{\mathsf{v}}$ is a measure equivalence
coupling between the stabilizers $\hat{G}_{\mathsf{v}}$ and $H_{\mathsf{v}}$.

Let $\mathcal{L}_{\mathsf{v}}$ be the set of all $\mathsf{v}$-lines in $G$.
Recall from Section 3 that the union $P_{\mathsf{v}}$ of all
$\mathsf{v}$-lines in $G$ is a left coset of form $gG_{\mathrm{st}(v)}$ for
some $v\in V\Gamma$. Recall also that the action
$H_{\mathsf{v}}\curvearrowright G$ is by flat-preserving bijections and that
$H_{\mathsf{v}}$ preserves $P_{\mathsf{v}}$ and maps $\mathsf{v}$-lines to
$\mathsf{v}$-lines. In particular, there is an induced action
$H_{\mathsf{v}}\curvearrowright\mathcal{L}_{\mathsf{v}}$. More generally the
action of $\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ on $G$ by flat-preserving
bijections preserves $P_{\mathsf{v}}$ and sends $\mathsf{v}$-line to
$\mathsf{v}$-line. We denote by
$\theta_{\mathsf{v}}:\hat{\Omega}_{\mathsf{v}}\to\mathrm{Bij}(P_{\mathsf{v}})$
and $\iota_{\mathsf{v}}:H_{\mathsf{v}}\to\mathrm{Bij}(P_{\mathsf{v}})$ the
induced maps.

The goal of this section is to understand the action
$H_{\mathsf{v}}\curvearrowright P_{\mathsf{v}}$ to the extent that we can
connect with the assumptions of Theorem 3.2.

###  8.1 Finite generation of $H_{\mathsf{v}}$

The main goal of this subsection is to prove that $H_{\mathsf{v}}$ is finitely
generated (Lemma 8.4 below).

Let $\mathbb{B}(\mathsf{v})$ be the union of all cubes in $\mathbb{B}$ whose
vertices correspond to standard flats that are contained in $P_{\mathsf{v}}$.
Then $\mathbb{B}(\mathsf{v})$ is isomorphic to the right-angled building
associated with $G_{\mathrm{st}(v)}$, in particular it is simply connected.
Moreover, $\mathbb{B}(\mathsf{v})$ is $H_{\mathsf{v}}$-invariant.

######  Lemma 8.1.

The action of $H_{\mathsf{v}}$ on $P_{\mathsf{v}}$ has finitely many orbits of
vertices. In particular,

  1. 1.

the action of $H_{\mathsf{v}}$ on $\mathcal{L}_{\mathsf{v}}$ has finitely many
orbits;

  2. 2.

for every $\mathsf{v}$-line $\ell\in\mathcal{L}_{\mathsf{v}}$, the action of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ on $\ell$ has finitely many orbits of
vertices;

  3. 3.

the action of $H_{\mathsf{v}}$ on $\mathbb{B}(\mathsf{v})$ is cocompact.

###### Proof.

The action of $G_{\mathsf{v}}$ on $P_{\mathsf{v}}$ is transitive. We can thus
apply Corollary 4.7 with $\mathsf{G}=G_{\mathsf{v}}$ and
$\mathsf{H}=H_{\mathsf{v}}$, with $K=P_{\mathsf{v}}$, and with the coupling
$\hat{\Omega}_{\mathsf{v}}$ and the maps $\iota_{\mathsf{v}}$ and
$\theta_{\mathsf{v}}$. Since vertices in $P_{\mathsf{v}}$ have trivial
$G_{\mathsf{v}}$-stabilizer, the first part of Corollary 4.7 shows that the
action of $H_{\mathsf{v}}$ on $P_{\mathsf{v}}$ has finitely many orbits.

The first two consequences follow because $H_{\mathsf{v}}$ sends
$\mathsf{v}$-lines to $\mathsf{v}$-lines. For Assertion 3, note that
$H_{\mathsf{v}}$ acts on the set of rank 0 vertices of
$\mathbb{B}(\mathsf{v})$, which is exactly $P_{\mathsf{v}}$, with finitely
many orbits. So Assertion 3 follows by exactly the same argument as in
Corollary 6.9. ∎

######  Lemma 8.2.

The $H$-stabilizer of any vertex in $\mathbb{B}$ is finitely generated.

###### Proof.

Recall that the $H$-action on $\mathbb{B}$ can be equivalently viewed as an
$H$-action on $G$ by flat-preserving bijections. Using this viewpoint, for
every vertex $v\in V\mathbb{B}$ with associated standard flat $F$, the
stabilizer $H_{v}$ coincides with the stabilizer $H_{F}$ of $F$ for this
$H$-action on $G$. We will use this viewpoint throughout the proof.

We first claim that for any standard flat $F^{\prime}\subset F$, the group
$H_{F^{\prime}}$ has a finite index subgroup $H^{0}_{F^{\prime}}$ which is
contained in $H_{F}$, and acts on $F^{\prime}$ with finitely many orbits.
Indeed, let $v_{F^{\prime}}$ be the vertex of $\mathbb{B}$ associated to
$F^{\prime}$. Lemma 6.8 ensures that $H_{F^{\prime}}$ acts on the set of rank
0 vertices of $\mathbb{B}$ that are smaller than $v_{F^{\prime}}$ with
finitely many orbits. Therefore $H_{F^{\prime}}$ acts on $F^{\prime}$ with
finitely many orbits. Any element in $H_{F^{\prime}}$ sends $F$ to another
standard flat containing $F^{\prime}$. As there are only finitely many
standard flats containing $F^{\prime}$, we can find a finite index subgroup
$H^{0}_{F^{\prime}}\subseteq H_{F^{\prime}}$ that stabilizes $F$. The action
$H^{0}_{F^{\prime}}\curvearrowright F^{\prime}$ still has finitely many
orbits, thus the claim is proved.

We now prove the lemma by induction on the rank of vertices of $\mathbb{B}$.
Stabilizers in $H$ of rank $0$ vertices are finite (Lemma 6.8), and the case
of rank 1 vertices is given by Proposition 7.1. Let now $v\in V\mathbb{B}$ be
a vertex of rank $n\geq 2$, corresponding to a standard flat $F$, and assume
by induction that the lemma is proven for all vertices of rank at most $n-1$.
Take a standard line $\ell\subset F$. By the above claim, there is a finite
index subgroup $H^{0}_{\ell}\subseteq H_{\ell}$ that stabilizes both $F$ and
$\ell$, and whose action on $\ell$ has finitely many orbits. Let $\mathcal{C}$
be the set of all standard flats in $F$ of dimension $\dim F-1$ that intersect
$\ell$ in exactly one point. As the $H$-action on $F$ is flat-preserving, the
group $H_{\ell}^{0}$ permutes elements in $\mathcal{C}$, and the permutation
action $H_{\ell}^{0}\curvearrowright\mathcal{C}$ has finitely many orbits. Let
$\\{F_{1},\dots,F_{k}\\}$ be a set consisting of exactly one representative in
each orbit of this permutation action. By the above claim, for each
$i\in\\{1,\dots,k\\}$, there is a finite index subgroup
$H^{0}_{F_{i}}\subseteq H_{F_{i}}$ that stabilizes both $F$ and $F_{i}$, and
acts on $F_{i}$ with finitely many orbits. Let $K_{i}$ be a finite subset of
$F_{i}$ such that $H^{0}_{F_{i}}K_{i}=F_{i}$. Let $K=\cup_{i=1}^{k}K_{i}$ and
let $H^{\prime}$ be the subgroup of $H_{F}$ generated by $H^{0}_{\ell}$ and
$\\{H^{0}_{F_{i}}\\}_{i=1}^{k}$. As $H^{0}_{\ell}$ and $H^{0}_{F_{i}}$ are
finitely generated by induction, it follows that $H^{\prime}$ is finitely
generated. Moreover, our construction implies that $H^{\prime}K=F$. Thus for
any $h\in H_{F}$, there exists $h^{\prime}\in H^{\prime}$ such that
$h^{\prime}hK\cap K\neq\emptyset$. On the other hand, as the action
$H_{F}\curvearrowright F$ has finite stabilizers (Lemma 6.8) and $K$ is
finite, there are only finitely many elements $h\in H_{F}$ such that $hK\cap
K\neq\emptyset$. Thus $H^{\prime}$ has finite index in $H_{F}$, and hence
$H_{F}$ is finitely generated. ∎

######  Corollary 8.3.

The $H$-stabilizer of any cube in $\mathbb{B}$ is finitely generated.

###### Proof.

Let $C$ be a cube of $\mathbb{B}$, and let $v$ be the (unique) vertex of
$\mathbb{B}$ of minimal rank. Then $v$ is the vertex of minimal rank in only
finitely many cubes of $\mathbb{B}$. As the $H$-action on $\mathbb{B}$
preserves ranks of vertices (Lemma 2.5), it follows that
$\mathrm{Stab}_{H}(C)$ is a finite-index subgroup of $\mathrm{Stab}_{H}(v)$.
The corollary thus follows from Lemma 8.2. ∎

Recall that the $H$-action on $\mathbb{B}$ induces an $H$-action on the
extension graph of $G$. We will also need the following fact.

######  Lemma 8.4.

For every $\mathsf{v}\in V\Gamma^{e}$, the stabilizer $H_{\mathsf{v}}$ is
finitely generated.

###### Proof.

First we claim that $H_{\mathsf{v}}$ is of finite index in
$\mathrm{Stab}_{H}(P_{\mathsf{v}})$. Indeed, recall that
$P_{\mathsf{v}}=gG_{\mathrm{st}(v)}$ for some $v\in V\Gamma$. Let
$\mathrm{st}(v)=\\{v\\}\circ\Gamma_{1}\circ\cdots\circ\Gamma_{k}$ be the join
decomposition of $\mathrm{st}(v)$. This gives
$P_{\mathsf{v}}=gG_{\mathrm{st}(v)}\cong g\langle v\rangle\times
gG_{\Gamma_{1}}\times\cdots\times gG_{\Gamma_{k}}$. As
$\mathrm{Stab}_{H}(P_{\mathsf{v}})$ sends standard flats to standard flats in
$P_{\mathsf{v}}$, its action on $P_{\mathsf{v}}$ respects this product
decomposition (it can possibly permute the factors). Then
$\mathrm{Stab}_{H}(P_{\mathsf{v}})$ has a finite index subgroup sending
$\mathsf{v}$-lines to $\mathsf{v}$-lines, which is $H_{\mathsf{v}}$.

If $F\subseteq P_{\mathsf{v}}$ is a standard flat (corresponding to a vertex
of $\mathbb{B}(\mathsf{v})$), then $F$ is contained in only finitely many
regions of the form $P_{\mathsf{w}}$ with $\mathsf{w}\in V\Gamma^{e}$. Then
$\mathrm{Stab}_{H}(F)$ has a finite index subgroup preserving
$P_{\mathsf{v}}$. By the claim in the previous paragraph, it follows that
$\mathrm{Stab}_{H_{\mathsf{v}}}(F)$ is a finite-index subgroup of
$\mathrm{Stab}_{H}(F)$, and is thus finitely generated by Lemma 8.2. More
generally, by the same argument (and using Corollary 8.3), for any cube $C$ in
$\mathbb{B}(\mathsf{v})$, the $H_{\mathsf{v}}$-stabilizer of $C$ has finite
index in its $H$-stabilizer, hence is finitely generated.

Now $H_{\mathsf{v}}$ acts by cubical automorphisms on the simply connected
complex $\mathbb{B}(\mathsf{v})$ cocompactly (Lemma 8.1(3)), with finitely
generated cell stabilizers, so it is finitely generated by [Bro84, Theorem 1].
∎

###  8.2 Commensurability of stabilizers

Let $Z_{\mathsf{v}}=g\langle v\rangle$. The splitting
$P_{\mathsf{v}}=gG_{\mathrm{st}(v)}=g(\langle v\rangle\times
G_{\mathrm{lk}(v)})$ gives a projection map $\pi_{1}:P_{\mathsf{v}}\to
Z_{\mathsf{v}}$. On the other hand, as $\mathcal{L}_{\mathsf{v}}$ is the
collection of $\mathsf{v}$-lines, there is a map
$\pi_{2}:P_{\mathsf{v}}\to\mathcal{L}_{\mathsf{v}}$ sending each point to the
$\mathsf{v}$-line containing this point. Moreover, $\mathcal{L}_{\mathsf{v}}$
can be naturally identified with $gG_{\mathrm{lk}(v)}$. Note that the map
$(\pi_{1},\pi_{2})$ gives an identification between $P_{\mathsf{v}}$ and
$Z_{\mathsf{v}}\times\mathcal{L}_{\mathsf{v}}$. Moreover, the action of
$G_{\mathsf{v}}$ on $P_{\mathsf{v}}$ respects this product decomposition,
hence gives two factor actions:

  1. 1.

$G_{\mathsf{v}}\curvearrowright Z_{\mathsf{v}}$, for which the stabilizer of
each element is $gG_{\mathrm{lk}(v)}g^{-1}$;

  2. 2.

$G_{\mathsf{v}}\curvearrowright\mathcal{L}_{\mathsf{v}}$, for which the
stabilizer of each element is $g\langle v\rangle g^{-1}$.

We let $Z_{1}=Z_{\mathsf{v}}$ and $Z_{2}=\mathcal{L}_{\mathsf{v}}$. We denote
by $\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ the stabilizer of $\mathsf{v}$ in
$\mathrm{Aut}(\mathbb{B})$ (under the canonical isomorphism between
$\mathrm{Aut}(\mathbb{B})$ and $\mathrm{Aut}(\Gamma^{e})$ recalled in Section
2.3). Note that

  1. 1.

$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ preserves $P_{\mathsf{v}}$;

  2. 2.

$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ sends standard flats in
$P_{\mathsf{v}}$ to standard flats in $P_{\mathsf{v}}$;

  3. 3.

$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ sends $\mathsf{v}$-lines to
$\mathsf{v}$-lines.

Thus $\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ preserves the splitting
$P_{\mathsf{v}}=Z_{1}\times Z_{2}$. For every $i\in\\{1,2\\}$, we have factor
actions of
$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})=\mathrm{Stab}_{\mathrm{Aut}(\Gamma^{e})}(\mathsf{v})$,
$G_{\mathsf{v}}$ and $H_{\mathsf{v}}$ on $Z_{i}$. Given $x,y\in Z_{i}$, we
denote by $\mathrm{Aut}_{\mathsf{v},x}(\mathbb{B})$, $G_{\mathsf{v},x}$ and
$H_{\mathsf{v},x}$ the stabilizers of $x$ in
$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$, $G_{\mathsf{v}}$ and $H_{\mathsf{v}}$
respectively, and by $G_{\mathsf{v},x,y}$ and $H_{\mathsf{v},x,y}$ the common
stabilizers of $x$ and $y$.

