# Strong primeness for equivalence relations arising from Zariski dense
subgroups

Daniel Drimbe  Mathematical Institute
University of Oxford
Oxford
OX2 6GG
United Kingdom<EMAIL_ADDRESS>and  Cyril Houdayer  École
Normale Supérieure
Département de Mathématiques et Applications
Université Paris-Saclay
45 rue d’Ulm
75230 Paris Cedex 05
France<EMAIL_ADDRESS>
###### Abstract.

We show that orbit equivalence relations arising from essentially free ergodic
probability measure preserving actions of Zariski dense discrete subgroups of
simple algebraic groups are strongly prime. As a consequence, we prove the
existence and the uniqueness of a prime factorization for orbit equivalence
relations arising from direct products of higher rank lattices. This extends
and strengthens Zimmer’s primeness result for equivalence relations arising
from actions of lattices in simple Lie groups. The proof of our main result
relies on a combination of ergodic theory of algebraic group actions and
Popa’s intertwining theory for equivalence relations.

###### Key words and phrases:

Algebraic groups; Equivalence relations; Higher rank lattices; Strong
primeness; von Neumann algebras; Zariski dense subgroups

###### 2020 Mathematics Subject Classification:

37A20, 20G25, 46L10

DD was supported by Engineering and Physical Sciences Research Council grant
EP/X026647/1

CH is supported by Institut Universitaire de France

##  1\. Introduction and statement of the main results

Throughout, by an equivalence relation $\mathscr{R}$, we simply mean a
probability measure preserving (pmp) countable Borel equivalence relation
defined on a standard probability space $(X,\nu)$.

### Introduction

We say that a type ${\rm II_{1}}$ ergodic equivalence relation $\mathscr{R}$
is prime if for any ergodic equivalence relations
$\mathscr{T}_{1},\mathscr{T}_{2}$ for which
$\mathscr{R}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, then there exists
$i\in\\{1,2\\}$ such that $\mathscr{T}_{i}$ has finite orbits almost
everywhere. Zimmer’s seminal work [Zi81] shows that for any lattice $\Gamma<G$
in a noncompact simple connected real Lie group with finite center and any
essentially free ergodic pmp action $\Gamma\curvearrowright(X,\nu)$, the orbit
equivalence relation $\mathscr{R}(\Gamma\curvearrowright X)$ is prime.

In order to motivate and state our main results, we introduce the following
terminology using the framework of Popa’s intertwining theory for equivalence
relations (see Section 2 for further details). We say that a type ${\rm
II_{1}}$ ergodic equivalence relation $\mathscr{R}$ is strongly prime if for
any ergodic equivalence relations $\mathscr{S}$, $\mathscr{T}_{1}$,
$\mathscr{T}_{2}$ for which
$\mathscr{R}\times\mathscr{S}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, then
there exists $i\in\\{1,2\\}$ such that as subequivalence relations,
$\mathscr{T}_{i}$ embeds into $\mathscr{S}$ inside
$\mathscr{R}\times\mathscr{S}$, that we write
$\mathscr{T}_{i}\preceq_{\mathscr{R}\times\mathscr{S}}\mathscr{S}$ (see
Definition 2.4 for a precise statement). Note that this implies in particular
that there exists an ergodic equivalence relation $\mathscr{U}$ such that
$\mathscr{T}_{i}\times\mathscr{U}$ and $\mathscr{S}$ are stably isomorphic
(see [Sp21]). We point out that the notion of strong primeness for type ${\rm
II_{1}}$ factors was studied by Isono in [Is16].

The ergodic theory of equivalence relations is intimately connected to the
theory of von Neumann algebras. By a well-known construction due to
Feldman–Moore [FM75], to any ergodic equivalence relation $\mathscr{R}$ on a
standard probability space $(X,\nu)$, one can associate a von Neumann factor
$\operatorname{L}(\mathscr{R})$ so that
$\operatorname{L}^{\infty}(X,\nu)\subset\operatorname{L}(\mathscr{R})$ sits as
a Cartan subalgebra. Recall that a type ${\rm II_{1}}$ factor $M$ is prime if
for any factors $M_{1},M_{2}$ for which $M\cong
M_{1}\mathbin{\overline{\otimes}}M_{2}$, then there exists $i\in\\{1,2\\}$
such that $M_{i}$ is finite dimensional. For any ergodic equivalence relation
$\mathscr{R}$, if $\operatorname{L}(\mathscr{R})$ is prime, then $\mathscr{R}$
is prime, but the converse need not hold in general (see Proposition 3.7
below). In relation with Zimmer’s primeness result [Zi81], in the case of
simple connected real Lie groups with finite center and real rank one, any
discrete subgroup $\Gamma<G$ is biexact in the sense of [BO08]. Then by
Ozawa’s relative solidity result [Oz04], it follows that for any essentially
free ergodic pmp action $\Gamma\curvearrowright(X,\nu)$ of a countable
discrete nonamenable biexact group, the group measure space factor
$\operatorname{L}(\Gamma\curvearrowright X)$ is prime. For other iconic
primeness results in von Neumann algebra theory, we refer to [Ge96, Oz03,
Pe06].

Over the last two decades, Popa’s deformation/rigidity theory (see the ICM
surveys [Po06, Va10, Io18]) has led to a plethora of primeness and
indecomposability results for von Neumann algebras and orbit equivalence
relations arising from essentially free ergodic pmp actions of negatively
curved groups such as free groups, hyperbolic groups, biexact groups, groups
with a positive first $\ell^{2}$-betti number and so on (see also [Ad92,
Ki05]). Except for very few results (see e.g. [DHI16, BIP18, IM19]), higher
rank lattices remain allergic to deformation/rigidity theory methods.

The key novelty of our paper is to combine methods from ergodic theory of
algebraic group actions and Popa’s intertwining theory to obtain strong
primeness and existence and uniqueness of prime factorization for orbit
equivalence relations arising from essentially free ergodic pmp actions of
higher rank lattices and more generally Zariski dense discrete subgroups of
simple algebraic groups.

### Statement of the main results

Our first main theorem is the following strong primeness result strengthening
Zimmer’s result [Zi81, Theorem 1.1].

######  Theorem A.

Let $k$ be a local field of characteristic zero, $\mathbf{G}$ a $k$-isotropic
almost $k$-simple algebraic $k$-group and $\Gamma<\mathbf{G}(k)$ a Zariski
dense discrete subgroup. Let $\Gamma\curvearrowright(X,\nu)$ be an essentially
free ergodic pmp action.

Then the orbit equivalence relation $\mathscr{R}(\Gamma\curvearrowright X)$ is
strongly prime.

By Borel’s density theorem [Bo60], Theorem A applies to all essentially free
ergodic pmp actions of lattices $\Gamma<\mathbf{G}(k)$, where $\mathbf{G}$ is
a $k$-isotropic almost $k$-simple algebraic $k$-group (e.g.
$\operatorname{SL}_{n}(\mathbb{Z})<\operatorname{SL}_{n}(\mathbb{R})$ for
every $n\geq 2$).

In that respect, denote by $\mathfrak{Z}$ the class of orbit equivalence
relations $\mathscr{R}=\mathscr{R}(\Gamma\curvearrowright X)$ that arise from
essentially free ergodic pmp actions $\Gamma\curvearrowright(X,\nu)$, where
$\Gamma<\mathbf{G}(k)$ is a Zariski dense discrete subgroup, $k$ is a local
field of characteristic zero and $\mathbf{G}$ is a $k$-isotropic almost
$k$-simple algebraic $k$-group.

Using Theorem A, we deduce the following unique prime factorization result for
direct products of ergodic equivalence relations that belong to
$\mathfrak{Z}$.

######  Corollary B.

Let $n\geq 2$ and $\mathscr{R}_{1},\dots,\mathscr{R}_{n}$ be ergodic
equivalence relations that belong to $\mathfrak{Z}$. Then the following
assertions hold:

  * $(\rm i)$

If
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$,
where $\mathscr{T}_{1},\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic
equivalence relations, then there exists a partition
$\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty subsets such that for every
$j\in\\{1,2\\}$, we have
$\mathscr{T}_{j}\preceq_{\mathscr{R}}\mathscr{R}_{T_{j}}$ as subequivalence
relations, where $\mathscr{R}_{T_{j}}=\prod_{i\neq j}\mathscr{R}_{i}$.

  * $(\rm ii)$

If
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\cdots\times\mathscr{T}_{p}$,
where $\mathscr{T}_{1},\dots,\mathscr{T}_{p}$ are type ${\rm II_{1}}$ ergodic
equivalence relations and $p\geq n$, then $n=p$ and upon permuting the
indices, $\mathscr{T}_{i}$ and $\mathscr{R}_{i}$ are stably isomorphic for
every $1\leq i\leq n$.

We point out that the first unique prime factorization result appeared in von
Neumann algebra theory in the pioneering work of Ozawa–Popa [OP03]. Indeed,
they showed that tensor products of type ${\rm II_{1}}$ factors arising from
countable discrete icc nonamenable biexact groups have a unique tensor product
decomposition upon taking amplifications and permuting the indices.

Assuming that the equivalence relations
$\mathscr{R}_{1},\dots,\mathscr{R}_{n}$ that belong to $\mathfrak{Z}$ are
moreover strongly ergodic, we obtain the following sharper unique prime
factorization result.

######  Corollary C.

Let $n\geq 2$ and $\mathscr{R}_{1},\dots,\mathscr{R}_{n}$ be strongly ergodic
equivalence relations that belong to $\mathfrak{Z}$. Then the following
assertions hold:

  * $(\rm i)$

If
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$,
where $\mathscr{T}_{1},\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic
equivalence relations, then there exist a partition
$\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty subsets and $t>0$ such that

$\mathscr{T}_{1}^{t}\cong\mathscr{R}_{T_{1}}\quad\text{and}\quad\mathscr{T}_{2}^{1/t}\cong\mathscr{R}_{T_{2}}.$

  * $(\rm ii)$

If
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\cdots\times\mathscr{T}_{p}$,
where $\mathscr{T}_{1},\dots,\mathscr{T}_{p}$ are type ${\rm II_{1}}$ ergodic
equivalence relations and $p\geq n$, then $n=p$ and there exist
$t_{1},\dots,t_{n}>0$ such that $t_{1}\cdots t_{n}=1$ and upon permuting the
indices, $\mathscr{T}_{i}^{t_{i}}\cong\mathscr{R}_{i}$ for every $1\leq i\leq
n$.