######  Proposition 8.5.

For every $i\in\\{1,2\\}$, and any $x,y\in Z_{i}$, the groups
$H_{\mathsf{v},x}$ and $H_{\mathsf{v},y}$ are commensurable in
$H_{\mathsf{v}}$.

###### Proof.

By symmetry, it is enough to prove that $H_{\mathsf{v},x,y}$ has finite index
in $H_{\mathsf{v},x}$. Our proof will use two facts regarding the action of
$G_{\mathsf{v}}$ on $Z_{i}$ (proved below), namely:

  * •

(Fact 1) The actions of $G_{\mathsf{v}}$ and of
$\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ have the same orbits on $Z_{i}$ (they
are both transitive).

  * •

(Fact 2) For any $x,z\in Z_{i}$, we have $G_{\mathsf{v},x}=G_{\mathsf{v},z}$.

For Fact 1, note that the action $G_{\mathsf{v}}\curvearrowright
P_{\mathsf{v}}$ is transitive and respects the splitting $P_{\mathsf{v}}\cong
Z_{\mathsf{v}}\times\mathcal{L}_{\mathsf{v}}$, so the action
$G_{\mathsf{v}}\curvearrowright Z_{i}$ is transitive for $i=1,2$. Hence the
same holds for the bigger group $\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$. Fact
2 follows from the discussion of stabilizers of factor actions before the
proposition.

As before, let $\hat{\Omega}$ be a measure equivalence coupling between
$\hat{G}$ and $H$, and let
$\theta:\hat{\Omega}\to\mathrm{Aut}(\mathbb{B})\cong\mathrm{Aut}(\Gamma^{e})$
be a measurable $(\hat{G}\times H)$-equivariant map. As recalled in the
introductory paragraph of Section 8, the space
$\hat{\Omega}_{\mathsf{v}}=\theta^{-1}(\mathrm{Aut}_{\mathsf{v}}(\mathbb{B}))$
is a measure equivalence coupling between $\hat{G}_{\mathsf{v}}$ and
$H_{\mathsf{v}}$. Hence $\hat{\Omega}_{\mathsf{v}}$ is also a measure
equivalence coupling between $G_{\mathsf{v}}$ and $H_{\mathsf{v}}$.

Let
$\hat{\Omega}_{\mathsf{v},x}=\theta^{-1}(\mathrm{Aut}_{\mathsf{v},x}(\mathbb{B}))$.
Since the orbits of $x$ under $\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$ and
$G_{\mathsf{v}}$ coincide (Fact 1 above), it follows from Corollary 4.4
(applied with $K=Z_{i}$, with $L=\mathrm{Aut}_{\mathsf{v}}(\mathbb{B})$, with
$\mathsf{G}=G_{\mathsf{v}}$, and with $\hat{\Omega}_{\mathsf{v}}$ in place of
$\Omega$) that $\hat{\Omega}_{\mathsf{v},x}$ is a measure equivalence coupling
between $G_{\mathsf{v},x}$ and $H_{\mathsf{v},x}$.

We want to apply Lemma 4.9 with
$\mathsf{G}=G_{\mathsf{v},x},\mathsf{H}=H_{\mathsf{v},x},\Sigma=\hat{\Omega}_{\mathsf{v},x}$
and $\mathsf{H}^{\prime}=H_{\mathsf{v},x,y}$. It remains to find
$\Sigma^{\prime}$ with desired properties. For the following discussion, we
refer to the statement of Lemma 6.6 for our convention of how $\mathsf{G}$ and
$\mathsf{H}$ act on $\mathrm{Aut}_{\mathsf{v},x}(\mathbb{B})$.

Given $z\in Z_{i}$, we let $\mathrm{Aut}_{\mathsf{v},x,y\to z}(\mathbb{B})$ be
the Borel subset of $\mathrm{Aut}_{\mathsf{v},x}(\mathbb{B})$ consisting of
all automorphisms that send $y$ to $z$, and let
$\hat{\Omega}_{\mathsf{v},x,y\to
z}=\theta^{-1}(\mathrm{Aut}_{\mathsf{v},x,y\to z}(\mathbb{B}))$. Then
$\hat{\Omega}_{\mathsf{v},x}=\sqcup_{z\in
Z_{i}}\hat{\Omega}_{\mathsf{v},x,y\to z}$. Therefore, we can (and will) choose
$z\in Z_{i}$ such that $\mu(\hat{\Omega}_{\mathsf{v},x,y\to z})>0$. We take
$\Sigma^{\prime}=\hat{\Omega}_{\mathsf{v},x,y\to z}$. It is invariant under
$G_{\mathsf{v},x,z}$ and $\mathsf{H}^{\prime}=H_{\mathsf{v},x,y}$, and the
former group is equal to $\mathsf{G}=G_{\mathsf{v},x}$ by Fact 2. Now take
$h\in\mathsf{H}\setminus\mathsf{H}^{\prime}$. Then $h(x)=x$ and $h(y)\neq y$.
Then $h\Sigma^{\prime}=\hat{\Omega}_{\mathsf{v},x,h(y)\to z}$, which is
disjoint from $\hat{\Omega}_{\mathsf{v},x,y\to z}$, as desired. ∎

###  8.3 From commensurated to normal

Let $\mathsf{v}\in V\Gamma^{e}$. Proposition 8.5, applied to
$Z_{2}=\mathcal{L}_{\mathsf{v}}$, shows that the $H_{\mathsf{v}}$-stabilizers
of any two $\mathsf{v}$-lines are commensurable (and they are virtually
infinite cyclic by Proposition 7.1). In this section, we will improve this by
showing that $H_{\mathsf{v}}$ contains a normal subgroup that preserves all
$\mathsf{v}$-lines: this is Proposition 8.7 below. We start with a lemma.

######  Lemma 8.6.

Let $\mathsf{v}\in V\Gamma^{e}$. Let $\ell_{1},\ell_{2}$ be two
$\mathsf{v}$-lines in the same $H_{\mathsf{v}}$-orbit. Let $A$ be any finite-
index infinite cyclic subgroup of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1})\cap\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2})$.

Then
$[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1}):A]=[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2}):A]$.

###### Proof.

Recall that every $\mathsf{v}$-line $\ell$ determines a rank $1$ vertex
$v_{\ell}$ in $\mathbb{B}$, and vertices of $\ell$ correspond to rank $0$
vertices adjacent to $v_{\ell}$. Therefore, for every $\mathsf{v}$-line
$\ell$, the action of $\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ on $\ell$ has
finite stabilizers (Lemma 6.8) and finitely many orbits (Lemma 8.1). As
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1})$ and
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2})$ are commensurable (Proposition 8.5
applied to $Z_{2}=\mathcal{L}_{\mathsf{v}}$), and $A$ is torsion-free and has
finite index in
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1})\cap\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2})$,
it follows that the actions $A\curvearrowright\ell_{1}$ and
$A\curvearrowright\ell_{2}$ are free with finitely many orbits.

As $A\subseteq H_{\mathsf{v}}$, the $A$-action on
$P_{\mathsf{v}}=Z_{\mathsf{v}}\times\mathcal{L}_{\mathsf{v}}$ preserves the
product structure. Therefore the two actions $A\curvearrowright\ell_{1}$ and
$A\curvearrowright\ell_{2}$ have the same number of orbits. Let $h\in
H_{\mathsf{v}}$ be such that $h(\ell_{1})=\ell_{2}$. Then $h$ induces a
bijection $h_{\ast}:\mathcal{O}_{1}^{j}\mapsto\mathcal{O}_{2}^{j}$ between the
set of orbits $\\{\mathcal{O}^{1}_{1},\dots,\mathcal{O}^{k}_{1}\\}$ of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1})\curvearrowright\ell_{1}$, and the
set of orbits $\\{\mathcal{O}^{1}_{2},\dots,\mathcal{O}^{k}_{2}\\}$ of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2})\curvearrowright\ell_{2}$. In
addition $h_{\ast}$ preserves the (finite) cardinalities of stabilizers, i.e.
the $\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1})$-stabilizer of any point in
$\mathcal{O}_{1}^{j}$ has the same cardinality (denoted by $k_{j}$) as the
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2})$-stabilizer of any point in
$\mathcal{O}_{2}^{j}$. Now, for every $i\in\\{1,2\\}$, the number of
$A$-orbits on $\ell_{i}$ is equal to
$[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{i}):A]\sum_{j=1}^{k}\frac{1}{k_{j}}$.
Therefore
$[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{1}):A]=[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell_{2}):A]$.
∎

######  Proposition 8.7.

For every $\mathsf{v}\in V\Gamma^{e}$, there exists an infinite cyclic normal
subgroup ${N_{\mathsf{v}}\unlhd H_{\mathsf{v}}}$ which preserves every
$\mathsf{v}$-line.

###### Proof.

As the action of $H_{\mathsf{v}}$ on $\mathcal{L}_{\mathsf{v}}$ has finitely
many orbits (Lemma 8.1), we can (and will) choose a finite subset
$\mathcal{L}_{0}\subseteq\mathcal{L}_{\mathsf{v}}$ such that
$\mathcal{L}_{\mathsf{v}}=H_{\mathsf{v}}\mathcal{L}_{0}$. Recall from Lemma
8.4 that $H_{\mathsf{v}}$ is finitely generated. Take a finite generating set
$S$ of $H_{\mathsf{v}}$ containing the trivial element, and let
$\mathcal{L}_{1}=\cup_{s\in S}s^{-1}\mathcal{L}_{0}$.

For every $\ell\in\mathcal{L}_{1}$, recall that
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ is virtually infinite cyclic by
Proposition 7.1. Let $k_{\ell}\in\mathbb{N}$ be the smallest integer such that
the intersection of all subgroups of $\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ of
index $k_{\ell}$ is infinite cyclic, and let $Z_{\ell}$ be this intersection.
Notice that $Z_{\ell}$ is a characteristic subgroup of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$. We observe that if
$\ell\in\mathcal{L}_{0}$ and $s\in S$, then $s^{-1}\ell\in\mathcal{L}_{1}$ and
$Z_{s^{-1}\ell}=s^{-1}Z_{\ell}s$: indeed $h\mapsto s^{-1}hs$ determines an
isomorphism between $\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ and
$\mathrm{Stab}_{H_{\mathsf{v}}}(s^{-1}\ell)$, so $k_{\ell}=k_{s^{-1}\ell}$ and
the isomorphism $h\mapsto s^{-1}hs$ sends $Z_{\ell}$ to $Z_{s^{-1}\ell}$ in
view of the definition of these subgroups.

Let $N_{\mathsf{v}}=\cap_{\ell\in\mathcal{L}_{1}}Z_{\ell}$. We claim that
$sN_{\mathsf{v}}s^{-1}=N_{\mathsf{v}}$ for every $s\in S$. Indeed, let $s\in
S$, and take $\ell\in\mathcal{L}_{0}$. Then $N_{\mathsf{v}}$ has finite index
in $\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$. As both $\ell$ and $s^{-1}\ell$
belong to $\mathcal{L}_{1}$, Lemma 8.6 shows that
$[\mathrm{Stab}_{H_{\mathsf{v}}}(s^{-1}\ell):N_{\mathsf{v}}]=[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell):N_{\mathsf{v}}]$.
As $N_{\mathsf{v}}\subseteq\mathrm{Stab}_{H_{\mathsf{v}}}(s^{-1}\ell)$, we
have $sN_{\mathsf{v}}s^{-1}\subseteq\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ and
$[\mathrm{Stab}_{H_{\mathsf{v}}}(s^{-1}\ell):N_{\mathsf{v}}]=[\mathrm{Stab}_{H_{\mathsf{v}}}(\ell):sN_{\mathsf{v}}s^{-1}]$.
Thus $sN_{\mathsf{v}}s^{-1}$ and $N_{\mathsf{v}}$ are subgroups of
$\mathrm{Stab}_{H_{\mathsf{v}}}(\ell)$ of the same finite index. In addition,
$N_{\mathsf{v}}$ is contained in $Z_{\ell}$ and in $Z_{s^{-1}\ell}$, so
$sN_{\mathsf{v}}s^{-1}\subseteq Z_{\ell}$. Therefore
$[Z_{\ell}:N_{\mathsf{v}}]=[Z_{\ell}:sN_{\mathsf{v}}s^{-1}]$. As $Z_{\ell}$ is
infinite cyclic, it follows that $sN_{\mathsf{v}}s^{-1}=N_{\mathsf{v}}$ for
every $s\in S$. As $S$ generates $H_{\mathsf{v}}$, it follows that
$N_{\mathsf{v}}$ is a normal subgroup of $H_{\mathsf{v}}$. As $N_{\mathsf{v}}$
preserves each line in $\mathcal{L}_{0}$, and
$H_{\mathsf{v}}\mathcal{L}_{0}=\mathcal{L}_{\mathsf{v}}$, it follows that
$N_{\mathsf{v}}$ preserves every $\mathsf{v}$-line. ∎

###  8.4 Conjugating the factor action to one by uniform quasi-isometries

We now consider the factor action
$\alpha_{\mathsf{v}}:H_{\mathsf{v}}\to\mathrm{Bij}(Z_{\mathsf{v}})$. Let
$N_{\mathsf{v}}\unlhd H_{\mathsf{v}}$ be a normal infinite cyclic subgroup
that preserves every $\mathsf{v}$-line, given by Proposition 8.7.

As $N_{\mathsf{v}}$ is torsion-free, by Lemma 6.8, the action of
$N_{\mathsf{v}}$ on each $\mathsf{v}$-line is free, and by Lemma 8.1 it has
finitely many orbits. Let $\\{\mathcal{O}_{1},\dots,\mathcal{O}_{n}\\}$ be the
set of orbits for the action of $N_{\mathsf{v}}$ on $Z_{\mathsf{v}}$. Since
$N_{\mathsf{v}}$ is normal in $H_{\mathsf{v}}$, the group $H_{\mathsf{v}}$
acts by permutations of the set $\\{\mathcal{O}_{1},\dots,\mathcal{O}_{n}\\}$.
Let $H_{\mathsf{v}}^{0}\subseteq H_{\mathsf{v}}$ be a finite index subgroup
which preserves every orbit $\mathcal{O}_{i}$.

Let $a$ be a generator of $N_{\mathsf{v}}$. We can (and will) identify each
$\mathcal{O}_{i}$ with $\mathbb{Z}$ in such a way that $a$ acts by translation
by $1$ on each $\mathcal{O}_{i}$. Since $N_{\mathsf{v}}$ is normal in
$H_{\mathsf{v}}$, the group $H^{0}_{\mathsf{v}}$ acts on each
$\mathcal{O}_{i}$ by isometries: indeed, every element $h\in
H^{0}_{\mathsf{v}}$ either commutes with $a$, and then satisfies
$h(x+1)=h(x)+1$ for $x\in\mathcal{O}_{i}\cong\mathbb{Z}$, whence acts by
translations; or else $hah^{-1}=a^{-1}$, in which case $h(x+1)=h(x)-1$ and $h$
acts by an orientation-reversing isometry.

Let $H_{\mathsf{v}}^{1}\subseteq H_{\mathsf{v}}^{0}$ be the finite-index
subgroup consisting of all elements that act by positive isometries (i.e.
translations) on each $\mathcal{O}_{i}$. Let
$\tau_{i}:H_{\mathsf{v}}^{1}\to\mathbb{Z}$ be the translation length
homomorphism on $\mathcal{O}_{i}$.

######  Lemma 8.8.

For any $i,j\in\\{1,\dots,n\\}$ and any $h\in H_{\mathsf{v}}^{1}$, one has
$\tau_{i}(h)=\tau_{j}(h)$.

###### Proof.

Arguing towards a contradiction, let $h\in H_{\mathsf{v}}^{1}$ be such that
$\tau_{i}(h)\neq\tau_{j}(h)$. Let $x\in\mathcal{O}_{i}$ and
$y\in\mathcal{O}_{j}$. Then all powers of $ha^{-\tau_{i}(h)}$ belong to
$H_{\mathsf{v},x}\setminus H_{\mathsf{v},y}$. This contradicts the fact that
$H_{\mathsf{v},x}$ and $H_{\mathsf{v},y}$ are commensurable (Proposition 8.5
applied to $Z_{1}=Z_{\mathsf{v}}$). ∎

######  Lemma 8.9.

The action of $H_{\mathsf{v}}$ on $Z_{\mathsf{v}}$ is conjugate to an action
on $\mathbb{Z}$ by uniform quasi-isometries.

###### Proof.

Let $\sigma:H_{\mathsf{v}}\to\mathfrak{S}(\\{1,\dots,n\\})$ be the
homomorphism given by the permutation of the orbits $\mathcal{O}_{i}$. Recall
that we have fixed a bijection between every orbit $\mathcal{O}_{i}$ and
$\mathbb{Z}$, such that the action of $a$ on each $\mathcal{O}_{i}$ is by
translation by $1$.

Take $h\in H_{\mathsf{v}}$. We claim that for each $i\in\\{1,\dots,n\\}$, the
map
$h_{|\mathcal{O}_{i}}:\mathcal{O}_{i}\cong\mathbb{Z}\to\mathcal{O}_{\sigma(h)(i)}\cong\mathbb{Z}$
is an isometry. Indeed, either $hah^{-1}=a$, in which case $h(x+1)=h(x)+1$ for
any $x\in\mathcal{O}_{1}\cup\dots\cup\mathcal{O}_{n}$. Or else
$hah^{-1}=a^{-1}$, in which case $h(x+1)=h(x)-1$ for any
$x\in\mathcal{O}_{1}\cup\dots\cup\mathcal{O}_{n}$. This proves the claim.
Moreover, either $h_{|\mathcal{O}_{i}}$ is a translation for all $i$, or
$h_{|\mathcal{O}_{i}}$ is a reflection for all $i$.

We now choose a bijection between $Z_{\mathsf{v}}$ and $\mathbb{Z}$ such that,
under this bijection, each $\mathcal{O}_{i}$ is identified to the subset
$\\{mn+i\\}_{m\in\mathbb{Z}}$, and the action of $a$ is by translation by $n$.
It follows from the previous paragraph that for every $h\in H_{\mathsf{v}}$,
there exist integers $c_{1}(h),\dots,c_{n}(h)$ such that either
$h(mn+i)=(m+c_{i}(h))n+\sigma(h)(i)$ for any $m\in\mathbb{Z}$ and any
$i\in\\{1,\dots,n\\}$, or else $h(mn+i)=(c_{i}(h)-m)n+\sigma(h)(i)$ for any
$x\in\mathbb{Z}$ and any $i\in\\{1,\dots,n\\}$. Thus each $h\in
H_{\mathsf{v}}$ acts on $\mathbb{Z}$ by a quasi-isometry. We are now left with
proving that the quasi-isometry constants are in fact uniform.

Lemma 8.8 implies that the action of $H_{\mathsf{v}}^{1}$ on
$Z_{\mathsf{v}}\cong\mathbb{Z}$ is by translations (in particular by uniform
quasi-isometries). Recall that $H_{\mathsf{v}}^{1}$ has finite index in
$H_{\mathsf{v}}$. Let $F$ be a finite set of representatives of the left
cosets of $H_{\mathsf{v}}^{1}$ in $H_{\mathsf{v}}$. Any $h\in H$ can then be
decomposed as $h=fh^{\prime}$ for some $f\in F$ and some $h^{\prime}\in
H_{\mathsf{v}}^{1}$. Thus the quasi-isometry constant of $h$ is bounded by a
constant that only depends on the quasi-isometry constants of elements of $F$.
This concludes our proof. ∎

##  9 Proof of the main theorem

We are now in position to complete the proof of our main theorem.

###### Proof of Theorem 1.

The conclusion is obvious if $G=\\{1\\}$. The case where $G$ is isomorphic to
$\mathbb{Z}$ is a consequence of Theorem 7.10. From now on we assume that $G$
is not cyclic.

Let $\mathbb{B}$ be the right-angled building of $G$. Lemma 6.6 yields an
action of $H$ on $\mathbb{B}$ by cubical automorphisms (equivalently, an
action of $H$ on $G$ by flat-preserving bijections). Lemma 6.8 ensures that
this flat-preserving action has finite point stabilizers, and finitely many
orbits of vertices. Finally, Lemma 8.9 ensures that for every $\mathsf{v}\in
V\Gamma^{e}$, the factor action $H_{\mathsf{v}}\curvearrowright
Z_{\mathsf{v}}$ is conjugate to an action on $\mathbb{Z}$ by uniform quasi-
isometries. Therefore, Theorem 3.2 applies and shows that $H$ is finitely
generated and quasi-isometric to $G$. ∎

We now show the importance of the bounded torsion assumption in our theorem,
by giving two examples of infinitely generated groups $H$ with unbounded
torsion such that there exists an $(L^{\infty},L^{0})$-measure equivalence
coupling $\Omega$ from $H$ to a right-angled Artin group $G$ (possibly with
$|\operatorname{Out}(G)|<+\infty$). The $(L^{\infty},L^{0})$-condition means
that there exists a Borel fundamental domain $X_{G}$ for the $G$-action on
$\Omega$ such that, denoting by $c:H\times X_{G}\to G$ the associated cocycle,
for every $h\in H$, $c(h,\cdot)$ takes essentially only finitely many values.

######  Proposition 9.1.

Let $K$ be a locally finite countable polyhedral complex, let $G$ be a
finitely generated group acting properly discontinuously and cocompactly on
$K$, and let $H$ be a lattice in $\mathrm{Aut}(K)$.

Then there exists an $(L^{\infty},L^{0})$-measure equivalence coupling from
$H$ to $G$.

In particular, let $G$ be a non-abelian right-angled Artin group, and let $H$
be an infinitely generated (non-uniform) lattice in the automorphism group of
the universal cover of the Salvetti complex of $G$ (examples were constructed
in [HH22a, Section 4.2], and these have unbounded torsion). Then there exists
an $(L^{\infty},L^{0})$-measure equivalence coupling from $H$ to $G$.

###### Proof.

We view $\mathrm{Aut}(K)$, equipped with its Haar measure, as a measure
equivalence coupling between $G$ and $H$, where the action of $G\times H$ is
via $(g,h)\cdot f=gfh^{-1}$. As $G$ acts cocompactly on $K$, we can find a
finite set $V_{0}$ of representatives of the $G$-orbits of vertices in $K$.
Fix $v_{0}\in V_{0}$. Then $X=\\{f\in\mathrm{Aut}(K)\mid f(v_{0})\in V_{0}\\}$
is a measurable fundamental domain for the $G$-action on $\mathrm{Aut}(K)$. We
let $c:H\times X\to G$ be the associated measure equivalence cocycle. We aim
to prove that for every $h\in H$, $c(h,\cdot)$ takes only finitely many
values.

Fix $h\in H$. Let $d$ be the metric on the $1$-skeleton of $K$ obtained by
assigning length $1$ to every edge, and considering the induced path metric.
For every $f\in X$, we have

$d(v_{0},f(h^{-1}v_{0}))\leq
d(v_{0},f(v_{0}))+d(v_{0},h^{-1}v_{0})\leq\mathrm{diam}(V_{0})+d(v_{0},h^{-1}v_{0}).$

Since the $G$-action on $K$ is properly discontinuous, we can find a finite
set $B_{h}\subseteq G$ such that for every $f\in X$, one has
$f(h^{-1}v_{0})\in B_{h}^{-1}V_{0}$. In particular, there exists $g(f)\in
B_{h}$ such that $g(f)fh^{-1}\in X$. This shows that $c(h,f)\in B_{h}$ and
concludes our proof. ∎

An _$(L^{\infty},L^{0})$ -orbit equivalence coupling_ from $H$ to $G$ is a
measure equivalence coupling $\Omega$ for which there exists a common Borel
fundamental domain $X$ for the actions of $G$ and $H$ on $\Omega$, such that
denoting by $c:H\times X\to G$ the associated cocycle, for every $h\in H$,
$c(h,\cdot)$ essentially takes only finitely many values. The existence of an
$(L^{\infty},L^{0})$-orbit equivalence coupling is equivalent to requiring
that $G$ and $H$ admit free, measure-preserving actions on a standard
probability space $X$ with the same orbits, so that for the natural orbit
equivalence cocycle $c:H\times X\to G$, for every $h\in H$, $c(h,\cdot)$
essentially only takes finitely many values.

######  Proposition 9.2.

Let $\Gamma$ be a finite simplicial graph. Let $H$ be any graph product over
$\Gamma$ with vertex groups isomorphic to
$\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$.

Then there exists an $(L^{\infty},L^{0})$-orbit equivalence coupling from $H$
to $G$.

###### Proof.

There exists an $(L^{\infty},L^{0})$-orbit equivalence coupling from
$\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$ to $\mathbb{Z}$, see Remark
7.11. It thus follows from [HH22a, Proposition 4.2] that $G$ and $H$ are orbit
equivalent. The proposition follows because the argument in [HH22a,
Proposition 4.2] preserves the integrability of the coupling (the computation
will be carried in detail in [EH]). ∎

##  10 Lattice embeddings of right-angled Artin groups

In this section, we first prove a theorem that gives restrictions on the
possible lattice embeddings of a countable group $H$ with bounded torsion
which is measure equivalent to a right-angled Artin group $G$ with
$|\operatorname{Out}(G)|<+\infty$ (Theorem 10.1). And we then describe all
possible lattice embeddings under an integrability condition on the coupling
between $G$ and $H$ (Theorem 10.5). For the second statement, we will first
need to introduce the language of quasi-actions, which will be also useful in
the next section.

###  10.1 Lattice embeddings and measure equivalence

######  Theorem 10.1.

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$, and let $H$ be a countable group with
bounded torsion that is measure equivalent to $G$. Let $\mathfrak{H}$ be a
locally compact second countable group, and let $\tau:H\to\mathfrak{H}$ be a
lattice embedding.

Then $\mathfrak{H}$ maps continuously with compact kernel to a totally
disconnected locally compact group, and $\tau$ is cocompact.

In the proof, we will make use of the extension of the notion of measure
equivalence to unimodular locally compact second countable groups given by
[BFS13, Definition 1.1].

###### Proof.

The group $\mathfrak{H}$ is measure equivalent to $G$. Let $(\Omega,\mu)$ be a
measure equivalence coupling between $G$ and $\mathfrak{H}$. Recall that the
inclusion $G\subseteq\mathrm{Aut}(\mathbb{B})$ is strongly ICC (Proposition
5.1). By Lemma 6.5, any self measure equivalence coupling of $G$ is taut
relative to the inclusion $G\subseteq\mathrm{Aut}(\mathbb{B})$ in the sense of
[BFS13, Definition 1.10] (the uniqueness of the tautening map in this
definition is ensured by [BFS13, Lemma A.8(1)]). Notice also that the locally
compact second countable group $\mathfrak{H}$ is unimodular because it
contains $H$ as a lattice. Therefore we can apply [BFS13, Theorem 2.6] and
deduce that there exists a continuous homomorphism
$\iota:\mathfrak{H}\to\mathrm{Aut}(\mathbb{B})$ with compact kernel $K$ and
closed image.

The intersection $H\cap K$ is finite, whence a lattice in the compact group
$K$. Let $\pi:\mathfrak{H}\to\mathfrak{H}/K$ be the projection map. By [Rag72,
Theorem 1.13] (as restated in [BFS20, Lemma 3.5]), the $\pi$-image of $H$ in
$\mathfrak{H}/K$ is again a lattice. Since $\mathrm{Aut}(\mathbb{B})$ is
totally disconnected, so is its closed subgroup $\mathfrak{H}/K$. Since every
lattice embedding of a countable group with bounded torsion in a totally
disconnected locally compact group is cocompact (see [BCGM19, Corollary
4.11]), it follows that the image of $H$ in $\mathfrak{H}/K$ is cocompact. Let
$K^{\prime}\subseteq\mathfrak{H}/K$ be a compact set whose $H$-translates
cover $\mathfrak{H}/K$. The map $\pi$ is closed, continuous, and it has
compact fibers because $K$ is compact, see [AT08, Theorem 1.5.7]. It follows
that $\pi^{-1}(K^{\prime})$ is compact, see [Eng89, Theorem 3.7.2], and
therefore $H$ is cocompact in $\mathfrak{H}$. ∎

###  10.2 Quasi-actions

In this section and the next, we will need the notion of a quasi-action of a
group. Given $L\geq 1$ and $A\geq 0$, an _$(L,A)$ -quasi-action_ of a group
$\mathfrak{H}$ on a metric space $(Z,d)$ is a map $\rho:\mathfrak{H}\times
Z\to Z$ such that

  * •

for every $h\in\mathfrak{H}$, the map $\rho(h,\cdot):Z\to Z$ is an
$(L,A)$-quasi-isometry,

  * •

for every $h_{1},h_{2}\in\mathfrak{H}$ and every $z\in Z$, one has
$d(\rho(h_{1},\rho(h_{2},z)),\rho(h_{1}h_{2},z))<A$, and

  * •

for every $z\in Z$, one has $d(\rho(e,z),z)<A$.

A _quasi-action_ of $\mathfrak{H}$ on $(Z,d)$ is an $(L,A)$-quasi-action for
some $L\geq 1$ and $A\geq 0$.