In particular, by Borel’s density theorem [Bo60] and Kazhdan’s theorem [Ka66],
Corollary C applies to direct products of equivalence relations arising from
essentially free ergodic pmp actions of lattices $\Gamma<\mathbf{G}(k)$, where
$\mathbf{G}$ is an almost $k$-simple algebraic $k$-group such that
$\operatorname{rk}_{k}(\mathbf{G})\geq 2$ (e.g.
$\operatorname{SL}_{n}(\mathbb{Z})<\operatorname{SL}_{n}(\mathbb{R})$ for
every $n\geq 3$). Corollary C should be compared with Hoff’s unique prime
factorization result for strongly ergodic equivalence relations with
nontrivial $1$-cohomology [Ho15]. Let us mention that Isono–Marrakchi [IM19]
obtained a unique prime factorization result for tensor products of group
measure space factors arising from compact pmp actions of higher rank lattices
with Kazhdan’s property (T). In that respect, Corollaries B and C provide the
first unique prime factorization result for orbit equivalence relations
arising from arbitrary pmp actions of higher rank lattices.

Our second main theorem gives a complete characterization of primeness for
orbit equivalence relations arising from essentially free ergodic pmp actions
of product groups
$\Gamma_{1}\times\cdots\times\Gamma_{n}\curvearrowright(X,\nu)$, where
$\Gamma_{i}$ is a discrete group as in Theorem A with Kazhdan’s property (T).

######  Theorem D.

Let $n\geq 1$. For every $1\leq i\leq n$, let $k_{i}$ be a local field of
characteristic zero, $\mathbf{G}_{i}$ a $k_{i}$-isotropic $k_{i}$-simple
algebraic $k_{i}$-group and $\Gamma_{i}<\mathbf{G}_{i}(k_{i})$ a Zariski dense
discrete subgroup with Kazhdan’s property (T). Set
$\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$. Let
$\Gamma\curvearrowright(X,\nu)$ be an essentially free ergodic pmp action and
set $\mathscr{R}=\mathscr{R}(\Gamma\curvearrowright X)$. Assume that
$\mathscr{R}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, where
$\mathscr{T}_{1},\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic equivalence
relations.

Then there exist a partition $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty
subsets, $t>0$, an essentially free ergodic pmp action
$\Gamma_{T_{j}}\curvearrowright(X_{j},\nu_{j})$, where
$\Gamma_{T_{j}}=\prod_{i\in T_{j}}\Gamma_{i}$ for every $j\in\\{1,2\\}$, such
that the following assertions hold:

  * $(\rm i)$

$\Gamma\curvearrowright(X,\nu)\cong\Gamma_{T_{1}}\times\Gamma_{T_{2}}\curvearrowright(X_{1}\times
X_{2},\nu_{1}\otimes\nu_{2})$ and

  * $(\rm ii)$

$\mathscr{T}_{1}^{t}\cong\mathscr{R}(\Gamma_{T_{1}}\curvearrowright X_{1})$
and $\mathscr{T}_{2}^{1/t}\cong\mathscr{R}(\Gamma_{T_{2}}\curvearrowright
X_{2})$.

Theorem D is an analogue of Drimbe’s primeness characterization for group
measure space factors arising from essentially free ergodic pmp actions of
products of hyperbolic groups with Kazhdan’s property (T) [Dr19]. By Borel’s
density theorem [Bo60] and Kazhdan’s theorem [Ka66], Theorem D applies to all
essentially free ergodic pmp actions $\Gamma\curvearrowright(X,\nu)$, where
$\Gamma_{i}<\mathbf{G}_{i}(k_{i})$ is a lattice and $\mathbf{G}_{i}$ is a
$k_{i}$-simple algebraic $k_{i}$-group such that
$\operatorname{rk}_{k_{i}}(\mathbf{G}_{i})\geq 2$ for every
$i\in\\{1,\dots,n\\}$.

As a consequence of Theorem D, we obtain the following existence and
uniqueness of a prime factorization for equivalence relations.

######  Corollary E.

Let $n\geq 1$. For every $1\leq i\leq n$, let $k_{i}$ be a local field of
characteristic zero, $\mathbf{G}_{i}$ a $k_{i}$-isotropic $k_{i}$-simple
algebraic $k_{i}$-group and $\Gamma_{i}<\mathbf{G}_{i}(k_{i})$ a Zariski dense
discrete subgroup with Kazhdan’s property (T). Set
$\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$. Let
$\Gamma\curvearrowright(X,\nu)$ be an essentially free ergodic pmp action.

Then there exist a unique partition $\\{1,\dots,n\\}=T_{1}\sqcup\cdots\sqcup
T_{r}$ into nonempty subsets (upon permuting the indices), an essentially free
ergodic pmp action $\Gamma_{T_{j}}\curvearrowright(X_{j},\nu_{j})$, where
$\Gamma_{T_{j}}=\prod_{i\in T_{j}}\Gamma_{i}$ for every $j\in\\{1,\dots,r\\}$,
such that the following assertions hold:

  * $(\rm i)$

$\Gamma\curvearrowright(X,\nu)\cong\Gamma_{T_{1}}\times\cdots\times\Gamma_{T_{r}}\curvearrowright(X_{1}\times\cdots\times
X_{r},\nu_{1}\otimes\cdots\otimes\nu_{r})$ and

  * $(\rm ii)$

$\mathscr{R}(\Gamma_{T_{j}}\curvearrowright X_{j})$ is prime for every
$j\in\\{1,\dots,r\\}$.

### Comments on the proofs

The proof of Theorem A builds upon Zimmer’s proof of [Zi81, Theorem 1.1] and
combines ergodic theory of algebraic group actions and Popa’s intertwining
theory for equivalence relations. Unlike Zimmer’s proof of [Zi81, Theorem
1.1], we do not use induction and we work directly with the orbit equivalence
relation $\mathscr{R}(\Gamma\curvearrowright X)$. This allows us to deal with
essentially free ergodic pmp action of arbitrary Zariski dense discrete
subgroups $\Gamma<\mathbf{G}(k)$ rather than lattices $\Gamma<\mathbf{G}(k)$
as in [Zi81]. The proof of Corollary B follows by combining Theorem A with
Popa’s intertwining theory.

Using Tucker-Drob’s results on the structure of inner amenable groups [TD14],
we observe in Proposition 2.7 that any Zariski dense discrete subgroup
$\Gamma<\mathbf{G}(k)$ as in Theorem A is not inner amenable. This implies
that for any essentially free strongly ergodic pmp action
$\Gamma\curvearrowright(X,\nu)$, the group measure space von Neumann factor
$\operatorname{L}(\Gamma\curvearrowright X)$ is full meaning that it has no
nontrivial central sequences (see [Ch81]). For the proof of Corollary C, we
use Theorem A and we exploit Popa’s intertwining theory in the setting of von
Neumann algebras in combination with Isono–Marrakchi’s results on tensor
product decompositions of full factors [IM19].

The proof of Theorem D and Corollary E follows the proof of Theorem A and
moreover exploits intertwining techniques from [Dr19].

### Acknowledgments

This work was initiated when CH was visiting the Mathematical Institute of the
University of Oxford during Hilary Term 2024. He is grateful towards Stuart
White for his kind invitation. We also thank Adrian Ioana for his valuable
comments.

###### Contents

  1. 1 Introduction and statement of the main results
  2. 2 Preliminaries
    1. 2.1 Equivalence relations
    2. 2.2 Popa’s intertwining theory for equivalence relations
    3. 2.3 Algebraic groups
    4. 2.4 Algebraic representations of equivalence relations
    5. 2.5 Chabauty topology and Wijsman topology
  3. 3 Proofs of the main results
    1. 3.1 Zimmer’s method for product equivalence relations
    2. 3.2 Proof of Theorem A
    3. 3.3 Proof of Corollary B
    4. 3.4 Proof of Corollary C
    5. 3.5 Proof of Theorem D
    6. 3.6 Proof of Corollary E

##  2\. Preliminaries

###  2.1. Equivalence relations

Let $\mathscr{R}$ be an equivalence relation defined on a standard probability
space $(X,\nu)$. Denote by $\mathscr{B}(X)$ (resp.
$\mathscr{B}(\mathscr{R})$) the $\sigma$-algebra of all Borel subsets of $X$
(resp. $\mathscr{R}$). For every $x\in X$, we denote by $[x]_{\mathscr{R}}$
the $\mathscr{R}$-equivalence class of $x\in X$. We define the $\sigma$-finite
Borel measure $m$ on $\mathscr{R}$ by the formula

$\forall\mathscr{W}\in\mathscr{B}(\mathscr{R}),\quad
m(\mathscr{W})=\int_{X}\sharp(\left\\{(x,y)\mid
y\in[x]_{\mathscr{R}}\right\\}\cap\mathscr{W})\,{\rm d}\nu(x).$

We denote by $\operatorname{Aut}(X,\nu)$ the group of all pmp Borel
automorphisms $\theta:(X,\nu)\to(X,\nu)$. We denote by
$\operatorname{Aut}(\mathscr{R})$ the automorphism group of $\mathscr{R}$ that
consists of all pmp Borel automorphisms $\theta\in\operatorname{Aut}(X,\nu)$
such that $(\theta(x),\theta(y))\in\mathscr{R}$ if and only if
$(x,y)\in\mathscr{R}$ for $(x,y)\in X\times X$. Then we denote by
$[\mathscr{R}]<\operatorname{Aut}(\mathscr{R})$ the full group of
$\mathscr{R}$ that consists of all automorphisms
$\theta\in\operatorname{Aut}(\mathscr{R})$ for which
$(\theta(x),x)\in\mathscr{R}$ for $\nu$-almost every $x\in X$. The uniform
metric

$d_{u}:[\mathscr{R}]\times[\mathscr{R}]\to\mathbb{R}_{+}:(\theta,\rho)\mapsto\nu(\\{x\in
X\mid\theta(x)\neq\rho(x)\\})$

is complete and separable and so the full group $[\mathscr{R}]$ is a Polish
group. By [FM75], there exists a countable subgroup $\Lambda<[\mathscr{R}]$
such that $\mathscr{R}=\mathscr{R}(\Lambda\curvearrowright X)$ is the orbit
equivalence relation of the pmp action $\Lambda\curvearrowright(X,\nu)$. The
full pseudogroup of $\mathscr{R}$ consists of all pmp Borel partial
isomorphisms $\theta:U\to V$, where $U,V\in\mathscr{B}(X)$ and
$(\theta(x),x)\in\mathscr{R}$ for $\nu$-almost every $x\in U$. For every
$\theta:U\to V\in[[\mathscr{R}]]$, there exists $\rho\in[\mathscr{R}]$ such
that $\rho|_{U}=\theta$. We say that $\mathscr{R}$ is ergodic if and only if
the Koopman representation
$\kappa:[\mathscr{R}]\to\mathscr{U}(\operatorname{L}^{2}(X,\nu)^{0})$ is
ergodic, where
$\operatorname{L}^{2}(X,\nu)^{0}=\operatorname{L}^{2}(X,\nu)\ominus\mathbb{C}\mathbf{1}_{X}$.