If $\mathfrak{H}$ is a topological group, the quasi-action $\rho$ is _proper_
if for any $z\in Z$ and $R>0$, the set $\\{h\in\mathfrak{H}\mid\rho(h,z)\in
B_{R}(z)\\}$ has compact closure. Notice that when $\mathfrak{H}$ is discrete,
this amounts to requiring that this set is finite. The quasi-action $\rho$ is
_cobounded_ if there exist $z\in Z$ and $L>0$ such that $Z$ is contained in
the $L$-neighborhood of $\rho(\mathfrak{H}\times\\{z\\})$.

Two quasi-actions $\rho,\rho^{\prime}$ of $\mathfrak{H}$ on the same metric
space $Z$ are _equivalent_ if

$\sup_{h\in\mathfrak{H}}\sup_{z\in Z}d(\rho(h,z),\rho^{\prime}(h,z))<+\infty.$

More generally, two quasi-actions $\rho,\rho^{\prime}$ of $\mathfrak{H}$ on
two metric spaces $Z,Z^{\prime}$ are _quasi-conjugate_ if there exists a
quasi-isometry $\varphi:Z\to Z^{\prime}$ such that

$\sup_{h\in\mathfrak{H}}\sup_{z\in
Z}d(\varphi\circ\rho(h,z),\rho^{\prime}(h,\varphi(z)))<+\infty.$

The study of quasi-actions on right-angled Artin groups relies on the
following theorem of the second-named author.

######  Theorem 10.2 ([Hua17, Theorem 4.18]).

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$. For every $L\geq 1,A\geq 0$, there exists
$D=D(A,L)\geq 0$ such that the following holds. For every $(L,A)$-quasi-
isometry $q:G\to G$, there exists a unique flat-preserving bijection
$q^{\prime}:G\to G$ such that $d(q(x),q^{\prime}(x))\leq D$ for every $x\in
G$.

In addition, for every $x\in G$, denoting by $\mathcal{F}_{x}$ the set of all
maximal standard flats that contain $x$, one has

$\\{q^{\prime}(x)\\}=\bigcap_{F\in\mathcal{F}_{x}}q_{\ast}(F),$

where $q_{\ast}(F)$ is the unique maximal standard flat at finite Hausdorff
distance of $q(F)$.

We refer to [Hua17, Lemma 4.12 and Equation 4.13] for the “in addition”
statement of the above theorem. We record the following corollary of Theorem
10.2.

######  Corollary 10.3.

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$, and let $\mathfrak{H}$ be a locally compact
second countable group. Then any quasi-action $\rho:\mathfrak{H}\times G\to G$
is equivalent to a unique action $\alpha:\mathfrak{H}\times G\to G$ by flat-
preserving bijections. Moreover, if $\rho$ is a measurable map, then $\alpha$
is a continuous map.

###### Proof.

The existence and uniqueness of $\alpha$ follow from Theorem 10.2. Its
measurability follows from the measurability of $\rho$ and the fact that
$\alpha(h,g)=g^{\prime}$ if and only if there exists $M\in\mathbb{N}$ such
that for every maximal standard flat $F$ containing $g$, there exists a
maximal standard flat $F^{\prime}$ containing $g^{\prime}$ such that for every
$x\in F$, there exists $x^{\prime}\in F^{\prime}$ such that
$d(\rho(h,x),x^{\prime})\leq M$. Equipped with the compact-open topology, the
group $\mathrm{Bij}_{\mathrm{FP}}(G)$ is secound-countable. So the measurable
map $\mathfrak{H}\to\mathrm{Bij}_{\mathrm{FP}}(G)$ induced by $\alpha$ is in
fact automatically continuous [Zim84, Theorem B.3]. Therefore $\alpha$ is
continuous. ∎

######  Proposition 10.4.

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$. Let $H$ be a finitely generated group that
is quasi-isometric to $G$, and let $\mathfrak{H}$ be a locally compact second
countable group that contains $H$ as a cocompact lattice.

Then $\mathfrak{H}$ has a proper, cobounded, continuous action on $G$ by flat-
preserving bijective quasi-isometries with uniform constants, and a proper,
cocompact, continuous action on a locally finite $\mathrm{CAT}(0)$ cube
complex $Y$ which is quasi-isometric to $G$.

###### Proof.

By [MSW03, Lemma 28], the group $\mathfrak{H}$ has a proper, cobounded,
measurable quasi-action $\rho$ on $H$. Notice that, while properness and
measurability of the quasi-action are not stated in [MSW03, Lemma 28], they
follow from the construction. Indeed, to construct $\rho$, one starts with an
enumeration $H=\\{h_{n}\\}_{n\in\mathbb{N}}$ and chooses a symmetric compact
subset $K\subseteq\mathfrak{H}$ containing the identity, such that
$HK=\mathfrak{H}$. For $g\in\mathfrak{H}$ and $h\in H$, one lets
$\rho(g,h)=h^{\prime}$, where $h^{\prime}$ is the first element of the
enumeration of $H$ such that $gh\in h^{\prime}K$. In particular $\mathfrak{H}$
has a proper, cobounded, measurable quasi-action on $G$, through a quasi-
isometry between $H$ and $G$.

By Corollary 10.3, there exists a unique continuous flat-preserving action
$\alpha:\mathfrak{H}\to\mathrm{Bij}_{\mathrm{FP}}(G)$ such that there exists
$D\geq 0$ such that for every $h\in\mathfrak{H}$ and every $g\in G$, one has
$d(\rho(h,g),\alpha(h)(g))\leq D$. In particular the action $\alpha$ is by
quasi-isometries, and it is proper and cobounded. This proves the first part
of the proposition.

The second part of the proposition then follows from [HK18, Theorem 6.2], and
$Y$ arises as a blow-up building as in Section 3. As in the proof of [HK18,
Theorem 6.2], the properness of the $\mathfrak{H}$-action on $Y$ follows from
the properness of its action on $G$, and the fact that there is an equivariant
quasi-isometry from $G$ to the set of rank $0$ vertices of $Y$ (see the
paragraphs after the proof of the claim in [HK18, p. 587]). The coboundedness
(in fact cocompactness as $Y$ is locally finite) of the action follows from
the same argument. And the measurability of the $\mathfrak{H}$-action on $Y$
also follows from the measurability of its action on $G$ by the same argument
(the argument gives the measurability at the level of rank $0$ vertices, but
the action on rank $0$ vertices determines the whole action). Finally, the
continuity of the homomorphism $\mathfrak{H}\to\mathrm{Aut}(Y)$ follows again
from an automatic continuity statement due to Mackey, see [Zim84, Theorem
B.3]. ∎

###  10.3 Lattice embeddings under an integrability assumption

######  Theorem 10.5.

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$, and let $H$ be a countable group with
bounded torsion. Assume that there exists an $(L^{1},L^{0})$-measure
equivalence coupling from $H$ to $G$. Let $\mathfrak{H}$ be a locally compact
second countable group in which $H$ embeds as a lattice.

Then there exists a uniformly locally finite $\mathrm{CAT}(0)$ cube complex
$Y$ quasi-isometric to $G$, and a continuous homomorphism
$\mathfrak{H}\to\mathrm{Aut}(Y)$ with compact kernel and cocompact image. Such
$Y$ can be chosen to be a blow-up building in the sense of Section 3.1.

###### Proof.

Since there is an $(L^{1},L^{0})$-measure equivalence coupling from $H$ to
$G$, Theorem 1 implies that $H$ is finitely generated and quasi-isometric to
$G$. Since $H$ is a cocompact lattice in $\mathfrak{H}$ by Theorem 10.1, it
follows from Proposition 10.4 that there exists a continuous proper cobounded
action by cubical automorphisms $\alpha:\mathfrak{H}\to\mathrm{Aut}(Y)$ on a
$\mathrm{CAT}(0)$ cube complex $Y$ that is quasi-isometric to $G$. The kernel
of $\alpha$ is then a closed subgroup of $\mathfrak{H}$, in fact compact by
properness of the action. And the cocompactness of the image of $\mathfrak{H}$
in $\mathrm{Aut}(Y)$ follows from the cocompactness of the
$\mathfrak{H}$-action on $Y$. ∎

##  11 Lack of virtual locally compact model for products

In this section, we prove Theorem 4 from the introduction. More precisely, we
start with a right-angled Artin group $G_{\Lambda}$ with
$|\operatorname{Out}(G_{\Lambda})|<+\infty$, which splits as a direct product
$G_{\Lambda}=G_{\Gamma_{1}}\times G_{\Gamma_{2}}$. We construct a sequence of
finite-index subgroups $G_{\Lambda_{n}}=G_{\Gamma_{1,n}}\times
G_{\Gamma_{2,n}}$, and cocompact torsion-free lattices $U_{n}$ in the
automorphism group $\mathrm{Aut}(X_{2n,1}\times X_{2n,2})$ of the universal
cover of the Salvetti complex of $\Lambda_{2n}$, such that the groups $U_{n}$
do not embed as lattices in a common locally compact group $\mathfrak{G}$,
even up to passing to finite-index subgroups (see Theorem 11.8).

There are several building blocks in our construction. We start from cocompact
lattices in the automorphism group of a product of two trees, coming from the
celebrated construction of Burger–Mozes [BM00b, BM00a] – this will be reviewed
in Section 11.2 below. We will then use two constructions, presented in
Section 11.1, which will allow us to extend the action on a product of trees,
to a cocompact action on the product $X_{2n,1}\times X_{2n,2}$. The
construction of the groups $U_{n}$ is completed in Section 11.3. Checking that
the groups $U_{n}$ do not admit any common lattice embedding, even virtually,
will require a very fine analysis on factor actions (which were already key in
the proof of the main theorem of this paper). This analysis is carried in
Section 11.4, and the proof of Theorem 4 is completed in Section 11.5.

###  11.1 Two arguments for extending actions to a bigger complex

For each $n\in\mathbb{N}$, let $T_{n}$ be the regular tree of valence $n$,
with each edge of length $2$. An action of a group $H$ on $T_{n}$ by
automorphisms is _even_ if it preserves the colours of vertices in any
bipartite colouring of $VT_{n}$.

Let $J_{n}$ be the Cayley graph of a rank $n$ free group with respect to a
free generating set $X$, with the usual orientation and labeling of edges by
elements of $X$, where edges have length $1$. Notice that $J_{n}$ is a regular
tree of valence $2n$. Let $\tilde{X}$ be a finite set in bijection with $X$.
We label edges of $J_{n}\times J_{n}$ by elements of $X\cup\tilde{X}$: edges
with a nondegenerate projection to the first (resp. second) factor are labeled
by elements of $X$ (resp. $\tilde{X}$). A _standard line_ in $J_{n}$ is a line
made of edges of the same label. A _standard flat_ in $J_{n}\times J_{n}$ is a
product of two standard lines, one in each factor.

An action $\alpha$ of a group $H$ on a product $T\times T^{\prime}$ of two
trees is _factor-preserving_ if there exist actions
$\beta:H\to\mathrm{Aut}(T)$ and $\beta^{\prime}:H\to\mathrm{Aut}(T^{\prime})$
such that $\alpha(h)=(\beta(h),\beta^{\prime}(h))$ for every $h\in H$ – in
particular $\alpha$ does not swap the two factors. The actions $\beta$ and
$\beta^{\prime}$ are called the _factor actions_ of $\alpha$.

######  Lemma 11.1.

Let $\alpha:H\curvearrowright T_{n}\times T_{n}$ be a free, cocompact, factor-
preserving action of a group $H$, whose factor actions are even.

Then there exist an isometric embedding $\theta:T_{n}\times T_{n}\to
J_{2n}\times J_{2n}$, a group $H^{\prime}$ with a factor-preserving, free and
cocompact action $\alpha^{\prime}:H^{\prime}\to\mathrm{Aut}(J_{2n}\times
J_{2n})$ which preserves the orientation of edges and sends standard flats to
standard flats, and an injective homomorphism $\varphi:H\to H^{\prime}$, such
that $\theta$ is equivariant with respect to $\alpha,\alpha^{\prime}$ and
$\varphi$.

In addition $\theta$ can be chosen so that for any $x\in V(T_{n}\times
T_{n})$, and any two distinct edges $e_{1},e_{2}\in E(T_{n}\times T_{n})$
containing $x$, the labels of $\theta(e_{1})$ and $\theta(e_{2})$ are
different.

###### Proof.

Let $S=\\{s_{1},\dots,s_{n}\\}$ be a finite set of cardinality $n$. We label
each edge of $T_{n}$ by a letter in $S$, in such a way that two edges incident
on the same vertex never have the same label. We fix a bipartition of $T_{n}$
into _black_ and _white_ vertices. Let $T^{\prime}_{n}$ be the barycentric
subdivision of $T_{n}$. Vertices of $T^{\prime}_{n}$ are of three types: black
vertices (in $V(T_{n})$), white vertices (in $V(T_{n})$), and _gray_ vertices,
corresponding to the midpoints of edges of $T_{n}$. We orient each edge of
$T^{\prime}_{n}$ so that it points towards its gray vertex. We fix a set
$\\{a_{1},\dots,a_{n},a^{\prime}_{1},\dots,a^{\prime}_{n}\\}$ of labels. Any
edge of $T^{\prime}_{n}$ from a white (resp. black) vertex of $T_{n}$ to the
midpoint of an $s_{i}$-labeled edge of $T_{n}$, is labeled by $a_{i}$ (resp.
$a^{\prime}_{i}$).

Let $W_{2n}$ be a wedge sum of $2n$ oriented circles, labeled by
$a_{1},\ldots,a_{n},a^{\prime}_{1},\ldots,a^{\prime}_{n}$. There is a unique
map $\pi:T^{\prime}_{n}\to W_{2n}$ which preserves labels and orientations of
edges. But $\pi$ is not a covering map. Now we enlarge $T^{\prime}_{n}$ to a
larger space $K_{n}$, and extend $\pi$ to a map $\pi:K_{n}\to W_{2n}$ which is
a covering map. This relies on Haglund and Wise’s _canonical completion_
procedure [HW08, Section 6], which we explain in this special case.

We enlarge $T^{\prime}_{n}$ as follows. For each edge $e$ of $T^{\prime}_{n}$
oriented from a vertex $x$ to another vertex $y$, we add an edge $e^{\prime}$
to $T^{\prime}_{n}$ such that $e^{\prime}$ and $e$ have the same label, but
$e^{\prime}$ is oriented from $y$ to $x$. Next for each white (resp. black)
vertex $x$ of $T^{\prime}_{n}$, we attach $n$ oriented loops based $x$,
labeled by $\\{a^{\prime}_{1},\ldots,a^{\prime}_{n}\\}$ (resp.
$\\{a_{1},\ldots,a_{n}\\}$). Finally for each gray vertex $x$ of
$T^{\prime}_{n}$ which is the midpoint of an $s_{i}$-labeled edge of $T_{n}$,
we attached $2(n-1)$ oriented loops based at $x$, labeled by
$\\{a_{1},\ldots,\hat{a}_{i},\ldots,a_{n},a^{\prime}_{1},\ldots,\hat{a}^{\prime}_{i},\ldots,a^{\prime}_{n}\\}$
– here $\hat{a}_{i},\hat{a}^{\prime}_{i}$ means that we remove these two
elements from the set of labels. Now one readily verifies that the map
$\pi:T^{\prime}_{n}\to W_{2n}$ extends to a label and orientation preserving
covering map $\pi:K_{n}\to W_{2n}$.