Whenever $U\subset X$ is a nonnull measurable subset, we define the
equivalence relation $\mathscr{R}|_{U}=\mathscr{R}\cap(U\times U)$ on the
standard probability space $(U,\nu_{U})$ where
$\nu_{U}=\frac{1}{\nu(U)}\nu|_{U}$. Assume that $\mathscr{R}$ is ergodic. For
every $t>0$, we define the amplification $\mathscr{R}^{t}$ as follows. Denote
by $c$ the counting measure on $\mathbb{N}$. Choose a measurable subset
$X_{t}\subset X\times\mathbb{N}$ such that $(\nu\otimes c)(X_{t})=t$ and set
$\nu_{t}=\frac{1}{(\nu\otimes c)(X_{t})}(\nu\otimes c)|_{X_{t}}$. Define the
ergodic equivalence relation $\mathscr{R}^{t}$ on the standard probability
space $(X_{t},\nu_{t})$ by declaring $((x,p),(y,q))\in\mathscr{R}^{t}$ if and
only if $(x,y)\in\mathscr{R}$ for $((x,p),(y,q))\in X_{t}\times X_{t}$.

For every $n\in\mathbb{N}\cup\\{+\infty\\}$, define the Borel subset

$X_{n}=\left\\{x\in X\mid\sharp([x]_{\mathscr{R}})=n\right\\}\subset X.$

We say that $\mathscr{R}$ is (essentially) finite if $\nu(X_{\infty})=0$. We
say that $\mathscr{R}$ is (essentially) bounded if there exists
$k\in\mathbb{N}$ such that $\nu(X_{n})=0$ for every $n\geq k$.

Let $\mathscr{S}$ be another equivalence relation defined on a standard
probability space $(Y,\eta)$. We say that $\mathscr{R}$ and $\mathscr{S}$ are
isomorphic if there exists a pmp Borel isomorphism $\theta:(X,\nu)\to(Y,\eta)$
such that for $m$-almost every $(x,y)\in\mathscr{R}$ we have
$(\theta(x),\theta(y))\in\mathscr{S}$. We say that $\mathscr{R}$ and
$\mathscr{S}$ are stably isomorphic if there exist nonnull measurable subsets
$U\subset X$ and $V\subset Y$ and a pmp Borel isomorphism
$\theta:(U,\nu_{U})\to(V,\eta_{V})$ such that for $m$-almost every
$(x,y)\in\mathscr{R}|_{U}$, we have
$(\theta(x),\theta(y))\in\mathscr{S}|_{V}$.

Let $\mathscr{S}\leq\mathscr{R}$ be a subequivalence relation. For every
$n\in\mathbb{N}\cup\\{+\infty\\}$, define the Borel subset

$Y_{n}=\left\\{x\in X\mid[x]_{\mathscr{R}}\text{ is the union of
}n\;\mathscr{S}\text{-classes}\right\\}\subset X.$

We say that $\mathscr{S}\leq\mathscr{R}$ has (essentially) finite index if
$\nu(Y_{\infty})=0$. We say that $\mathscr{S}\leq\mathscr{R}$ has
(essentially) bounded index if there exists $k\in\mathbb{N}$ such that
$\nu(Y_{n})=0$ for every $n\geq k$.

Following [FM75], we denote by $\operatorname{L}(\mathscr{R})$ the tracial von
Neumann algebra associated with $\mathscr{R}$. Then
$\operatorname{L}^{\infty}(X)\subset\operatorname{L}(\mathscr{R})$ is a Cartan
subalgebra. We simply denote by
$\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}\subset\mathscr{U}(\operatorname{L}(\mathscr{R}))$
the full group of $\mathscr{R}$ regarded as a subgroup of the unitary group
$\mathscr{U}(\operatorname{L}(\mathscr{R}))$. Then we have

$\operatorname{L}(\mathscr{R})=\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}^{\prime\prime}\vee\operatorname{L}^{\infty}(X).$

######  Facts 2.1.

Keep the same notation as above. We will be using the following useful
properties.

  * $(\rm i)$

The equivalence relation $\mathscr{R}$ is ergodic if and only if the Koopman
representation
$\kappa:[\mathscr{R}]\to\mathscr{U}(\operatorname{L}^{2}(X,\nu)^{0})$ is
weakly mixing, where
$\operatorname{L}^{2}(X,\nu)^{0}=\operatorname{L}^{2}(X,\nu)\ominus\mathbb{C}\mathbf{1}_{X}$.

  * $(\rm ii)$

If $\mathscr{R}$ is ergodic and $(X,\nu)$ is diffuse, then
$\operatorname{L}(\mathscr{R})=\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}^{\prime\prime}$.

  * $(\rm iii)$

Let $\Psi\in\operatorname{Aut}(X,\nu)$ be a pmp Borel automorphism and denote
by $\mathscr{R}_{\Psi}$ the equivalence relation generated by $\Psi$. Then for
any $\varepsilon>0$, there exist a measurable subset $X_{0}\subset X$ with
$\nu(X\setminus X_{0})\leq\varepsilon$, a positive integer $p$ and
$\Psi_{0}\in[\mathscr{R}_{\Psi}]$ such that $\Psi_{0}=\Psi$ on $X_{0}$ and
$\Psi_{0}^{p}=\operatorname{id}_{X}$.

###### Proof.

$(\rm i)$ This follows from [LM24, Theorem 11.20].

$(\rm ii)$ Indeed, it suffices to prove that
$\operatorname{L}^{\infty}(X)\subset\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}^{\prime\prime}$.
Let $Y\subset X$ be a measurable subset such that $\nu(Y)<1$. Set
$Z=X\setminus Y$ and $\mathscr{S}=\mathscr{R}|_{Z}$. Since $\mathscr{S}$ is
ergodic and $(Z,\nu_{Z})$ is diffuse, there exists an ergodic type ${\rm
II_{1}}$ hyperfinite subequivalence relation $\mathscr{T}\leq\mathscr{S}$ (see
e.g. [Zi84, Proposition 9.3.2]). By [CFW81], choose a free ergodic pmp action
$\mathbb{Z}\curvearrowright(Z,\nu_{Z})$ such that
$\mathscr{T}=\mathscr{R}(\mathbb{Z}\curvearrowright Z)$. Denote by
$\rho\in[\mathscr{T}]$ the generator of
$\mathbb{Z}\curvearrowright(Z,\nu_{Z})$. For every $n\in\mathbb{N}$, define
$\theta_{n}\in[\mathscr{R}]$ by $\theta_{n}(x)=x$ if $x\in Y$ and
$\theta_{n}(x)=\rho^{n}(x)$ if $x\in Z$. Then it is plain to see that
$u_{\theta_{n}}\to\mathbf{1}_{Y}$ weakly as $n\to\infty$. This shows that
$\operatorname{L}(\mathscr{R})=\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}^{\prime\prime}$.

$(\rm iii)$ Since $\mathscr{R}_{\Psi}$ is hyperfinite by Rokhlin’s lemma, it
follows that there exists an increasing sequence of finite equivalence
relations $\mathscr{T}_{1}\leq\mathscr{T}_{2}\leq\dots$ such that (up to a
conull Borel subset) we have $\mathscr{R}_{\Psi}=\bigcup_{k\geq
1}\mathscr{T}_{k}$. Then for $\nu$-almost every $x\in X$, we have
$[x]_{\mathscr{R}_{\Psi}}=\bigcup_{k\geq 1}[x]_{\mathscr{T}_{k}}$, and thus,
there exists $k\geq 1$ such that $\Psi(x)\in[x]_{\mathscr{T}_{k}}$. Next,
define the measurable sets $X_{k}=\\{x\in
X\mid\Psi(x)\in[x]_{\mathscr{T}_{k}}\\}$, $k\geq 1$ and note that
$X=\bigcup_{k\geq 1}X_{k}$. Take $N\geq 1$ such that $\nu(X\setminus
X_{N})\leq\varepsilon$ and let
$\Psi_{0}\in[\mathscr{T}_{N}]\leq[\mathscr{R}_{\Psi}]$ be such that
$\Psi_{0}=\Psi$ on $X_{N}$. By construction, it follows that the equivalence
relation generated by $\Psi_{0}$ is finite, hence, the lemma follows by
letting $X_{0}=X_{N}$. ∎

###  2.2. Popa’s intertwining theory for equivalence relations

We review Popa’s criterion for intertwining von Neumann subalgebras [Po01,
Po03]. Let $(M,\tau)$ be a tracial von Neumann algebra and $A\subset
1_{A}M1_{A}$, $B\subset 1_{B}M1_{B}$ be von Neumann subalgebras. Let
$\mathcal{G}\subset\mathscr{U}(A)$ be a subgroup such that
$\mathscr{G}^{\prime\prime}=A.$ By [Po03, Corollary 2.3] and [Po01, Theorem
A.1] (see also [Va06, Proposition C.1]), the following conditions are
equivalent:

  * $(\rm i)$

There exist $d\geq 1$, a projection $q\in\operatorname{M}_{d}(B)$, a nonzero
partial isometry $v\in\operatorname{M}_{1,d}(1_{A}M)q$ and a unital normal
$\ast$-homomorphism $\pi:A\to q\operatorname{M}_{d}(B)q$ such that
$av=v\pi(a)$ for all $a\in A$.

  * $(\rm ii)$

There is no net of unitaries $(w_{n})_{n}$ in $\mathscr{G}$ such that

$\forall x,y\in
1_{A}M1_{B},\quad\lim_{n}\|\operatorname{E}_{B}(x^{*}w_{n}y)\|_{2}=0.$

If one of the previous equivalent conditions is satisfied, we say that $A$
embeds into $B$ inside $M$ and write $A\preceq_{M}B$. Moreover, if the
inclusion $B\subset M$ is unital, then the above are also equivalent to:

  * $(\rm iii)$

The unitary representation
$\pi:\mathscr{G}\to\mathscr{U}(\operatorname{L}^{2}(\langle
M,e_{B}\rangle,\operatorname{Tr}))$ given by $\pi_{u}(\xi)=u\xi u^{*}$ for
$u\in\mathscr{G}$, $\xi\in\operatorname{L}^{2}(\langle
M,e_{B}\rangle,\operatorname{Tr})$, is not weakly mixing. Here, we denoted by
${\rm Tr}$ the associated semifinite trace on Jones’ basic construction
$\langle M,e_{B}\rangle$.

If for every nonzero projection $p\in A^{\prime}\cap 1_{A}M1_{A}$ we have
$Ap\preceq_{M}B$, then we write $A\preceq^{s}_{M}B$.

Popa’s criterion for intertwining von Neumann subalgebras was adapted to the
setting of equivalence relations by Ioana [Io11]. Let $\mathscr{R}$ be an
equivalence relation defined on a standard probability space $(X,\nu)$. Let
$Y,Z\subset X$ be nonnull measurable subsets and
$\mathscr{S}\leq\mathscr{R}|_{Y}$ and $\mathscr{T}\leq\mathscr{R}|_{Z}$ be
subequivalence relations. Following [IKT08], define the map

$\varphi_{\mathscr{T}}:[[\mathscr{R}]]\to[0,1]:\theta\mapsto\nu(\\{x\in\operatorname{dom}(\theta)\mid(\theta(x),x)\in\mathscr{T}\\}).$

By [Io11, Lemma 1.7] (see also [Sp21, Lemma 3.1]), the following conditions
are equivalent:

  * $(\rm i)$

There exist a $\mathscr{S}$-invariant nonnull measurable subset $Y_{0}\subset
Y$ and a subequivalence relation $\mathscr{S}_{0}\leq\mathscr{S}$ such that
for any nonnull measurable subset $Y_{1}\subset Y_{0}$, there is a nonnull
measurable subset $U\subset Y_{1}$ and $\theta\in[[\mathscr{R}]]$ with
$\theta:U\to V$, such that

    * –

$\mathscr{S}_{0}|_{U}\leq\mathscr{S}|_{U}$ has bounded index, and

    * –

$(\theta\times\theta)(\mathscr{S}_{0}|_{U})\leq\mathscr{T}|_{V}$.

  * $(\rm ii)$

There is no sequence $(\theta_{n})_{n}$ in $[\mathscr{S}]$ such that

$\forall\psi,\rho\in[[\mathscr{R}]],\quad\lim_{n}\varphi_{\mathscr{T}}(\psi\theta_{n}\rho)=0.$

If one of the previous equivalent conditions is satisfied, we say that
$\mathscr{S}$ embeds into $\mathscr{T}$ inside $\mathscr{R}$ and write
$\mathscr{S}\preceq_{\mathscr{R}}\mathscr{T}$.

######  Facts 2.2.

Keep the same notation as above. We will be using the following useful
properties.

  * $(\rm i)$

We have $\mathscr{S}\preceq_{\mathscr{R}}\mathscr{T}$ if and only
$\operatorname{L}(\mathscr{S})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{T})$.

  * $(\rm ii)$

If $\mathscr{S}_{0}\leq\mathscr{S}$ is a subequivalence relation and
$\mathscr{S}\preceq_{\mathscr{R}}\mathscr{T}$, then we have
$\mathscr{S}_{0}\preceq_{\mathscr{R}}\mathscr{T}$.

For every $i\in\\{1,2\\}$, let $(Z_{i},\nu_{i})$ be a standard probability
space such that $(X,\nu)\cong(Z_{1}\times Z_{2},\nu_{1}\otimes\nu_{2})$. Let
$Z\subset Z_{1}$ be a nonnull measurable subset and consider
$\nu_{Z}=\frac{1}{\nu_{1}(Z)}\nu_{1}|_{Z}$. Let $\mathscr{S}$ be an ergodic
equivalence relation on $(Z,\nu_{Z})$ such that
$\mathscr{S}\times\operatorname{id}_{Z_{2}}\leq\mathscr{R}|_{Z\times Z_{2}}$.
Denote by
$\kappa:[\mathscr{S}]\to\mathscr{U}(\operatorname{L}^{2}(Z,\nu_{Z})^{0})$ the
Koopman representation. Let $\mathscr{T}\leq\mathscr{R}$ be a subequivalence
relation.

  * $(\rm iii)$

Assume that
$\mathscr{S}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{R}}\mathscr{T}$.
Then for all $\varepsilon>0$, $n\geq 1$,
$f_{1},\dots,f_{n}\in\operatorname{L}^{2}(Z,\nu_{Z})^{0}$ and
$\psi_{1},\dots,\psi_{n}\in[\mathscr{R}]$, there exists
$\theta\in[\mathscr{S}]$ such that
$|\langle\kappa_{\theta}(f_{i}),f_{j}\rangle|\leq\varepsilon$ and
$\varphi_{\mathscr{T}}(\psi_{i}(\theta\times\operatorname{id}_{Z_{2}})\psi_{j})\leq\varepsilon$
for all $1\leq i,j\leq n$.

For every $i\in\\{1,2\\}$, let $\mathscr{R}_{i}$ be an ergodic equivalence
relation on $(X_{i},\nu_{i})$ and $\mathscr{S}_{i}$ an ergodic equivalence
relation on $(Y_{i},\eta_{i})$. Assume that
$\mathscr{R}_{1}\times\mathscr{S}_{1}\cong\mathscr{R}_{2}\times\mathscr{S}_{2}$.
Set $(Z,\zeta)=(X_{1}\times Y_{1},\nu_{1}\otimes\eta_{1})\cong(X_{2}\times
Y_{2},\nu_{2}\otimes\eta_{2})$ and
$\mathscr{R}=\mathscr{R}_{1}\times\mathscr{S}_{1}\cong\mathscr{R}_{2}\times\mathscr{S}_{2}$.
We simply write $\mathscr{R}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{2}$ as
subequivalence relations if
$\mathscr{R}_{1}\times\operatorname{id}_{Y_{1}}\preceq_{\mathscr{R}}\mathscr{R}_{2}\times\operatorname{id}_{Y_{2}}$.

  * $(\rm iv)$

If $\mathscr{S}_{1}\preceq_{\mathscr{R}}\mathscr{S}_{2}$ as subequivalence
relations, then we have

$\operatorname{L}^{\infty}(X_{2})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}^{\infty}(X_{1}).$

  * $(\rm v)$

If $\mathscr{T}_{1}\leq\mathscr{S}_{1}$ is an ergodic subequivalence relation
and $\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{S}_{2}$ as subequivalence
relations, then we have $\mathscr{S}_{1}\preceq_{\mathscr{R}}\mathscr{S}_{2}$
as subequivalence relations.

  * $(\rm vi)$

If $\mathscr{R}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{2}$ as subequivalence
relations, then there exists an ergodic equivalence relation $\mathscr{V}$
such that $\mathscr{R}_{1}\times\mathscr{V}$ and $\mathscr{R}_{2}$ are stably
isomorphic.

###### Proof.

$(\rm i)$ This follows from [Io11, Lemma 1.8].

$(\rm ii)$ It is obvious from the definition.

$(\rm iii)$ Since $\mathscr{S}$ is ergodic, it follows by Facts 2.1$(\rm
i),(\rm ii)$ that $\operatorname{L}(\mathscr{S})=[\mathscr{S}]^{\prime\prime}$
and that the Koopman representation
$\kappa:[\mathscr{S}]\to\mathscr{U}(\operatorname{L}^{2}(Z,\nu_{Z})^{0})$ is
weakly mixing. Denote by
$\left\\{u_{\theta}\mid\theta\in[\mathscr{R}]\right\\}\subset\mathscr{U}(\operatorname{L}(\mathscr{R}))$
the full group of $\mathscr{R}$ regarded as a subgroup of the unitary group
$\mathscr{U}(\operatorname{L}(\mathscr{R}))$. By assumption, we have
$\operatorname{L}(\mathscr{S})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Z_{2})\npreceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{T})$,
which implies that there exist two sequences of unitaries $(v_{n})_{n\geq
1}\subset\mathscr{U}(\operatorname{L}(\mathscr{S}))$ and $(w_{n})_{n\geq
1}\subset\mathscr{U}(\operatorname{L}^{\infty}(Z_{2}))$ such that
$\|\operatorname{E}_{\operatorname{L}(\mathscr{T})}(xv_{n}w_{n}u_{\theta})\|_{2}\to
0$, for all $x\in\operatorname{L}(\mathscr{R})$ and $\theta\in[\mathscr{R}]$.
Since
$u_{\theta}^{*}w_{n}u_{\theta}=w_{n}\circ\theta\in\mathscr{U}(\operatorname{L}^{\infty}(X))$,
for any $n\geq 1$, and
$\operatorname{L}^{\infty}(X)\subset\operatorname{L}(\mathscr{T})$, it follows
that
$\|\operatorname{E}_{\operatorname{L}(\mathscr{T})}(xv_{n}au_{\theta})\|_{2}\to
0$, for all $x\in\operatorname{L}(\mathscr{R})$,
$a\in\operatorname{L}^{\infty}(X)$ and $\theta\in[\mathscr{R}]$. This implies
that
$\operatorname{L}(\mathscr{S})\npreceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{T})$.
This further implies that the unitary representation
$\pi:[\mathscr{S}]\to\mathscr{U}(\operatorname{L}^{2}(\langle\operatorname{L}(\mathscr{R}),e_{\operatorname{L}(\mathscr{T})}\rangle,\operatorname{Tr}))$
given by $\pi_{u}(\xi)=u\xi u^{*}$ for $u\in[\mathscr{S}]$,
$\xi\in\operatorname{L}^{2}(\langle\operatorname{L}(\mathscr{R}),e_{\operatorname{L}(\mathscr{T})}\rangle,\operatorname{Tr})$,
is weakly mixing. Since
$\|\operatorname{E}_{\operatorname{L}(\mathscr{T})}(u_{\theta})\|_{2}^{2}=\varphi_{\mathscr{T}}(\theta)$
for any $\theta\in[\mathscr{R}]$ and since the direct sum $\kappa\oplus\pi$ is
a weakly mixing unitary representation of $[\mathscr{S}]$ as well, the
conclusion follows.

$(\rm iv)$ This follows from [Sp21, Corollary 3.4]

$(\rm v)$ If $\mathscr{T}_{1}\leq\mathscr{S}_{1}$ is an ergodic subequivalence
relation and $\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{S}_{2}$ as
subequivalence relations, then we have
$\operatorname{L}^{\infty}(X_{1})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{T}_{1})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}^{\infty}(X_{2})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{S}_{2})$
and so
$\operatorname{L}^{\infty}(X_{2})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}^{\infty}(X_{1})$
by [Va07, Lemma 3.5]. By applying again [Va07, Lemma 3.5], we infer that
$\operatorname{L}^{\infty}(X_{1})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{S}_{1})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}^{\infty}(X_{2})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{S}_{2})$
and so $\mathscr{S}_{1}\preceq_{\mathscr{R}}\mathscr{S}_{2}$ as subequivalence
relations.