A _standard circle_ in $K_{n}$ is an embedded copy of $\mathbb{S}^{1}$ made of
edges with the same label. Each standard circle in $K_{n}$ has either one or
two edges.

Now consider the action $H\curvearrowright T_{n}\times T_{n}$, which gives two
factor actions $H\curvearrowright T_{n}$ preserving the bipartite vertex
coloring of $T_{n}$. Any factor action induces an action $H\curvearrowright
T^{\prime}_{n}$ which preserves orientation of edges and colors of vertices of
$T^{\prime}_{n}$ (though it might not preserve labels of edges of
$T^{\prime}_{n}$). And since the number of edge-loops at vertices of a given
colour is constant, the action $H\curvearrowright T^{\prime}_{n}$ extends to
an action $H\curvearrowright K_{n}$ which preserves the orientation of edges
and sends standard circles to standard circles.

Now the free and cocompact action $H\curvearrowright T_{n}\times T_{n}$
extends to a free and cocompact action $H\curvearrowright K_{n}\times K_{n}$.
A _standard torus_ in $K_{n}\times K_{n}$ is a product of two standard
circles, one from each factor. Then the action $H\curvearrowright K_{n}\times
K_{n}$ preserves the edge orientation and sends every standard torus to a
standard torus. Moreover, the subspace $T^{\prime}_{n}\times T^{\prime}_{n}$
is invariant under this action. Let $J_{2n}\times J_{2n}$ be the universal
cover of $K_{n}\times K_{n}$ with the induced label and orientation of edges
(where the edges coming from the second factor are labeled by
$\tilde{a}_{i},\tilde{a}^{\prime}_{i}$). Then standard flats in $J_{2n}\times
J_{2n}$ are lifts of standard tori in $K_{n}\times K_{n}$ – for this it is
important to notice that components of $K_{n}$ consisting of edges with the
same label are reduced to circles.

Let $H^{\prime}\subseteq\mathrm{Aut}(J_{2n}\times J_{2n})$ be the subgroup
consisting of all automorphisms that lift automorphisms in $H$. Then the
$H^{\prime}$-action on $J_{2n}\times J_{2n}$ is free and cocompact, preserves
the orientation of edges, and sends standard flats to standard flats. In
addition, the inclusion map $T^{\prime}_{n}\times T^{\prime}_{n}\to
K_{n}\times K_{n}$ lifts to an isometric embedding
$\theta:T^{\prime}_{n}\times T^{\prime}_{n}\to J_{2n}\times J_{2n}$. As
$T^{\prime}_{n}\times T^{\prime}_{n}$ is invariant under the $H$-action on
$K_{n}\times K_{n}$, such a lift gives an injective homomorphism $\varphi:H\to
H^{\prime}$ such that $\theta$ is equivariant with respect to $\varphi$.
Finally, the additional part of the lemma follows from our construction. ∎

We need another extension criterion, which is based on an idea of Hughes in
[Hug21, Section 7.2]. Our next lemma is a variation over a statement to be
found in ongoing work of Mj and the second-named author [HM].

Recall that $X_{\Gamma}$ is the locally finite cube complex canonically
associated to $G_{\Gamma}$, in other words $X_{\Gamma}$ is the universal cover
of the Salvetti complex of $G_{\Gamma}$. Given a group $H$ acting by flat-
preserving bijections on $G_{\Gamma}$, we have a _type cocycle_ $c:H\times
VX_{\Gamma}\to\mathrm{Aut}(\Gamma)$ where $c(h,x)$ with $h\in H$ and $x\in
VX_{\Gamma}$ is defined as follows. Given a vertex $v\in V\Gamma$, let $\ell$
be the standard line of type $v$ containing $x$. Then $c(h,x)(v)$ is defined
to be the type of $h(\ell)$. One readily vertices that $c(h,x):V\Gamma\to
V\Gamma$ preserves adjacency of vertices, hence extends to an automorphism of
$\Gamma$, and that $c(h,x)$ is indeed a cocycle.

######  Lemma 11.2.

Let $\Gamma_{1}$ be an induced subgraph of $\Gamma_{2}$. Suppose $H$ is a
group acting freely and cocompactly on $X_{\Gamma_{1}}$, preserving the
orientation of edges, and sending standard flats to standard flats. Let
$c_{1}:H\times VX_{\Gamma_{1}}\to\mathrm{Aut}(\Gamma_{1})$ be the associated
type cocycle. Suppose that there exists a cocycle $c_{2}:H\times
VX_{\Gamma_{1}}\to\mathrm{Aut}(\Gamma_{2})$ such that
$c_{2}(h,x)_{\mid\Gamma_{1}}=c_{1}(h,x)$ for any $h\in H$ and $x\in
VX_{\Gamma_{1}}$.

Then there exist a group $H^{\prime}$ acting freely and compactly on
$X_{\Gamma_{2}}$ sending standard flats to standard flats, an injective group
homomorphism $\phi:H\to H^{\prime}$, and a $\phi$-equivariant embedding
$j:X_{\Gamma_{1}}\to X_{\Gamma_{2}}$ preserving labels and orientations of
edges. Moreover, each element of $H^{\prime}$ sends standard lines labeled by
$v\in V\Gamma_{2}$ to standard lines whose labels belong to the orbit of $v$
under the action of elements in $\\{c_{2}(h,x)\\}_{h\in H,x\in
VX_{\Gamma_{1}}}$.

###### Proof.

The inclusion of $\Gamma_{1}$ as an induced subgraph of $\Gamma_{2}$ yields an
isometric embedding $S_{\Gamma_{1}}\hookrightarrow S_{\Gamma_{2}}$ between the
Salvetti complexes. By pre-composing this with the covering map
$X_{\Gamma_{1}}\to S_{\Gamma_{1}}$, we obtain a local isometric embedding
$X_{\Gamma_{1}}\to S_{\Gamma_{2}}$. While $X_{\Gamma_{1}}\to S_{\Gamma_{2}}$
is not a covering map, we can “complete” it to a covering map as follows,
using (as in the previous proof) a special case of a construction by
Haglund–Wise [HW08, Section 6]. Consider the homomorphism $G_{\Gamma_{2}}\to
G_{\Gamma_{1}}$ fixing each generator in $\Gamma_{1}$ and sending all
generators in $\Gamma_{2}\setminus\Gamma_{1}$ to identity. Let $K$ be the
kernel of this homomorphism and $Z$ be the cover of $S_{\Gamma_{2}}$
corresponding to $K$. Then there is an embedding $X_{\Gamma_{1}}\to Z$. Under
such an embedding, $X_{\Gamma_{1}}$ and $Z$ have the same vertex set.
Moreover, we can obtain the 1-skeleton of $Z$ from the 1-skeleton of
$X_{\Gamma_{1}}$ by attaching a collection of edge loops to each of the
vertices of $X_{\Gamma_{1}}$, one edge loop for each vertex outside
$\Gamma_{2}\setminus\Gamma_{1}$. The complex $Z$ is called the _canonical
completion_ of $X_{\Gamma_{1}}$ with respect to the local isometric embedding
$X_{\Gamma_{1}}\to S_{\Gamma_{2}}$.

Now we define an action of $H$ on the 1-skeleton $Z^{(1)}$ of $Z$ extending
the existing action of $H$ on $X^{(1)}_{\Gamma_{1}}$ as follows. For $h\in H$,
we let $h$ send an edge loop based at $x\in X_{\Gamma_{1}}$ labeled by $v\in
V\Gamma_{2}\setminus V\Gamma_{1}$ to an edge loop based at $h(x)$ labeled by
$c_{2}(h,x)(v)$, moreover, we require that $h$ respects the orientation of
edges. It follows from the construction of the action $H\curvearrowright
Z^{(1)}$ that it sends a pair of edges with commuting labels based at the same
vertex to another pair of edges with commuting labels. Thus $H\curvearrowright
Z^{(1)}$ extends to an action of $H$ on the 2-skeleton of $Z$. One readily
checks that the action extends to higher skeleta as well. This gives an edge
orientation preserving action $H\curvearrowright Z$ extending the existing
action $H\curvearrowright X_{\Gamma_{1}}$. Note that $H\curvearrowright Z$ is
also free and cocompact.

Let $H^{\prime}$ be the subgroup of $\mathrm{Aut}(X_{\Gamma_{2}})$ consisting
of all lifts of automorphisms of $Z$ coming from the action $H\curvearrowright
Z$. Then $H^{\prime}$ fits into an exact sequence $1\to\pi_{1}(Z)\to
H^{\prime}\to H\to 1$. As $H\curvearrowright Z$ is free and cocompact, the
same holds for the action $H^{\prime}\curvearrowright X_{\Gamma_{2}}$. The
moreover statement of the lemma follows from the construction of $H^{\prime}$.
Let $j:X_{\Gamma_{1}}\to X_{\Gamma_{2}}$ be a lift of the embedding
$X_{\Gamma_{1}}\to Z$ with respect to the covering map $X_{\Gamma_{2}}\to Z$.
Now we define an injective homomorphism $\phi:H\to H^{\prime}$ as follows.
Given $h\in H$, let $\alpha_{h}$ be the automorphism of $Z$ coming from the
action $H\curvearrowright Z$, and let
$\beta_{h}=(\alpha_{h})_{|X_{\Gamma_{1}}}$, an automorphism of
$X_{\Gamma_{1}}$. Let $\beta^{\prime}_{h}$ be the map $j\circ\beta_{h}\circ
j^{-1}$ defined on $j(X_{\Gamma_{1}})$. We define $\phi(h):X_{\Gamma_{2}}\to
X_{\Gamma_{2}}$ to be the unique lift of $\alpha_{h}:Z\to Z$ with respect to
the covering $X_{\Gamma_{2}}\to Z$ such that $\phi(h)$ is an extension of
$\beta^{\prime}_{h}$. Then $\phi$ is indeed a group homomorphism, and $j$ is
equivariant with respect to $\phi$. ∎

###  11.2 On local actions

Given a factor-preserving action of a group $H$ on $T_{n}\times T_{n}$, and a
vertex $x$ in one of the factor trees, the _local action_ of $H$ at $x$ is the
action of the $H$-stabilizer of $x$ (with respect to the factor action) on the
collection of edges of the factor tree containing $x$. The _local group_ at
$x$ is the subgroup of the permutation group of edges containing $x$ coming
from the local action.

The following can be deduced from work of Lazarovich–Levcovitz–Margolis
[LLM22], relying on earlier works by Burger–Mozes [BM00a, BM00b] and Radu
[Rad20].

######  Lemma 11.3.

For each $n_{0}>0$, there exist $n\geq n_{0}$ and a simple torsion-free group
$H$ which acts on $T_{n}\times T_{n}$ freely, cocompactly, in a factor-
preserving way, with even factor actions, such that for any vertex $x$ in one
of the tree factors, the local action of $H$ at $x$ has an orbit of size at
least $n/4$.

###### Proof.

By [LLM22, Theorem 4.2 and Lemma 4.3], there is a group $H$ acting on
$T_{n}\times T_{n}$ as a uniform lattice such that all the local groups are
the full symmetry groups on $n$ letters. Moreover, $H$ has an index $4$
subgroup $H^{+}$ which is simple and preserves the bipartition of each factor
tree (see the beginning of [LLM22, Section 2] for the definition of $H^{+}$).
In addition $H^{+}$ is torsion-free by [Rad20, Lemma 3.1]. Thus the group
$H^{+}$ satisfies the requirement of the lemma. ∎

###  11.3 The family of examples

Let $\Gamma_{1}$ and $\Gamma_{2}$ be finite simplicial non-complete graphs –
at this point we are not making any extra assumption, but in the next section
our discussion will be applied to graphs $\Gamma_{1},\Gamma_{2}$ such that
$|\operatorname{Out}(G_{\Gamma_{i}})|<+\infty$ for every $i\in\\{1,2\\}$.

For every $i\in\\{1,2\\}$, we fix a choice of two vertices $v_{i},w_{i}\in
V\Gamma_{i}$, with $w_{i}\notin\mathrm{st}(v_{i})$. For every $n\geq 1$, let
$\Gamma_{n,i}$ be the finite simplicial graph obtained by gluing $n$ copies of
$\Gamma_{i}$ along $\mathrm{st}(v_{i})$. We will denote by $v_{n,i}$ the image
of $v_{i}$ in $\Gamma_{n,i}$. Let $\Gamma^{w}_{n,i}$ be the totally
disconnected subgraph of $\Gamma_{n,i}$ whose vertices are the $n$ copies of
$w_{i}$ in $\Gamma_{n,i}$, denoted by $w_{n,i}[1],\ldots,w_{n,i}[n]$.

Let $\Lambda=\Gamma_{1}\circ\Gamma_{2}$ and
$\Lambda_{n}=\Gamma_{n,1}\circ\Gamma_{n,2}$. Let
$\Lambda^{w}_{n}=\Gamma^{w}_{n,1}\circ\Gamma^{w}_{n,2}$, a subgraph of
$\Lambda_{n}$.

Notice that for every $i\in\\{1,2\\}$ and every $n\geq 1$, the group
$G_{\Gamma_{n,i}}$ is isomorphic to the kernel of the homomorphism
$\varphi_{n,i}:G_{\Gamma_{i}}\to\mathbb{Z}/n\mathbb{Z}$ that sends $v_{i}$ to
$1$ and all other generators to $0$. This gives an injective homomorphism
$q_{n,i}:G_{\Gamma_{n,i}}\to G_{\Gamma_{i}}$ with finite index image. More
precisely, let $\Gamma_{n,i}[1],\dots,\Gamma_{n,i}[n]$ be the $n$ copies of
$\Gamma_{i}$ in $\Gamma_{n,i}$, where $\Gamma_{n,i}[j]$ is the copy that
contains $w_{n,i}[j]$. Then (up to reordering the $\Gamma_{n,i}[j]$, which we
do once and for all), for any vertex $u\in\Gamma_{n,i}[j]$ with $u\neq
v_{n,i}$, we have $q_{n,i}(u)=v_{i}^{j-1}\bar{u}v_{i}^{-j+1}$, where $\bar{u}$
denotes the vertex of $\Gamma_{i}$ that corresponds to $u$ (when writing the
above equality, we identify vertices of $\Gamma_{i}$ and $\Gamma_{n,i}$ with
the corresponding elements of $G_{\Gamma_{i}}$ and $G_{\Gamma_{n,i}}$). And
$q_{n,i}(v_{n,i})=v_{i}^{n}$. In particular
$q_{n,i}(w_{n,i}[j])=v_{i}^{j-1}w_{i}v_{i}^{-j+1}$.