$(\rm vi)$ This follows from [Sp21, Proposition 3.6]. ∎

The following result and its proof are inspired by [BO08, Corollary F.14]. It
will turn out to be useful in the proof of Theorem D.

######  Lemma 2.3.

Let $\mathscr{R}$ be an equivalence relation on $(X,\nu)$. For every
$i\in\\{1,2\\}$, let $(Z_{i},\nu_{i})$ be a standard probability space such
that $(X,\nu)\cong(Z_{1}\times Z_{2},\nu_{1}\otimes\nu_{2})$. Let
$\mathscr{S}$ be an ergodic equivalence relation on $(Z_{1},\nu_{1})$ such
that $\mathscr{S}\times\operatorname{id}_{Z_{2}}\leq\mathscr{R}$ and
$\mathscr{T}\leq\mathscr{R}$ a subequivalence relation.

If
$\mathscr{S}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{R}}\mathscr{T}$,
then there exists a hyperfinite ergodic subequivalence relation
$\mathscr{S}_{0}\leq\mathscr{S}$ such that
$\mathscr{S}_{0}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{R}}\mathscr{T}$.

###### Proof.

Choose a countable dense subset $\\{g_{n}\mid n\geq 1\\}$ in $[\mathscr{R}]$
with respect to the uniform metric $d_{u}$. Let $(f_{n})_{n\geq 1}$ be a dense
sequence in $\operatorname{L}^{2}(Z_{1},\nu_{1})^{0}$. For any measurable
subset $Z\subset Z_{1}$, we set $\nu_{Z}=\frac{1}{\nu_{1}(Z)}\nu_{1}|_{Z}$ and
we let
$\kappa^{Z}:[\mathscr{S}|_{Z}]\to\mathscr{U}(\operatorname{L}^{2}(Z,\nu_{Z})^{0})$
be the associated Koopman representation. The proof of the lemma will be
concluded by constructing an increasing sequence $(\mathscr{S}_{n})_{n}$ of
bounded subequivalence relations of $\mathscr{S}$ and a sequence
$(\theta_{n})_{n}$ in $[\mathscr{S}]$ such that for all $n\geq 1$ and all
$1\leq i,j\leq n$, we have $\theta_{n}\in[\mathscr{S}_{n}]$ and

$\varphi_{\mathscr{T}}(g_{i}(\theta_{n}\times\operatorname{id}_{Z_{2}})g_{j})\leq\frac{1}{n}\quad\text{and}\quad|\langle\kappa_{\theta_{n}}(f_{i}),f_{j}\rangle|\leq\frac{1}{n}.$

Let $\mathscr{S}_{1}\leq\mathscr{S}$ be the trivial subequivalence relation,
$\theta_{1}=\operatorname{id}_{Z_{1}}$ and assume that
$\mathscr{S}_{1},\dots,\mathscr{S}_{n-1}$ and $\theta_{1},\dots,\theta_{n-1}$
have been constructed. Since $\mathscr{S}_{n-1}$ is bounded, there exists a
finite measurable partition $Z_{1}=X_{1}\sqcup\dots\sqcup X_{N}$ into
$\mathscr{S}_{n-1}$-invariant sets such that $\mathscr{S}_{n-1}|_{X_{k}}$ is
of type ${\rm I}_{t_{k}}$ for any $1\leq k\leq N$. For any $1\leq k\leq N$,
let $Y_{k}\subset X_{k}$ be a fundamental domain for
$\mathscr{S}_{n-1}|_{X_{k}}$ and let
$\omega_{k}\in[\mathscr{S}_{n-1}|_{X_{k}}]$ such that
$X_{k}=\sqcup_{r=0}^{t_{k}-1}\omega_{k}^{r}Y_{k}$.

Let $1\leq k\leq N$. Since
$\mathscr{S}|_{Y_{k}}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{R}}\mathscr{T}$,
Facts 2.2$(\rm iii)$ implies that there exists
$\Psi_{k}\in[\mathscr{S}|_{Y_{k}}]$ such that for all $0\leq r\leq t_{k}-1$
and $1\leq i,j\leq n$, we have

(2.1)
$\varphi_{\mathscr{T}}(g_{i}(\omega_{k}^{r}\Psi_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})\leq\frac{1}{2nt_{k}N}\quad$

and

(2.2)
$\quad|\langle\kappa^{Y_{k}}_{\Psi_{k}}((f_{i}\circ\omega_{k}^{r})|_{Y_{k}}),(f_{j}\circ\omega_{k}^{r})|_{Y_{k}}\rangle|\leq\frac{1}{n}.$

Next, by applying Facts 2.1$(\rm iii)$, there exist a measurable subset
$\tilde{Y}_{k}\subset Y_{k}$ with
$\nu_{1}(Y_{k}\setminus\tilde{Y}_{k})\leq\frac{1}{2nt_{k}N}$, a positive
integer $p_{k}$ and $\tilde{\Psi}_{k}\in[\mathscr{S}|_{Y_{k}}]$ such that
$\tilde{\Psi}_{k}=\Psi_{k}$ on $\tilde{Y}_{k}$ and
$\tilde{\Psi}_{k}^{p_{k}}=\operatorname{id}_{Y_{k}}$. Fix $0\leq r\leq
t_{k-1}$. Since
$g_{i}(\omega_{k}^{r}\tilde{\Psi}_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j}=g_{i}(\omega_{k}^{r}\Psi_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j}$
on $g_{j}^{-1}(\omega_{k}^{r}\tilde{Y}_{k}\times Z_{2})$, it follows that for
all $1\leq i,j\leq n$, we have

$\displaystyle\varphi_{\mathscr{T}}(g_{i}(\omega_{k}^{r}\tilde{\Psi}_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})\leq\nu((Y_{k}\setminus\tilde{Y}_{k})\times
Z_{2})+$ $\displaystyle\nu(\\{x\in g_{j}^{-1}(\tilde{Y}_{k}\times
Z_{2})\mid((g_{i}(\omega_{k}^{r}\Psi_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})(x),x)\in\mathscr{T}\\}),$

and thus, using (2.1) we have

(2.3)
$\varphi_{\mathscr{T}}(g_{i}(\omega_{k}^{r}\tilde{\Psi}_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})\leq\frac{1}{nt_{k}N}.$

Next, since $Y_{k}\subset X_{k}$ is a fundamental domain for
$\mathscr{S}_{n-1}|_{X_{k}}$, we can extend
$\tilde{\Psi}_{k}\in[\mathscr{S}|_{Y_{k}}]$ naturally to an element
$\Omega_{k}\in[\mathscr{S}|_{X_{k}}]$. More precisely, we let
$\Omega_{k}(\omega_{k}^{r}y)=\omega_{k}^{r}\tilde{\Psi}_{k}(y)$, for all $y\in
Y_{k}$ and $0\leq r\leq t_{k}-1$. In this way, since
$\nu_{X_{k}}(Y_{k})=\frac{1}{t_{k}}$, for all $1\leq i,j\leq n$, we deduce
from (2.3) and (2.2) that

(2.4)
$\varphi_{\mathscr{T}}(g_{i}(\Omega_{k}\times\operatorname{id}_{Z_{2}})g_{j})\leq\frac{1}{nN}\quad\text{and}\quad|\langle\kappa^{X_{k}}_{\Omega_{k}}(f_{i}|_{X_{k}}),f_{j}|_{X_{k}}\rangle|\leq\frac{1}{n}.$

Indeed, note that (2.3) implies that

$\displaystyle\varphi_{\mathscr{T}}(g_{i}(\Omega_{k}\times\operatorname{id}_{Z_{2}})g_{j})$
$\displaystyle=\sum_{r=0}^{t_{k}-1}\nu(\\{x\in
g_{j}^{-1}(\omega_{k}^{r}Y_{k}\times
Z_{2})\mid((g_{i}(\Omega_{k}\times\operatorname{id}_{Z_{2}})g_{j})(x),x)\in\mathscr{T}\\})$
$\displaystyle=\sum_{r=0}^{t_{k}-1}\nu(\\{x\in
g_{j}^{-1}(\omega_{k}^{r}Y_{k}\times
Z_{2})\mid((g_{i}(\omega_{k}^{r}\tilde{\Psi}_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})(x),x)\in\mathscr{T}\\})$
$\displaystyle\leq\sum_{r=0}^{t_{k}-1}\varphi_{\mathscr{T}}(g_{i}(\omega_{k}^{r}\tilde{\Psi}_{k}\omega_{k}^{-r}\times\operatorname{id}_{Z_{2}})g_{j})\leq\frac{1}{nN}.$

The second part of (2.4) follows in a similar way and we omit the details.

Next, define $\theta_{n}\in\operatorname{Aut}(Z_{1},\nu_{1})$ by
$\theta_{n}=\Omega_{k}$ on $X_{k}$ for any $1\leq k\leq N$. By construction,
we deduce that $\theta_{n}\in[\mathscr{S}]$ and that there exists an integer
$p$ (e.g. take $p=p_{1}\cdots p_{N}$) for which
$\theta_{n}^{p}=\operatorname{id}_{Z_{1}}$. Note that for all $1\leq i,j\leq
n$, it follows from (2.4) that

$\displaystyle\varphi_{\mathscr{T}}(g_{i}(\theta_{n}\times\operatorname{id}_{Z_{2}})g_{j})$
$\displaystyle=\nu(\bigcup_{k=1}^{N}\\{x\in g_{j}^{-1}(X_{k}\times
Z_{2})\mid((g_{i}(\theta_{n}\times\operatorname{id}_{Z_{2}})g_{j})(x),x)\in\mathscr{T}\\})$
$\displaystyle\leq\sum_{k=1}^{N}\nu(\\{x\in g_{j}^{-1}(X_{k}\times
Z_{2})\mid((g_{i}(\Omega_{k}\times\operatorname{id}_{Z_{2}})g_{j})(x),x)\in\mathscr{T}\\})$
$\displaystyle=\sum_{k=1}^{N}\varphi_{\mathscr{T}}(g_{i}(\Omega_{k}\times\operatorname{id}_{Z_{2}})g_{j})\leq\sum_{k=1}^{N}\frac{1}{nN}=\frac{1}{n}.$

We derive in a similar way using (2.4) that for all $1\leq i,j\leq n$, we have

(2.5)
$|\langle\kappa^{Z_{1}}_{\theta_{n}}(f_{i}),f_{j}\rangle|\leq\frac{1}{n}.$

We can now define $\mathscr{S}_{n}$ as the subequivalence relation of
$\mathscr{S}$ generated by $\mathscr{S}_{n-1}$ and $\theta_{n}$. Since
$\theta_{n}|_{X_{k}}$ commutes with $\omega_{k}$ for any $1\leq k\leq N$, one
can deduce that $\mathscr{S}_{n}$ is bounded. By letting
$\mathscr{S}_{0}=\bigcup_{k\geq 1}\mathscr{S}_{k}$, we derive that
$\mathscr{S}_{0}$ is hyperfinite and that
$\mathscr{S}_{0}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{R}}\mathscr{T}$.
By (2.5), the Koopman representation associated with $[\mathscr{S}_{0}]$ is
weakly mixing. This implies by Facts 2.1(i) that $\mathscr{S}_{0}$ is ergodic.
∎

The notion of strong primeness for type ${\rm II_{1}}$ factors was introduced
by Isono [Is16]. We adapt this notion to the setting of ergodic equivalence
relations.