For every $i\in\\{1,2\\}$ and every $n\geq 1$, the $q_{n,i}$-image of any
standard line of $G_{\Gamma_{n,i}}$ is Hausdorff close to a standard line of
$G_{\Gamma_{i}}$. In addition $q_{n,i}$ sends parallel standard lines to
parallel standard lines, up to finite Hausdorff distance. Thus $q_{n,i}$
induces a map from the vertex set of the extension graph $\Gamma_{n,i}^{e}$ to
the vertex set of $\Gamma_{i}^{e}$, which extends to a graph isomorphism
$(q_{n,i})_{*}:\Gamma_{n,i}^{e}\to\Gamma_{i}^{e}$. Likewise we have a map
$q_{n}:G_{\Lambda_{n}}\to G_{\Lambda}$.

For every $i\in\\{1,2\\}$, let $X_{n,i}$ be the universal cover of the
Salvetti complex of $G_{\Gamma_{n,i}}$. We orient edges of $X_{n,i}$ and label
them by vertices of $\Gamma_{n,i}$. As $\\{w_{n,i}[1],\ldots,w_{n,i}[n]\\}$
generates a free subgroup of $G_{\Gamma_{n,i}}$, we have an embedding
$j_{n,i}:J_{n}\to X_{n,i}$ which preserves the orientation and the labeling of
edges, which gives $j_{n}:J_{n}\times J_{n}\to X_{n,1}\times X_{n,2}$. We will
often refer to the first product factor as the _horizontal_ factor, and the
second factor as _vertical_.

######  Corollary 11.4.

There exists an infinite subset $\mathcal{C}\subset\mathbb{Z}_{\geq 0}$ such
that the following is true. For each $n\in\mathcal{C}$, there exists a group
$V_{n}$ acting on $T_{n}\times T_{n}$ satisfying the requirements of Lemma
11.3, a group $U_{n}$ acting on $X_{2n,1}\times X_{2n,2}$ freely and
cocompactly sending standard flats to standard flats, an isometric embedding
$\theta_{n}:T_{n}\times T_{n}\to X_{2n,1}\times X_{2n,2}$, and an injective
group homomorphism $\phi_{n}:V_{n}\to U_{n}$ such that

  1. 1.

$\theta_{n}$ sends each edge of $T_{n}\times T_{n}$ (of length 2) to a
concatenation of two edges in $X_{2n,1}\times X_{2n,2}$;

  2. 2.

$\theta_{n}$ is equivariant with respect to $\phi_{n}$;

  3. 3.

for each $x\in T_{n}$, there exists $y\in X_{2n,1}$ such that
$\theta(\\{x\\}\times T_{n})\subset\\{y\\}\times X_{2n,2}$;

  4. 4.

for any vertex $x\in T_{n}\times T_{n}$ and any two different edges $e_{1}$
and $e_{2}$ containing $x$, the edges $\theta_{n}(e_{1})$ and
$\theta_{n}(e_{2})$ have different labels, both contained in
$\Lambda^{w}_{2n}$;

  5. 5.

for every $i\in\\{1,2\\}$, each element of $U_{n}$ sends standard lines
labeled by $v_{2n,i}$ to standard lines labeled by $v_{2n,i}$.

###### Proof.

Let $V_{n}$ be a group given by Lemma 11.3. By Lemma 11.1, there exist a group
$V^{\prime}_{n}$ acting geometrically on $J_{2n}\times J_{2n}$, preserving the
orientation of edges, and sending standard flats to standard flats, an
injective homomorphism $\varphi_{n}:V_{n}\to V^{\prime}_{n}$, and a
$\varphi_{n}$-equivariant isometric embedding $T_{n}\times T_{n}\to
J_{2n}\times J_{2n}$. Since the action of $V^{\prime}_{n}$ is factor-
preserving, the type cocycle of $V^{\prime}_{n}\curvearrowright J_{2n}\times
J_{2n}$ is defined and takes its values in automorphisms of $\Lambda^{w}_{2n}$
that preserves the two factors $\Gamma^{w}_{2n,1}$ and $\Gamma^{w}_{2n,2}$ in
$\Lambda^{w}_{2n}$. Any such automorphism of $\Lambda^{w}_{2n}$ extends
naturally to an automorphism of $\Lambda_{2n}$, because any permutation of the
vertex set of $\Gamma^{w}_{2n,i}$ extends to an automorphism of
$\Gamma_{2n,i}$ permuting the copies
$\Gamma_{2n,i}[1],\dots,\Gamma_{2n,i}[2n]$. This gives the required extension
of cocycles as in the assumption of Lemma 11.2. Now the corollary follows. ∎

###  11.4 Auxiliary facts about star projections

Given a finite simplicial graph $\Gamma$ with
$|\operatorname{Out}(G_{\Gamma})|<+\infty$, we collect several facts about
certain projections on $G_{\Gamma}$ for later use. Recall that we identify
$G_{\Gamma}$ as the $0$-skeleton of $X_{\Gamma}$. Given $\mathsf{v}\in
V\Gamma^{e}$, take a $\mathsf{v}$-line $\ell$. Then $\ell$ is the 0-skeleton
of a convex subcomplex of $X_{\Gamma}$. Then there is nearest point projection
$\pi_{\ell}:G_{\Gamma}\to\ell$ sending each point to the nearest point in
$\ell$ with respect to the word distance [HW08, Lemma 13.8]. It is known that
if $\ell_{1},\ell_{2}$ are standard lines with
$\Delta(\ell_{1})=\Delta(\ell_{2})\notin\mathrm{st}(\mathsf{v})$, then
$\pi_{\ell}(\ell_{i})$ is a single point for every $i\in\\{1,2\\}$, and
$\pi_{\ell}(\ell_{1})=\pi_{\ell}(\ell_{2})$ (see [Hua17, Lemma 6.2]). This
gives a well-defined map
$\pi_{\mathsf{v}}:V(\Gamma^{e}\setminus\mathrm{st}(\mathsf{v}))\to\ell$.

Recall that the left action $G_{\Gamma}\curvearrowright\Gamma^{e}$ induces a
projection map $\Gamma^{e}\to\Gamma$. We say a vertex $\mathsf{v}$ of
$\Gamma^{e}$ is of _type $v$_ with $v\in\Gamma$, if the map
$\Gamma^{e}\to\Gamma$ sends $\mathsf{v}$ to $v$.

Now we continue with the notations from the previous section, and assume in
addition that $\operatorname{Out}(G_{\Gamma_{1}})$ and
$\operatorname{Out}(G_{\Gamma_{2}})$ are finite. In particular, for
$i\in\\{1,2\\}$, let $\Gamma_{n,i}$, $q_{n,i}:G_{\Gamma_{n,i}}\to
G_{\Gamma_{i}}$ and $(q_{n,i})_{*}:\Gamma^{e}_{n,i}\to\Gamma_{i}^{e}$ be as in
the previous section.

######  Lemma 11.5.

Assume that $\operatorname{Out}(G_{\Gamma_{1}})$ and
$\operatorname{Out}(G_{\Gamma_{2}})$ are finite. Let $i\in\\{1,2\\}$. Let
$\ell$ be a standard line in $G_{\Gamma_{n,i}}$ such that
$\mathsf{v}=\Delta(\ell)$ is of type $v_{n,i}$. For $1\leq j\neq k\leq n$, let
$\ell_{j}$ and $\ell_{k}$ be two standard lines in $G_{\Gamma_{n,i}}$
intersecting $P_{\mathsf{v}}$ non-trivially such that
$\Delta(\ell_{j})=\mathsf{w}_{j}$ is of type $w_{n,i}[j]$ and
$\Delta(\ell_{k})=\mathsf{w}_{k}$ is of type $w_{n,i}[k]$. Let
$\mathsf{v}^{\prime}=(q_{n,i})_{*}(\mathsf{v})$. Then

  1. 1.

$\pi_{\mathsf{v}^{\prime}}((q_{n,i})_{*}(\mathsf{w}_{j}))\neq\pi_{\mathsf{v}^{\prime}}((q_{n,i})_{*}(\mathsf{w}_{k}))$.

  2. 2.

Write $P_{\mathsf{v}}=gG_{\mathrm{st}(v_{n,i})}$ and let $\mathcal{C}$ be the
collection of standard lines $\ell^{\prime}$ which have non-trivial
intersection with $gG_{\mathrm{lk}(v_{n,i})}$ and satisfy
$\Delta(\ell^{\prime})\notin\mathrm{st}(\mathsf{v})$. Let
$\mathcal{W}=\\{\Delta(\ell^{\prime})\\}_{\ell^{\prime}\in\mathcal{C}}$. Then
$\pi_{\mathsf{v}^{\prime}}((q_{i})_{*}(\mathcal{W}))$ is a finite set of
cardinality at most $n$.

###### Proof.

As $(q_{n,i})_{\ast}$ is $G_{\Gamma_{n,i}}$-equivariant, up to translation, we
can assume that $P_{\mathsf{v}}=G_{\mathrm{st}(v_{n,i})}$, that
$\ell_{j}=\langle w_{n,i}[j]\rangle$, and that
$\ell_{k}=g_{1}v_{n,i}^{m}\langle w_{n,i}[k]\rangle$ for some $m\in\mathbb{Z}$
and some $g_{1}\in G_{\mathrm{lk}(v_{n,i})}$. Then $q_{n,i}(\ell_{j})$ is
Hausdorff close to the standard line $v_{i}^{j-1}\langle w_{i}\rangle$, and
$q_{n,i}(\ell_{k})$ is Hausdorff close to the standard line
$g_{1}v_{i}^{mn+k-1}\langle w_{i}\rangle$. Identifying
$Z_{\mathsf{v}^{\prime}}$ with $\langle v_{i}\rangle$, we get that
$\pi_{\mathsf{v}^{\prime}}((q_{n,i})_{*}(\mathsf{w}_{j}))=v_{i}^{j-1}$ and
$\pi_{\mathsf{v}^{\prime}}((q_{n,i})_{*}(\mathsf{w}_{k}))=v_{i}^{nm+k-1}$, and
Assertion 1 follows. Assertion 2 follows from a similar computation. ∎

######  Lemma 11.6.

Let $\Gamma$ be a finite simplicial graph such that
$|\operatorname{Out}(G_{\Gamma})|<+\infty$. Suppose $H\curvearrowright
G_{\Gamma}$ is an action by flat-preserving bijections. Take $\mathsf{v}\in
V\Gamma^{e}$ and let $\alpha_{\mathsf{v}}:H_{\mathsf{v}}\curvearrowright
Z_{\mathsf{v}}$ be the associated factor action. Let $\ell\subset G_{\Gamma}$
be a $\mathsf{v}$-line. Take another standard line $\ell^{\prime}$ with
$\Delta(\ell^{\prime})\notin\mathrm{st}(\mathsf{v})$.

Then
$\alpha_{\mathsf{v}}(h)(\pi_{\mathsf{v}}(\Delta(\ell^{\prime})))=\pi_{\mathsf{v}}(\Delta(h(\ell^{\prime})))$
for any $h\in H_{\mathsf{v}}$.

###### Proof.

As $P_{\mathsf{v}}$ is the vertex set of a convex subcomplex $C_{\mathsf{v}}$
of $X_{\Gamma}$, by [HW08, Lemma 13.8], there is a nearest point projection
map $\pi_{P_{\mathsf{v}}}:G_{\Gamma}\to P_{\mathsf{v}}$. The lemma follows
from the fact, established below, that the nearest point projection map
$\pi_{P_{\mathsf{v}}}:G_{\Gamma}\to P_{\mathsf{v}}$ is
$H_{\mathsf{v}}$-equivariant.

Now we prove the fact. Let $x\in G_{\Gamma}$ and $h\in H_{\mathsf{v}}$. Let
$y=\pi_{P_{\mathsf{v}}}(x)$. Let $\omega$ be a shortest edge path in the
1-skeleton of $X_{\Gamma}$ connecting the vertices $x$ and $y\in
P_{\mathsf{v}}$. Let $x_{0},\dots,x_{n}$ be vertices in $\omega$ such that for
$0\leq i\leq n-1$, $[x_{i},x_{i+1}]$ is a maximal sub-segment of $\omega$ that
is contained in a standard line ($x_{0}=x$ and $x_{n}=y$). Denote the
corresponding standard line by $\ell_{i}$. As $y$ can be alternatively
characterized as the unique point in $P_{\mathsf{v}}$ such that any hyperplane
separating $x$ and $y$ does not cross $C_{\mathsf{v}}$ (see [HW08, Lemmas 13.1
and 13.8]), $y=\pi_{P_{\mathsf{v}}}(x)$ if and only if for any $\omega$ and
$(\ell_{i})$ as above, $\Delta(\ell_{i})\notin\mathrm{st}(\mathsf{v})$ for any
$0\leq i\leq n-1$.

For $0\leq i\leq n-1$, let $\ell^{\prime}_{i}$ be the standard line with
$h(\ell_{i})=\ell^{\prime}_{i}$ and let $\omega^{\prime}_{i}$ be a geodesic
segment in $\ell^{\prime}_{i}$ from $h(x_{i})$ to $h(x_{i+1})$. Let
$\omega^{\prime}$ be the concatenation of all the $\omega^{\prime}_{i}$ with
$0\leq i\leq n-1$. By considering the automorphism
$h_{*}:\Gamma^{e}\to\Gamma^{e}$ induced by the flat-preserving bijection $h$,
we see that $\Delta(\ell^{\prime}_{i})\notin\mathrm{st}(\mathsf{v})$ for any
$0\leq i\leq n-1$. Thus none of the hyperplanes that have nonempty
intersection with $\omega^{\prime}$ will intersect $C_{\mathsf{v}}$. As any
hyperplane separating $h(x)$ and $h(y)$ is dual to an edge in
$\omega^{\prime}$, this hyperplane does not cross $C_{\mathsf{v}}$. Thus
$h(y)=\pi_{P_{\mathsf{v}}}(h(x))$ by the previous paragraph. ∎

###  11.5 Conclusion

As usual, given an action $\alpha$ of a group $H$ by flat-preserving
bijections on $G=G_{\Gamma}$, for every $\mathsf{v}\in V\Gamma^{e}$, we let
$\alpha_{\mathsf{v}}:H_{\mathsf{v}}\curvearrowright Z_{\mathsf{v}}$ be the
factor action.

######  Proposition 11.7.

Let $G$ be a non-cyclic right-angled Artin group with
$|\operatorname{Out}(G)|<+\infty$. Let $\rho$ be a quasi-action of $H$ on $G$,
and let $\alpha:H\curvearrowright G$ be the unique $H$-action on $G$ by flat-
preserving bijections which is equivalent to $\rho$ (see Corollary 10.3).