######  Definition 2.4.

Let $\mathscr{R}$ be an ergodic equivalence relation. We say that
$\mathscr{R}$ is strongly prime if for any ergodic equivalence relations
$\mathscr{S}$, $\mathscr{T}_{1}$, $\mathscr{T}_{2}$ for which
$\mathscr{R}\times\mathscr{S}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, then
there exists $i\in\\{1,2\\}$ such that we have
$\mathscr{T}_{i}\preceq_{\mathscr{R}\times\mathscr{S}}\mathscr{S}$ as
subequivalence relations.

###  2.3. Algebraic groups

Let $k$ be a local field, that is, a nondiscrete locally compact field.
Throughout, we assume that the characteristic of $k$ is zero. Then $k$ is
$\mathbb{R}$, $\mathbb{C}$ or a finite extension of $\mathbb{Q}_{p}$ for some
prime $p$. In this paper, by a $k$-group $\mathbf{G}$, we simply mean a linear
algebraic group defined over $k$. We denote by $\mathbf{G}^{0}$ the Zariski
connected component of the identity element $e\in\mathbf{G}$. We say that
$\mathbf{G}$ is Zariski connected if $\mathbf{G}=\mathbf{G}^{0}$. We say that
$\mathbf{G}$ is almost $k$-simple if $\mathbf{G}$ is not abelian and the only
proper normal $k$-closed subgroups are finite. In that case, since
$\mathbf{G}$ is connected, finite normal subgroups are contained in
$\mathscr{Z}(\mathbf{G})$. We say that $\mathbf{G}$ is $k$-simple if
$\mathbf{G}$ is not abelian and the only normal $k$-closed subgroups are
$\\{e\\}$ and $\mathbf{G}$. If $\mathbf{G}$ is (almost) $k$-simple, then we
moreover say that $\mathbf{G}$ is $k$-isotropic if
$\operatorname{rk}_{k}(\mathbf{G})\geq 1$. We denote by $G=\mathbf{G}(k)$ the
locally compact second countable group of its $k$-points. Then $G$ is a
$k$-analytic Lie group and we naturally have
$\operatorname{Lie}(\mathbf{G})(k)=\operatorname{Lie}(G)$ as $k$-Lie algebras.

By a $k$-$\mathbf{G}$-variety $\mathbf{V}$, we simply mean an algebraic
variety defined over $k$ that is endowed with an algebraic action
$\mathbf{G}\curvearrowright\mathbf{V}$ so that the action map
$\mathbf{G}\times\mathbf{V}\to\mathbf{V}:(g,v)\mapsto gv$ is a $k$-morphism.
We denote by $V=\mathbf{V}(k)$ the Hausdorff locally compact second countable
topological space of its $k$-points. If $\mathbf{V}$ is smooth, then $V$ is a
$k$-analytic manifold. In the case $\mathbf{V}=\mathbf{G}/\mathbf{H}$, where
$\mathbf{H}<\mathbf{G}$ is a $k$-subgroup, we denote by
$\pi:\mathbf{G}\to\mathbf{G}/\mathbf{H}$ the $\mathbf{G}$-equivariant
$k$-morphism such that $\pi(e)=\mathbf{H}$. Then
$v=\pi(e)\in(\mathbf{G}/\mathbf{H})(k)$ and
$H=\mathbf{H}(k)=\operatorname{Stab}_{G}(v)$.

######  Facts 2.5.

Keep the same notation as above. We will be using the following useful
properties.

  * $(\rm i)$

The continuous action $G\curvearrowright V$ has locally closed orbits for the
$k$-analytic topology and so the quotient space

$G\backslash V=\left\\{Gv\mid v\in V\right\\}$

is a standard Borel space.

  * $(\rm ii)$

The continuous action $G\curvearrowright\operatorname{Prob}(V)$ has locally
closed orbits and so the quotient space

$G\backslash\operatorname{Prob}(V)=\left\\{G\mu\mid\mu\in\operatorname{Prob}(V)\right\\}$

is a standard Borel space.

  * $(\rm iii)$

Assume that $\mathbf{V}=\mathbf{G}/\mathbf{H}$, where $\mathbf{H}<\mathbf{G}$
is a $k$-subgroup. Set $v=\pi(e)\in(\mathbf{G}/\mathbf{H})(k)$ and
$H=\mathbf{H}(k)=\operatorname{Stab}_{G}(v)$. The orbit map $G/H\to Gv:gH\to
gv$ is a homeomorphism, we may identify the $G$-orbit $Gv$ with $G/H$ and we
may regard $G/H\subset(\mathbf{G}/\mathbf{H})(k)$ as a closed and open subset.

###### Proof.

$(\rm i)$ This follows from [Zi84, Theorem 2.1.14, Proposition 3.1.3].

$(\rm ii)$ This follows from [Zi84, Theorem 2.1.14, Proposition 3.2.6] and
more generally [BDL14, Theorem 1.7].

$(\rm iii)$ This follows from [Ma91, Proposition I.2.1.4]. ∎

###  2.4. Algebraic representations of equivalence relations

Let $\mathscr{R}$ be an equivalence relation on a standard probability space
$(X,\nu)$, $H$ a locally compact second countable group and
$\alpha:\mathscr{R}\to H$ a measurable $1$-cocycle. Let $Z$ be a standard
Borel space endowed with a Borel action $H\curvearrowright Z$ and $f:X\to Z$ a
Borel map. We say that $f:X\to Z$ is $(\mathscr{R},\alpha)$-equivariant if for
$m$-almost every $(x,y)\in\mathscr{R}$, we have $f(x)=\alpha(x,y)f(y)$. We say
that $f:X\to Z$ is $\mathscr{R}$-invariant if for $m$-almost every
$(x,y)\in\mathscr{R}$, we have $f(x)=f(y)$.

The following dichotomy theorem generalizes Zimmer’s result [Zi84, Theorem
9.2.3] (see also [BDL14, Theorem 6.1] for the case of measurable $1$-cocycles
associated with nonsingular group actions). For the sake of completeness, we
give a short proof.

######  Theorem 2.6.

Let $k$ be a local field of characteristic zero and $\mathbf{G}$ a
$k$-isotropic almost $k$-simple algebraic $k$-group. Let $\mathscr{R}$ be an
amenable ergodic equivalence relation on $(X,\nu)$ and $\alpha:\mathscr{R}\to
G$ a measurable $1$-cocycle. Then at least one of the following assertions
holds:

  * $(\rm i)$

There exist a proper $k$-subgroup $\mathbf{H}<\mathbf{G}$ and an
$(\mathscr{R},\alpha)$-equivariant measurable map $\psi:X\to G/H$, where we
regard $G/H\subset(\mathbf{G}/\mathbf{H})(k)$ with $H=\mathbf{H}(k)$.

  * $(\rm ii)$

There exist a compact subgroup $L<G$ and an $(\mathscr{R},\alpha)$-equivariant
measurable map $f:X\to G/L$.

Observe that in the case $k=\mathbb{R}$, assertion $(\rm i)$ always holds.

###### Proof.

Choose a minimal parabolic $k$-subgroup $\mathbf{P}<\mathbf{G}$ and denote by
$\mathbf{V}=\mathbf{G}/\mathbf{P}$ the homogeneous $k$-$\mathbf{G}$-variety.
Write $G=\mathbf{G}(k)$, $P=\mathbf{P}(k)$ and
$V=(\mathbf{G}/\mathbf{P})(k)=\mathbf{G}(k)/\mathbf{P}(k)$ (see [Bo91,
Proposition 20.5]). Since $V$ is compact and since $\mathscr{R}$ is amenable,
there exists an $(\mathscr{R},\alpha)$-equivariant measurable map
$\beta:X\to\operatorname{Prob}(V)$ (see [Zi84, Proposition 4.3.9]). By Facts
2.5$(\rm ii)$, the continuous action $G\curvearrowright\operatorname{Prob}(V)$
has locally closed orbits and so the quotient space
$G\backslash\operatorname{Prob}(V)$ is a standard Borel space. Denote by
$p:\operatorname{Prob}(V)\to G\backslash\operatorname{Prob}(V)$ the quotient
Borel map. Then the measurable map $p\circ\beta:X\to
G\backslash\operatorname{Prob}(V)$ is $\mathscr{R}$-invariant. Since
$\mathscr{R}$ is ergodic, $p\circ\beta:X\to G\backslash\operatorname{Prob}(V)$
is $\nu$-almost everywhere constant and so there exists
$\mu\in\operatorname{Prob}(V)$ such that $\beta(X)$ is essentially contained
in $G\mu$. Set $L=\operatorname{Stab}_{G}(\mu)<G$. By [Zi84, Theorem 2.1.14],
the orbit map $G/L\to G\mu:gL\to g\mu$ is a homeomorphism and so we may regard
$\beta:X\to G/L$ as an $(\mathscr{R},\alpha)$-equivariant measurable map.

Denote by $\mathbf{H}$ the Zariski closure of $L$ in $\mathbf{G}$. Then
$\mathbf{H}<\mathbf{G}$ is a $k$-subgroup and we set $H=\mathbf{H}(k)$. By
[BDL14, Proposition 1.9], there exists a $k$-subgroup
$\mathbf{H}_{0}<\mathbf{H}<\mathbf{G}$ such that
$\mathbf{H}_{0}\lhd\mathbf{H}$ is normal, the image of $L$ is precompact in
$(\mathbf{H}/\mathbf{H}_{0})(k)$ and $\mu$ is supported on
$\mathbf{V}^{\mathbf{H}_{0}}\cap V$. Since
$\mathbf{V}^{\mathbf{G}}=\emptyset$, we have $\mathbf{H}_{0}\neq\mathbf{G}$.