Then there exists $C\geq 0$ (depending on $\rho$) such that for every
$\mathsf{v}\in V\Gamma^{e}$ and every subgroup
$H^{\prime}_{\mathsf{v}}\subseteq H_{\mathsf{v}}$, if some
$H^{\prime}_{\mathsf{v}}$-orbit of the factor action $\alpha_{\mathsf{v}}$ is
finite, then every $H^{\prime}_{\mathsf{v}}$-orbit for this action has
cardinality at most $C$.

###### Proof.

Note that there are constants $L,A>0$, depending on $\rho$, such that for any
$\mathsf{v}\in V\Gamma^{e}$, the factor action
$\alpha_{\mathsf{v}}:H_{\mathsf{v}}\curvearrowright Z_{\mathsf{v}}$ is by
bijections which are $(L,A)$-quasi-isometries. Then by [HK18, Proposition
6.3], there exist a constant $D$ depending only on $L$ and $A$, an isometric
action $\alpha^{\prime}_{\mathsf{v}}:H_{\mathsf{v}}\curvearrowright\mathbb{Z}$
and a surjective equivariant map $f:Z_{\mathsf{v}}\to\mathbb{Z}$ such that
each point inverse of $f$ has cardinality at most $D$. If
$(\alpha_{\mathsf{v}})_{|H^{\prime}_{\mathsf{v}}}$ has a finite orbit, then
$(\alpha^{\prime}_{\mathsf{v}})_{|H^{\prime}_{\mathsf{v}}}$ has a finite
orbit, hence a fixed point. Hence any orbit of
$(\alpha^{\prime}_{\mathsf{v}})_{|H^{\prime}_{\mathsf{v}}}$ has cardinality at
most $2$, and therefore any orbit of
$(\alpha_{\mathsf{v}})_{|H^{\prime}_{\mathsf{v}}}$ has cardinality at most
$2D$. ∎

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two finite simplicial graphs such that
$\operatorname{Out}(G_{\Gamma_{1}})$ and $\operatorname{Out}(G_{\Gamma_{2}})$
are finite. Given this choice, let $\mathcal{C}\subseteq\mathbb{Z}_{\geq 0}$
and the groups $U_{n},V_{n}$ be as in Corollary 11.4. Recall that $U_{n}$ acts
properly and cocompactly on $X_{2n,1}\times X_{2n,2}$, and so does
$G_{\Gamma_{2n,1}}\times G_{\Gamma_{2n,2}}$. In particular $U_{n}$ and
$G_{\Lambda}=G_{\Gamma_{1}}\times G_{\Gamma_{2}}$ are strongly commable in the
sense recalled before the statement of Theorem 4 in the introduction.
Therefore, Theorem 4 is a consequence of the following statement – recall that
$U_{n}$ is torsion-free, so we can focus on lattice embeddings instead of
lattice representations with finite kernel.

######  Theorem 11.8.

There does not exist a locally compact second countable topological group
$\mathfrak{G}$ such that for each $n\in\mathcal{C}$, there exists a finite
index subgroup $U^{\prime}_{n}$ of $U_{n}$ which admits an embedding
$U^{\prime}_{n}\to\mathfrak{G}$ whose image is a lattice.

###### Proof.

We argue by contradiction and assume such $\mathfrak{G}$ exists. Recall that
$|\operatorname{Out}(G_{\Lambda})|<+\infty$. Since $G_{\Lambda_{2n}}$ has
finite index in $G_{\Lambda}$, and since $U_{n}$ and its finite-index subgroup
$U^{\prime}_{n}$ are uniform lattices in $\mathrm{Aut}(X_{2n,1}\times
X_{2n,2})$, we can apply Theorem 10.1 and deduce that the lattice embedding
$U^{\prime}_{n}\to\mathfrak{G}$ is cocompact.

Fix $n_{0}\in\mathcal{C}$. Since $U^{\prime}_{n_{0}}$ is quasi-isometric to
$G_{\Lambda}$, we can apply Proposition 10.4 and get a proper, cobounded
action $\alpha_{1}$ of $\mathfrak{G}$ on $G_{\Lambda}$ by flat-preserving
bijective quasi-isometries with uniform constants. This gives, for every
$n\in\mathcal{C}$, an action by flat-preserving bijections
$\alpha_{1,n}:U^{\prime}_{n}\curvearrowright G_{\Lambda}$ through the lattice
embedding $U^{\prime}_{n}\to\mathfrak{G}$. We apply Proposition 11.7 to the
action $\alpha_{1}:\mathfrak{G}\curvearrowright G_{\Lambda}$, and let $C$ be
the resulting constant.

On the other hand, for every $n\in\mathcal{C}$, the group $U^{\prime}_{n}$ has
a properly discontinuous cocompact action on $X_{2n,1}\times X_{2n,2}$. Let
$q_{2n}:X_{2n,1}\times X_{2n,2}\to G_{\Lambda}$ be as in the previous section,
after identifying the $0$-skeleton of $X_{2n,1}\times X_{2n,2}$ with
$G_{\Lambda_{2n}}$. Then $q_{2n}$ gives a quasi-action of $U^{\prime}_{n}$ on
$G_{\Lambda}$, which is equivalent to a unique action by flat-preserving
bijections $\alpha_{2,n}:U^{\prime}_{n}\curvearrowright G_{\Lambda}$ (see
Corollary 10.3). Note that both $\alpha_{1,n}$ and $\alpha_{2,n}$ are proper
and cobounded actions of $U^{\prime}_{n}$ on $G_{\Lambda}$ by quasi-isometries
with uniform constants. Thus $\alpha_{1,n}$ and $\alpha_{2,n}$ are quasi-
conjugate (as any two proper and cobounded quasi-actions of the same finitely
generated group on the same space are quasi-conjugate). By Theorem 10.2,
$\alpha_{1,n}$ and $\alpha_{2,n}$ are actually conjugate via a flat-preserving
bijection of $G_{\Lambda}$.

Through $\alpha_{2,n}$, the group $U^{\prime}_{n}$ acts on the extension graph
$\Lambda^{e}$. Since the action of $U^{\prime}_{n}$ on $X_{2n,1}\times
X_{2n,2}$ sends standard flats to standard flats (Corollary 11.4), we also
have an action of $U^{\prime}_{n}$ on $\Lambda_{2n}^{e}$. The map
$q_{2n}:X_{2n,1}\times X_{2n,2}\to G_{\Lambda}$ induces an isomorphism
$(q_{2n})_{\ast}:\Lambda_{2n}^{e}\to\Lambda^{e}$, which is
$U^{\prime}_{n}$-equivariant with respect to the above actions.

Let $\\{\alpha_{2,n,\mathsf{v}}:U^{\prime}_{n,\mathsf{v}}\curvearrowright
Z_{\mathsf{v}}\\}_{\mathsf{v}\in V\Lambda^{e}}$ be the collection of factor
actions for $\alpha_{2,n}:U^{\prime}_{n}\curvearrowright G_{\Lambda}$. We
observe that for every subgroup $U^{\prime\prime}_{n,\mathsf{v}}\subseteq
U^{\prime}_{n,\mathsf{v}}$, every finite orbit of
$(\alpha_{2,n,\mathsf{v}})_{|U^{\prime\prime}_{n,\mathsf{v}}}$ has cardinality
at most $C$: indeed, this follows from the same property for the action
$\alpha_{1,n,\mathsf{v}}$, which comes from our choice of constant $C$, and
from the fact that the actions $\alpha_{1,n}$ and $\alpha_{2,n}$ are conjugate
via a flat-preserving bijection. In the rest of the proof, we will show that
as $n$ becomes larger and larger, we can find some $\mathsf{v}\in
V\Lambda^{e}$ and some subgroup $U^{\prime\prime}_{n,\mathsf{v}}$ of
$U^{\prime}_{n,\mathsf{v}}$ acting on $Z_{\mathsf{v}}$ with finite orbits of
larger and larger size, which will be a contradiction.

Consider the simple subgroup $V_{n}$ of $U_{n}$ as above. Then $V_{n}\subseteq
U^{\prime}_{n}$. We view $T_{n}\times T_{n}$ as a $V_{n}$-invariant subcomplex
of $X_{2n,1}\times X_{2n,2}$ via the embedding $\theta_{n}$ provided by
Corollary 11.4. A _vertical_ (resp.  _horizontal_) _$T_{n}$ -copy_ in
$X_{2n,1}\times X_{2n,2}$ is the $\theta_{n}$-image of $\\{z\\}\times T_{n}$
(resp. $T_{n}\times\\{z\\}$) for some vertex $z$ in the first (resp. second)
tree factor. Let $\ell\subset X_{2n,1}\times X_{2n,2}$ be a standard line of
type $v_{2n,1}$ such that $\ell$ intersects $T_{n}\times T_{n}$ in a vertex
$x\in X_{2n,1}\times X_{2n,2}$. Let $p_{1}:T_{n}\times T_{n}\to T_{n}$ be the
projection to the first factor. Let $V_{n,x}$ be the $V_{n}$-stabilizer of
$p_{1}(x)$ with respect to the action of $V_{n}$ on the first factor of
$T_{n}\times T_{n}$.

We claim that each element in $V_{n,x}$ sends $\ell$ to a standard line which
is parallel to $\ell$. Indeed, by Corollary 11.4(5), for any $g\in V_{n,x}$,
edges in $g(\ell)$ are also labeled by $v_{2n,1}$. On the other hand, $x$ and
$g(x)$ are connected by an edge path in a vertical $T_{n}$-copy whose edge
labels commute with $v_{2n,1}$, thus $\ell$ and $g(\ell)$ are parallel.

Let $P_{\ell}$ be the parallel set of $\ell$ in $G_{\Lambda_{2n}}$. We write
$P_{\ell}=g_{0}G_{\mathrm{st}(v_{2n,1})}$ and assume without loss of
generality that $x\in g_{0}G_{\mathrm{lk}(v_{2n,1})}$. By the previous
paragraph, $V_{n,x}$ stabilizes both $P_{\ell}$ and
$g_{0}G_{\mathrm{lk}(v_{2n,1})}$.

Let $T_{n,x}^{h}$ be the horizontal $T_{n}$-copy that contains $x$. Let
$\\{\ell_{1},\dots,\ell_{k}\\}$ be the set of standard lines in
$X_{2n,1}\times X_{2n,2}$ passing through $x$ whose intersection with
$T_{n,x}^{h}$ contains at least one edge. By Corollary 11.4(4), we have $k=n$,
and the types of the lines $\ell_{1},\dots,\ell_{n}$ are contained in
$\\{w_{2n,1}[1],\ldots,w_{2n,1}[2n]\\}$. For every $i\in\\{1,\dots,n\\}$, let
$e_{i}\subseteq\ell_{i}$ be an edge based at $x$ and contained in
$\ell_{i}\cap T_{n,x}^{h}$. Since the action of $U_{n}$ on $X_{2n,1}\times
X_{2n,2}$ sends standard flat to standard flat (Corollary 11.4), for every
$g\in V_{n,x}$ and every $i\in\\{1,\dots,n\\}$, the image $g\ell_{i}$ is a
standard line, and $g(e_{i})$ is an edge based at $gx$ contained in
$g\ell_{i}\cap T_{n,gx}^{h}$. It thus follows from Lemma 11.3 and Corollary
11.4(4) that, up to permuting the lines $\ell_{i}$, the set of all types of
$g\ell_{1}$ as $g$ varies in $V_{n,x}$, is a subset of
$\\{w_{2n,1}[1],\dots,w_{2n,1}[2n]\\}$ of cardinality at least $n/4$.

Let $\ell^{\prime}_{1}$ (resp. $\ell^{\prime}$) be a standard line in
$G_{\Lambda}$ at finite Hausdorff distance from $q_{2n}(\ell_{1})$ (resp.
$q_{2n}(\ell)$). Let $\mathsf{v}=\Delta(\ell^{\prime})$. We consider the
action $\alpha_{2,n,\mathsf{v}}:U^{\prime}_{n,\mathsf{v}}\curvearrowright
Z_{\mathsf{v}}$. The fact that every element of $V_{n,x}$ sends $\ell$ to a
parallel line ensures that $V_{n,x}\subset U^{\prime}_{n,\mathsf{v}}$.

Notice that for every $g\in V_{n,x}$, we have
$(q_{2n})_{\ast}(\Delta(g\ell_{1}))=\Delta(g\ell^{\prime}_{1})$. Thus, using
Lemma 11.6, we deduce that the projections
$\pi_{\mathsf{v}}((q_{2n})_{*}(\Delta(g\ell_{1})))$, as $g$ varies in
$V_{n,x}$, form exactly one orbit of the action $V_{n,x}\curvearrowright
Z_{\mathsf{v}}$ under $\alpha_{2,n,\mathsf{v}}$. As the types of the lines
$g\ell_{1}$, with $g$ varying in $V_{n,x}$, form a subset of
$\\{w_{2n,1}[1],\ldots,w_{2n,1}[2n]\\}$ of cardinality at least $n/4$, Lemma
11.5(1) implies that the cardinality of the set
$\\{\pi_{\mathsf{v}}((q_{2n})_{*}(\Delta(g\ell_{1})))\\}_{g\in V_{n,x}}$ is at
least $n/4$. In addition, since $\Delta(\ell_{1})$ and $\Delta(\ell)$ are not
adjacent in $\Lambda_{2n}^{e}$, and $(q_{2n})_{\ast}$ is an isomorphism, we
have $\Delta(\ell^{\prime}_{1})\notin\mathrm{st}(\Delta(\ell^{\prime}))$. We
can thus apply Lemma 11.5(2) and deduce that the above set (which as we
explained is exactly one orbit of $V_{n,x}$ for the action
$\alpha_{2,n,\mathsf{v}}$) is finite. We have thus found a subgroup of
$U^{\prime}_{n,\mathsf{v}}$ whose action on $Z_{\mathsf{v}}$ via
$\alpha_{2,n,\mathsf{v}}$ has a finite orbit of size at least $n/4$, as
desired. ∎

## References

  * [AB08] P. Abramenko and K.S. Brown.  Buildings. Theory and applications, volume 248 of Graduate Texts in Mathematics.  Springer, New York, 2008.
  * [Ago13] I. Agol.  The virtual Haken conjecture.  Doc. Math., 18:1045–1087, 2013.
  * [AT08] A. Arhangel’skii and M. Tkachenko.  Topological groups and related structures, volume 1 of Atlantis Studies in Mathematics.  Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
  * [Aus16a] T. Austin.  Behaviour of entropy under bounded and integrable orbit equivalence.  Geom. Funct. Anal., 26(6):1483–1525, 2016.
  * [Aus16b] T. Austin.  Integrable measure equivalence for groups of polynomial growth.  Groups Geom. Dyn., 10(1):117–154, 2016.
  * [BC12] J. Behrstock and R. Charney.  Divergence and quasimorphisms of right-angled Artin groups.  Math. Ann., 352(2):339–356, 2012.
  * [BCG+09] J. Brodzki, S.J. Campbell, E. Guentner, G.A. Niblo, and N.J. Wright.  Property A and $\mathrm{CAT}(0)$ cube complexes.  J. Funct. Anal., 256(5):1408–1431, 2009.
  * [BCGM19] U. Bader, P.-E. Caprace, T. Gelander, and S. Mozes.  Lattices in amenable groups.  Fund. Math., 246(3):217–255, 2019.
  * [Bel68] R. M. Belinskaja.  Partitionings of a Lebesgue space into trajectories which may be defined by ergodic automorphisms.  Funkcional. Anal. i Priložen., 2(3):4–16, 1968.
  * [BFS13] U. Bader, A. Furman, and R. Sauer.  Integrable measure equivalence and superrigidity of hyperbolic lattices.  Invent. Math., 194(2):313–379, 2013.
  * [BFS20] U. Bader, A. Furman, and R. Sauer.  Lattice envelopes.  Duke Math. J., 169(2):213–278, 2020.
  * [BH99] M.R. Bridson and A. Haefliger.  Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften.  Springer-Verlag, Berlin, 1999.
  * [BJN10] J.A. Behrstock, T. Januszkiewicz, and W.D. Neumann.  Quasi-isometric classification of some high dimensional right-angled Artin groups.  Groups Geom. Dyn., 4(4):681–692, 2010.
  * [BKMM12] J. Behrstock, B. Kleiner, Y. Minsky, and L. Mosher.  Geometry and rigidity of mapping class groups.  Geom. Topol., 16(2):781–888, 2012.
  * [BKS08] M. Bestvina, B. Kleiner, and M. Sageev.  The asymptotic geometry of right-angled Artin groups. I.  Geom. Topol., 12(3):1653–1699, 2008.
  * [BM00a] M. Burger and S. Mozes.  Groups acting on trees: from local to global structure.  Inst. Hautes Études Sci. Publ. Math., (92):113–150 (2001), 2000\.
  * [BM00b] M. Burger and S. Mozes.  Lattices in product of trees.  Inst. Hautes Études Sci. Publ. Math., (92):151–194 (2001), 2000\.
  * [BN08] J.A. Behrstock and W.D. Neumann.  Quasi-isometric classification of graph manifold groups.  Duke Math. J., 141(2):217–240, 2008.
  * [Bro84] K.S. Brown.  Presentations for groups acting on simply-connected complexes.  J. Pure Appl. Algebra, 32(1):1–10, 1984.
  * [CCV07] R. Charney, J. Crisp, and K. Vogtmann.  Automorphisms of $2$-dimensional right-angled Artin groups.  Geom. Topol., 11:2227–2264, 2007.
  * [CD95] R. Charney and M.W. Davis.  Finite $K(\pi,1)$s for Artin groups.  In Prospects in topology (Princeton, NJ, 1994), volume 138 of Ann. of Math. Stud., pages 110–124. Princeton Univ. Press, Princeton, NJ, 1995.
  * [CF12] R. Charney and M. Farber.  Random groups arising as graph products.  Algebr. Geom. Topol., 12(2):979–995, 2012.
  * [CH15] P.-E. Caprace and D. Hume.  Orthogonal forms of Kac-Moody groups are acylindrically hyperbolic.  Ann. Inst. Fourier (Grenoble), 65(6):2613–2640, 2015.
  * [Cha07] R. Charney.  An introduction to right-angled Artin groups.  Geom. Dedic., 125(1):141–158, 2007.
  * [CL10] P.-E. Caprace and A. Lytchak.  At infinity of finite-dimensional CAT(0) spaces.  Math. Ann., 346(1):1–21, 2010.
  * [Cor15] Y. Cornulier.  Commability and focal locally compact groups.  Indiana Univ. Math. J., 64(1):115–150, 2015.
  * [CS11] P.-E. Caprace and M. Sageev.  Rank rigidity for CAT(0) cube complexes.  Geom. Funct. Anal., 21(4):851–891, 2011.
  * [Dav98] M.W. Davis.  Buildings are $\mathrm{CAT}(0)$.  In Geometry and cohomology in group theory (Durham, 1994), volume 252 of London Math. Soc. Lecture Note Ser., pages 108–123, Cambridge, 1998. Cambridge Univ. Press.
  * [Day12] M.B. Day.  Finiteness of outer automorphism groups of random right-angled Artin groups.  Algebr. Geom. Topol., 12(3):1553–1583, 2012.
  * [DKLMT22] T. Delabie, J. Koivisto, F. Le Maître, and R. Tessera.  Quantitative measure equivalence between amenable groups.  Ann. H. Lebesgue, 5:1417–1487, 2022.
  * [Dye59] H. A. Dye.  On groups of measure preserving transformations. I.  Amer. J. Math., 81:119–159, 1959.
  * [Dye63] H. A. Dye.  On groups of measure preserving transformations. II.  Amer. J. Math., 85:551–576, 1963.
  * [EF97] A. Eskin and B. Farb.  Quasi-flats and rigidity in higher rank symmetric spaces.  J. Amer. Math. Soc., 10(3):653–692, 1997.
  * [EH] A. Escalier and C. Horbez.  Graph products and measure equivalence: classification, rigidity, and quantitative aspects.  In preparation.
  * [Eng89] R. Engelking.  General topology, volume 6 of Sigma Series in Pure Mathematics.  Heldermann Verlag, Berlin, second edition, 1989.
  * [Esk98] A. Eskin.  Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces.  J. Amer. Math. Soc., 11(2):321–361, 1998.
  * [Fer18] T. Fernós.  The Furstenberg-Poisson boundary and ${\rm CAT}(0)$ cube complexes.  Ergodic Theory Dynam. Systems, 38(6):2180–2223, 2018.
  * [FHT11] B. Farb, C. Hruska, and A. Thomas.  Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices.  In Geometry, rigidity, and group actions, Chicago Lectures in Math., pages 515–560. Univ. Chicago Press, Chicago, IL, 2011.
  * [FLM18] T. Fernós, J. Lécureux, and F. Mathéus.  Random walks and boundaries of $\mathrm{CAT}(0)$ cubical complexes.  Comment. Math. Helv., 93(2):291–333, 2018.
  * [Fur63] H. Furstenberg.  A Poisson formula for semi-simple Lie groups.  Ann. of Math. (2), 77:335–386, 1963.
  * [Fur99a] A. Furman.  Gromov’s measure equivalence and rigidity of higher-rank lattices.  Ann.of Math. (2), 150(3):1059–1081, 1999.
  * [Fur99b] A. Furman.  Orbit equivalence rigidity.  Ann.of Math. (2), 150(3):1083–1108, 1999.
  * [Fur01] A. Furman.  Mostow-Margulis rigidity with locally compact targets.  Geom. Funct. Anal., 11(1):30–59, 2001.
  * [Gab02] D. Gaboriau.  Invariants $l^{2}$ de relations d’équivalence et de groupes.  Publ. Math. Inst. Hautes Études Sci., (95):93–150, 2002.
  * [GH21] V. Guirardel and C. Horbez.  Measure equivalence rigidity of $\mathrm{Out}({F}_{N})$.  arXiv:2103.03696, 2021.
  * [God03] E. Godelle.  Parabolic subgroups of Artin groups of type FC.  Pacific J. Math., 208(2):243–254, 2003.
  * [Gre90] E.R. Green.  Graph products of groups.  PhD thesis, University of Leeds, 1990.
  * [Gro93] M. Gromov.  Asymptotic invariants of infinite groups.  In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. Cambridge Univ. Press, Cambridge, 1993.
  * [Ham07] U. Hamenstädt.  Geometry of the mapping class groups III: Quasi-isometric rigidity.  arXiv:math/0512429, 2007.
  * [HH20] C. Horbez and J. Huang.  Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type.  arXiv:2004.09325, 2020.
  * [HH22a] C. Horbez and J. Huang.  Measure equivalence classification of transvection-free right-angled Artin groups.  J. Éc. polytech. Math., 9:1021–1067, 2022.
  * [HH22b] C. Horbez and J. Huang.  Measure equivalence rigidity among the Higman groups.  arXiv:2206.00884, 2022.
  * [HHI23] C. Horbez, J. Huang, and A. Ioana.  Orbit equivalence rigidity of irreducible actions of right-angled Artin groups.  Compos. Math., 159(4):860–887, 2023.
  * [HK18] J. Huang and B. Kleiner.  Groups quasi-isometric to right-angled Artin groups.  Duke Math. J., 167(3):537–602, 2018.
  * [HM] J. Huang and M. Mj.  Indiscrete common commensurators.  In preparation.
  * [Hua16] J. Huang.  Quasi-isometry classification of right-angled Artin groups II: several infinite out cases.  arXiv:1603.02372, 2016.
  * [Hua17] J. Huang.  Quasi-isometric classification of right-angled Artin groups I: the finite out case.  Geom. Topol., 21(6):3467–3537, 2017.
  * [Hua18] J. Huang.  Commensurability of groups quasi-isometric to raags.  Invent. Math., 213(3):1179–1247, 2018.
  * [Hug21] S. Hughes.  Graphs and complexes of lattices.  arXiv:2104.13728, 2021.
  * [HW08] F. Haglund and D.T. Wise.  Special cube complexes.  Geom. Funct. Anal., 17(5):1551–1620, 2008.
  * [HW12] F. Haglund and D.T. Wise.  A combination theorem for special cube complexes.  Ann. of Math. (2), 176(3):1427–1482, 2012.
  * [Iva97] N.V. Ivanov.  Automorphism of complexes of curves and of Teichmüller spaces.  Internat. Math. Res. Notices, (14):651–666, 1997.
  * [Jus71] J. Justin.  Groupes et semi-groupes à croissance linéaire.  C. R. Acad. Sci. Paris Sér. A-B, 273:A212–A214, 1971.
  * [Kec95] A.S. Kechris.  Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1995.
  * [Kid09] Y. Kida.  Introduction to measurable rigidity of mapping class groups.  In Handbook of Teichmüller theory, volume II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 297–367. Eur. Math. Soc., 2009.
  * [Kid10] Y. Kida.  Measure equivalence rigidity of the mapping class group.  Ann. of Math. (2), 171(3):1851–1901, 2010.
  * [Kid11] Y. Kida.  Rigidity of amalgamated free products in measure equivalence.  J. Topol., 4(3):687–735, 2011.
  * [KK13] S.-h. Kim and T. Koberda.  Embedability between right-angled Artin groups.  Geom. Topol., 17(1):493–530, 2013.
  * [KL97] B. Kleiner and B. Leeb.  Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings.  Inst. Hautes Études Sci. Publ. Math., (86):115–197 (1998), 1997\.
  * [KL21] D. Kerr and H. Li.  Entropy, Shannon orbit equivalence, and sparse connectivity.  Math. Ann., 380(3-4):1497–1562, 2021.
  * [KL22] D. Kerr and H. Li.  Entropy, virtual Abelianness, and Shannon orbit equivalence.  arXiv:2202.10795, 2022.
  * [KS16] A. Kar and M. Sageev.  Ping pong on $\rm CAT(0)$ cube complexes.  Comment. Math. Helv., 91(3):543–561, 2016.
  * [Lau95] M.R. Laurence.  A generating set for the automorphism group of a graph group.  J. London Math. Soc. (2), 52(2):318–334, 1995.
  * [LLM22] N. Lazarovich, I. Levcovitz, and A. Margolis.  Counting lattices in products of trees.  arXiv:2202.00378, 2022.
  * [Mar91] G. A. Margulis.  Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)].  Springer-Verlag, Berlin, 1991.
  * [Mar20] A. Margolis.  Quasi-isometry classification of right-angled Artin groups that split over cyclic subgroups.  Groups Geom. Dyn., 14(4):1351–1417, 2020.
  * [MS06] N. Monod and Y. Shalom.  Orbit equivalence rigidity and bounded cohomology.  Ann. of Math. (2), 164(3):825–878, 2006.
  * [MSW03] L. Mosher, M. Sageev, and K. Whyte.  Quasi-actions on trees. I. Bounded valence.  Ann. of Math. (2), 158(1):115–164, 2003.
  * [OW80] D.S. Ornstein and B. Weiss.  Ergodic theory of amenable group actions. I. The Rohlin lemma.  Bull. Amer. Math. Soc. (N.S.), 2(1):161–164, 1980.
  * [Rad20] N. Radu.  New simple lattices in products of trees and their projections.  Canad. J. Math., 72(6):1624–1690, 2020.
  * [Rag72] M. S. Raghunathan.  Discrete subgroups of Lie groups. Springer-Verlag, New York-Heidelberg,, 1972.
  * [Rol98] M.A. Roller.  Poc sets, median algebras and group actions. an extended study of Dunwoody’s construction and Sageev’s theorem.  1998\.  preprint.
  * [Ré05] B. Rémy.  Integrability of induction cocycles for Kac-Moody groups.  Math. Ann., 333(1):29–43, 2005.
  * [Sag14] M. Sageev.  $\rm CAT(0)$ cube complexes and groups.  In Geometric group theory, volume 21 of IAS/Park City Math. Ser., pages 7–54. Amer. Math. Soc., Providence, RI, 2014.
  * [Sal87] M. Salvetti.  Topology of the complement of real hyperplanes in ${\bf C}^{N}$.  Invent. Math., 88(3):603–618, 1987.
  * [Ser89] H. Servatius.  Automorphisms of graph groups.  J. Algebra, 126(1):34–60, 1989.
  * [Sha00] Y. Shalom.  Rigidity of commensurators and irreducible lattices.  Invent. Math., 141(1):1–54, 2000.
  * [Sha04] Y. Shalom.  Harmonic analysis, cohomology, and the large-scale geometry of amenable groups.  Acta Math., 192(2):119–185, 2004.
  * [She22] S. Shepherd.  Commensurability of lattices in right-angled buildings.  arXiv:2203.01210, 2022.
  * [Zim80] R.J. Zimmer.  Strong rigidity for ergodic actions of semisimple Lie groups.  Ann. of Math. (2), 112(3):511–529, 1980.
  * [Zim84] R.J. Zimmer.  Ergodic Theory and Semisimple Groups, volume 81 of Monographs in Mathematics.  Birkhäuser, Basel; Boston; Stuttgart, 1984.

Camille Horbez

Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405,
Orsay, France

_e-mail:_<EMAIL_ADDRESS>

Jingyin Huang

Department of Mathematics

The Ohio State University, 100 Math Tower

231 W 18th Ave, Columbus, OH 43210, U.S.

_e-mail:_<EMAIL_ADDRESS>