Assume that $\mathbf{H}_{0}$ is infinite. Since
$\mathbf{H}_{0}\lhd\mathbf{H}$, $\mathbf{H}_{0}\neq\mathbf{G}$ and
$\mathbf{G}$ is almost $k$-simple, we have $\mathbf{H}\neq\mathbf{G}$. Since
$L<H$, we may consider the $G$-equivariant measurable factor map $q:G/L\to
G/H$. Then the measurable map $f=q\circ\beta:X\to G/H$ is
$(\mathscr{R},\alpha)$-equivariant.

Assume that $\mathbf{H}_{0}$ is finite. Set $H_{0}=\mathbf{H}_{0}(k)$. The
image of $L$ in $(\mathbf{H}/\mathbf{H}_{0})(k)$ is contained in
$H/H_{0}\subset(\mathbf{H}/\mathbf{H}_{0})(k)$, which is closed in
$(\mathbf{H}/\mathbf{H}_{0})(k)$. Then the image of $L$ in $H/H_{0}$ is
precompact. Since $H_{0}<H$ is finite, this further implies that $L<H$ is
compact. ∎

We use [BDL14, Theorem 6.1] in combination with [TD14, Theorem 14] to show
that Zariski dense discrete subgroups are not inner amenable.

######  Proposition 2.7.

Let $k$ be a local field of characteristic zero and $\mathbf{G}$ a
$k$-isotropic almost $k$-simple algebraic $k$-group. Let
$\Gamma<\mathbf{G}(k)$ be a Zariski dense discrete subgroup. Then $\Gamma$ is
not inner amenable.

###### Proof.

By contradiction, assume that $\Gamma<\mathbf{G}(k)$ is inner amenable. By
[TD14, Theorem 14], there exists a short exact sequence $1\to N\to\Gamma\to
K\to 1$, where $K$ is amenable and either

  * $(\rm i)$

$\mathscr{Z}(N)$ is infinite, or

  * $(\rm ii)$

$N=LM$, where $L$ and $M$ are commuting normal subgroups of $\Gamma$ such that
$M$ is infinite and amenable, and $L\cap M$ is finite.

In case $(\rm i)$, since $\mathscr{Z}(N)\lhd\Gamma$ is an infinite normal
subgroup, it follows that $\mathscr{Z}(N)$ is Zariski dense in $\mathbf{G}$.
This would imply that $\mathbf{G}$ is abelian, which is absurd.

In case $(\rm ii)$, since $M\lhd\Gamma$ is an infinite normal subgroup, it
follows that $M$ is Zariski dense in $\mathbf{G}$. Since $M$ is amenable,
applying [BDL14, Theorem 6.1] to the trivial action
$M\curvearrowright\\{\bullet\\}$ which is amenable, at least one of the
following assertions holds:

  * •

There exist a proper $k$-subgroup $\mathbf{H}<\mathbf{G}$ and an
$M$-equivariant measurable map $\psi:\\{\bullet\\}\to G/H$, where we regard
$G/H\subset(\mathbf{G}/\mathbf{H})(k)$ with $H=\mathbf{H}(k)$. Then there
exists $g\in G$ such that $M<gHg^{-1}$ and so the Zariski closure of $M$ is a
proper $k$-subgroup of $\mathbf{G}$, which is a contradiction.

  * •

There exist a compact subgroup $L<G$ and an $M$-equivariant measurable map
$f:\\{\bullet\\}\to G/L$. Then there exists $g\in G$ such that $M<gLg^{-1}$
and so $M$ is compact, which is a contradiction.

This shows that $\Gamma$ is not inner amenable. ∎

######  Remark 2.8.

Let $k$ be a local field of characteristic zero and $\mathbf{G}$ a
$k$-isotropic $k$-simple algebraic $k$-group. Let $\Gamma<\mathbf{G}(k)$ be a
Zariski dense discrete subgroup. Then $\Gamma$ is icc meaning that $\Gamma$
has infinite conjugacy classes. Indeed, let $\gamma\in\Gamma$ be such that its
conjugacy class $C(\gamma)=\left\\{h\gamma h^{-1}\mid h\in\Gamma\right\\}$ is
finite. Then $\mathscr{Z}_{\Gamma}(\gamma)<\Gamma$ has finite index and so
$\mathscr{Z}_{\Gamma}(\gamma)$ is Zariski dense in $\mathbf{G}$ by Borel’s
density theorem [Bo60]. This implies that $\gamma\in\mathscr{Z}(\mathbf{G})$.
Since $\mathbf{G}$ is $k$-simple, we have $\gamma=e$.

###  2.5. Chabauty topology and Wijsman topology

Let $G$ be a locally compact second countable group. Consider the space
$\operatorname{Sub}(G)$ of all closed subgroups of $G$. Endowed with the
Chabauty topology, $\operatorname{Sub}(G)$ is a compact metrizable space and
the conjugation action $G\curvearrowright\operatorname{Sub}(G)$ is continuous
(see [Ch50]). Recall that for any sequence $(H_{n})_{n\in\mathbb{N}}$ in
$\operatorname{Sub}(G)$ and any element $H\in\operatorname{Sub}(G)$, we have
$H_{n}\to H$ in $\operatorname{Sub}(G)$ if and only if the following two
conditions hold:

  * $(\rm i)$

For every $x\in H$, there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ in $G$
such that $x_{n}\in H_{n}$ for every $n\in\mathbb{N}$ and $x_{n}\to x$ in $G$.

  * $(\rm ii)$

For every increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$,
every sequence $(x_{j})_{j\in\mathbb{N}}$ in $G$ such that $x_{j}\in
H_{n_{j}}$ for every $j\in\mathbb{N}$, and every element $x\in G$, if
$x_{j}\to x$ in $G$, then $x\in H$.

We refer the reader to [BP92, Section E.1] for further details. Then for every
$K\in\operatorname{Sub}(G)$, the subset

(2.6) $\left\\{H\in\operatorname{Sub}(G)\mid
H<K\right\\}\subset\operatorname{Sub}(G)\quad\text{is closed}.$

Let $\mathscr{G}$ be a Polish group and choose a compatible complete metric
$d:\mathscr{G}\times\mathscr{G}\to\mathbb{R}_{+}$. Denote by
$\operatorname{CL}(\mathscr{G})$ the space of all nonempty closed subsets of
$\mathscr{G}$. For all $x\in\mathscr{G}$,
$A\in\operatorname{CL}(\mathscr{G})$, set $d(x,A)=\inf\left\\{d(x,a)\mid a\in
A\right\\}$. Following [Be90], the Wijsman topology $\tau_{W(d)}$ on
$\operatorname{CL}(\mathscr{G})$ is defined as the weakest topology on
$\operatorname{CL}(\mathscr{G})$ that makes the maps
$\operatorname{CL}(\mathscr{G})\to\mathbb{R}_{+}:A\mapsto d(x,A)$ continuous
for all $x\in\mathscr{G}$. As explained in [Be90], the Borel structure induced
by the Wijsman topology $\tau_{W(d)}$ coincides with the Effros-Borel
structure on $\operatorname{CL}(\mathscr{G})$. By [Be90, Theorem 4.3],
$(\operatorname{CL}(\mathscr{G}),\tau_{W(d)})$ is a Polish space. Consider the
space $\operatorname{Sub}(\mathscr{G})$ of all closed subgroups of
$\mathscr{G}$. Then
$\operatorname{Sub}(\mathscr{G})\subset\operatorname{CL}(\mathscr{G})$ is
closed with respect to $\tau_{W(d)}$ and so
$(\operatorname{Sub}(\mathscr{G}),\tau_{W(d)})$ is a Polish space. Observe
that the subset

(2.7)
$\left\\{(h,\mathscr{H})\in\mathscr{G}\times\operatorname{Sub}(\mathscr{G})\mid
h\in\mathscr{H}\right\\}\subset\mathscr{G}\times\operatorname{Sub}(\mathscr{G})\quad\text{is
closed}.$

When $\mathscr{G}=G$ is a locally compact second countable group and
$d_{G}:G\times G\to\mathbb{R}_{+}$ is a left invariant compatible proper
complete metric (see [St73]), the Wijsman topology $\tau_{W(d_{G})}$ and the
Chabauty topology coincide on $\operatorname{Sub}(G)$ (see [BLLN89, Section
5]).

Let now $G$ be a locally compact second countable group. Choose a left
invariant compatible proper complete metric $d_{G}:G\times G\to\mathbb{R}_{+}$
(see [St73]). Then $d=\min(d_{G},1):G\times G\to\mathbb{R}_{+}$ is a left
invariant compatible complete metric such that $d\leq 1$. Let $(X,\nu)$ be a
standard probability space. Denote by
$\mathscr{G}=\operatorname{L}^{0}(X,\nu,G)$ the space of all $\nu$-equivalence
classes of measurable maps $f:X\to G$. Endowed with the topology of
convergence in measure, $\mathscr{G}$ is a Polish group (see e.g. [Ke10,
Section 19]). More precisely, define the metric
$d_{\mathscr{G}}:\mathscr{G}\times\mathscr{G}\to\mathbb{R}_{+}$ by the formula

$\forall f_{1},f_{2}\in\mathscr{G},\quad
d_{\mathscr{G}}(f_{1},f_{2})=\int_{X}d(f_{1}(x),f_{2}(x))\,{\rm d}\nu(x).$

Then $d_{\mathscr{G}}$ is a left invariant compatible complete metric on
$\mathscr{G}$. For every $H\in\operatorname{Sub}(G)$, we have that
$\mathscr{H}=\operatorname{L}^{0}(X,\nu,H)\in\operatorname{Sub}(\mathscr{G})$.
The following technical result will be very useful in the proof of Theorem
3.3.

######  Proposition 2.9.

Keep the same notation as above. Then the map

$\Psi:\operatorname{Sub}(G)\to\operatorname{Sub}(\mathscr{G}):H\mapsto\operatorname{L}^{0}(X,\nu,H)$

is Borel.

###### Proof.

Consider the Wijsman topology $\tau=\tau_{W(d_{\mathscr{G}})}$ on
$\operatorname{Sub}(\mathscr{G})$. Fix a countable dense subset
$D\subset\mathscr{G}$. For every $f\in D$ and all
$\alpha,\beta\in\mathbb{Q}\cap[0,1]$ such that $\alpha<\beta$, define the open
subsets

$\displaystyle O(f,\alpha)^{+}$
$\displaystyle=\left\\{\mathscr{K}\in\operatorname{Sub}(\mathscr{G})\mid
d_{\mathscr{G}}(f,\mathscr{K})>\alpha\right\\}$ $\displaystyle O(f,\beta)^{-}$
$\displaystyle=\left\\{\mathscr{K}\in\operatorname{Sub}(\mathscr{G})\mid
d_{\mathscr{G}}(f,\mathscr{K})<\beta\right\\}$ $\displaystyle
O(f,\alpha,\beta)$ $\displaystyle=O(f,\alpha)^{+}\cap O(f,\beta)^{-}.$

Then the countable set $\left\\{O(f,\alpha,\beta)\mid f\in
D,\alpha,\beta\in\mathbb{Q}\cap[0,1],\alpha<\beta\right\\}$ is a subbase for
the topology $\tau$ on $\operatorname{Sub}(\mathscr{G})$. Thus, in order to
show that the map
$\Psi:\operatorname{Sub}(G)\to\operatorname{Sub}(\mathscr{G})$ is Borel, it
suffices to show that $\Psi^{-1}(O(f,\alpha)^{+})$ and
$\Psi^{-1}(O(f,\alpha)^{-})$ are Borel subsets of $\operatorname{Sub}(G)$ for
every $f\in D$ and every $\alpha\in\mathbb{Q}\cap[0,1]$.

######  Claim 2.10.

Let $(H_{n})_{n\in\mathbb{N}}$ be a sequence in $\operatorname{Sub}(G)$ and
$H\in\operatorname{Sub}(G)$ such that $H_{n}\to H$ in $\operatorname{Sub}(G)$.
Set $\mathscr{H}_{n}=\operatorname{L}^{0}(X,\nu,H_{n})$ for every
$n\in\mathbb{N}$ and $\mathscr{H}=\operatorname{L}^{0}(X,\nu,H)$. Then for
every $f\in\mathscr{H}$, there exists a sequence $(f_{n})_{n\in\mathbb{N}}$ in
$\mathscr{G}$ such that $f_{n}\in\mathscr{H}_{n}$ for every $n\in\mathbb{N}$
and $\lim_{n}d_{\mathscr{G}}(f,f_{n})=0$.

Indeed, let $f\in\mathscr{H}$. Without loss of generality, we may assume that
$X$ is a compact metrizable space and $\nu\in\operatorname{Prob}(X)$ is a
Borel probability measure. Regard $f:X\to H$ as a measurable function. By
Lusin’s theorem, for every $n\geq 1$, there exists a closed subset
$X_{n}\subset X$ such that $\nu(X_{n})\geq 1-1/n$ and $f|_{X_{n}}:X_{n}\to H$
is continuous. By compactness, there exist $r\geq 1$ and $h_{1},\dots,h_{r}\in
H$ such that $X_{n}\subset\bigcup_{j=1}^{r}f^{-1}(B_{d}(h_{j},1/n))$, where
$B_{d}(h,\varepsilon)\subset G$ denotes the open ball in $G$ of center $h\in
G$ and radius $\varepsilon>0$. Set $Y_{1}=X_{n}\cap f^{-1}(B_{d}(h_{1},1/n))$
and $Y_{j}=X_{n}\cap
f^{-1}(B_{d}(h_{j},1/n))\setminus\bigcup_{i=1}^{j-1}Y_{i}$ for every $2\leq
j\leq r$. Then $(Y_{j})_{1\leq j\leq r}$ is a measurable partition of $X_{n}$.
Since $H_{n}\to H$ in $\operatorname{Sub}(G)$, for every $1\leq j\leq r$, we
may find $h_{j}^{n}\in B_{d}(h_{j},1/n)\cap H_{n}$. Define the measurable map
$f_{n}:X\to H_{n}$ by $f_{n}|_{X\setminus X_{n}}=e\mathbf{1}_{X\setminus
X_{n}}$ and $f_{n}|_{Y_{j}}=h_{j}^{n}\mathbf{1}_{Y_{j}}$ for every $1\leq
j\leq r$. Then we have

$\displaystyle\limsup_{n}d_{\mathscr{G}}(f,f_{n})$
$\displaystyle=\limsup_{n}\int_{X}d(f(x),f_{n}(x))\,{\rm d}\nu(x)$
$\displaystyle\leq\limsup_{n}\left(\nu(X\setminus
X_{n})+\frac{2}{n}\nu(X_{n})\right)=0.$

This finishes the proof of Claim 2.10.

######  Claim 2.11.

Let $(H_{n})_{n\in\mathbb{N}}$ be a sequence in $\operatorname{Sub}(G)$ and
$H\in\operatorname{Sub}(G)$ such that $H_{n}\to H$ in $\operatorname{Sub}(G)$.
Set $\mathscr{H}_{n}=\operatorname{L}^{0}(X,\nu,H_{n})$ for every
$n\in\mathbb{N}$ and $\mathscr{H}=\operatorname{L}^{0}(X,\nu,H)$. Then for
every $f\in\mathscr{G}$, we have

$\limsup_{n}d_{\mathscr{G}}(f,\mathscr{H}_{n})\leq
d_{\mathscr{G}}(f,\mathscr{H}).$

Indeed, set $\alpha=d_{\mathscr{G}}(f,\mathscr{H})$ and
$\beta=\limsup_{n}d_{\mathscr{G}}(f,\mathscr{H}_{n})$. Choose an increasing
sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}^{*}$ such that
$\beta=\lim_{j}d_{\mathscr{G}}(f,\mathscr{H}_{n_{j}})$. Choose a sequence
$(f_{n})_{n\in\mathbb{N}}$ in $\mathscr{H}$ such that $\alpha\leq
d_{\mathscr{G}}(f,f_{n})<\alpha+1/n$ for every $n\geq 1$. By Claim 2.10, we
may choose a sequence $(h_{n})_{n\geq 1}$ in $\mathscr{G}$ such that
$h_{n}\in\mathscr{H}_{n}$ and $d_{\mathscr{G}}(f_{n},h_{n})\leq 1/n$ for every
$n\geq 1$. In particular, for every $j\in\mathbb{N}$, we have

$d_{\mathscr{G}}(f,\mathscr{H}_{n_{j}})\leq d_{\mathscr{G}}(f,h_{n_{j}})\leq
d_{\mathscr{G}}(f,f_{n_{j}})+d_{\mathscr{G}}(f_{n_{j}},h_{n_{j}})\leq\alpha+\frac{2}{n_{j}}.$

This implies that $\beta\leq\alpha$ and finishes the proof of Claim 2.11.

Let $f\in D$ and $\alpha\in\mathbb{Q}\cap[0,1]$. Claim 2.11 implies that
$\Psi^{-1}(O(f,\alpha)^{-})\subset\operatorname{Sub}(G)$ is open. Observe that

$O(f,\alpha)^{+}=\bigcup_{j=1}^{\infty}\left\\{\mathscr{K}\in\operatorname{Sub}(\mathscr{G})\mid
d_{\mathscr{G}}(f,\mathscr{K})\geq\alpha+1/j\right\\}.$

Claim 2.11 implies that
$\Psi^{-1}(O(f,\alpha)^{+})\subset\operatorname{Sub}(G)$ is a countable union
of closed subsets and so
$\Psi^{-1}(O(f,\alpha)^{+})\subset\operatorname{Sub}(G)$ is Borel. Therefore,
we have showed that the map
$\Psi:\operatorname{Sub}(G)\to\operatorname{Sub}(\mathscr{G})$ is Borel. ∎

##  3\. Proofs of the main results

We will frequently use the following version of the Jankov–von Neumann
selection theorem.

######  Theorem 3.1.

Let $(X,\nu)$ be a standard probability space, $Y$ a Polish space and
$B\subset X\times Y$ a Borel subset such that $\pi_{X}(B)=X$, where
$\pi_{X}:X\times Y\to X$ is the projection map. Then there exists a Borel map
$f:X\to Y$ such that $(x,f(x))\in B$ for $\nu$-almost every $x\in X$.

###### Proof.

By the Jankov–von Neumann selection theorem, there exists a
$\sigma(\Sigma_{1}^{1})$-measurable map $f:X\to Y$ such that $(x,f(x))$ for
every $x\in X$ (see [Ke95, Theorem 18.1]). We then use two well-known facts.
Firstly, any $\sigma(\Sigma_{1}^{1})$-measurable set is Lebesgue measurable
(see [Ke95, Theorem 29.7]). Secondly, any Lebesgue measurable map $X\to Y$
coincides $\nu$-almost everywhere with a Borel map. Therefore, upon modifying
$f:X\to Y$ on a $\nu$-null Borel subset, we may assume that the map $f:X\to Y$
is Borel and satisfies $(x,f(x))\in B$ for $\nu$-almost every $x\in X$. ∎

Next, let $(X,\nu)$ and $(Y,\eta)$ be two standard probability spaces. Recall
that the spaces $\operatorname{L}^{0}(X\times Y,\nu\otimes\eta,\mathbb{C})$,
$\operatorname{L}^{0}(Y,\eta,\mathbb{C})$ and
$\operatorname{L}^{0}(X,\nu,\operatorname{L}^{0}(Y,\eta,\mathbb{C}))$ are
Polish spaces when endowed with the topology of convergence in measure.
Moreover, the well-defined map

$\iota:\operatorname{L}^{0}(X\times
Y,\nu\otimes\eta,\mathbb{C})\to\operatorname{L}^{0}(X,\nu,\operatorname{L}^{0}(Y,\eta,\mathbb{C})):f\mapsto(x\mapsto
f(x,\cdot\,))$

is isometric (hence injective) and surjective. Therefore, we may and will
identify $\operatorname{L}^{0}(X\times Y,\nu\otimes\eta,\mathbb{C})$ and
$\operatorname{L}^{0}(X,\nu,\operatorname{L}^{0}(Y,\eta,\mathbb{C}))$.

###  3.1. Zimmer’s method for product equivalence relations

Let $k$ be a local field of characteristic zero and $\mathbf{G}$ a $k$-group.
Set $G=\mathbf{G}(k)$.

Let $\mathscr{U}$ be an ergodic equivalence relation on $(U,\nu)$ and
$\mathscr{V}$ an ergodic equivalence relation on $(V,\eta)$. Set
$\mathscr{W}=\mathscr{U}\times\mathscr{V}$ and $(W,\zeta)=(U\times
V,\nu\otimes\eta)$. Let $\alpha:\mathscr{W}\to G$ be a Borel $1$-cocycle.
Following [Zi81], we define the set $\mathfrak{A}(\alpha)$ that consists of
all pairs $(\varphi,\mathfrak{H})$, where $\varphi:W\to G$ is a Borel map,
$\mathfrak{H}:U\to\operatorname{Sub}(G):u\mapsto H_{u}$ is a Borel map such
that for $\nu$-almost every $u\in U$, there exists a $k$-subgroup
$\mathbf{H}_{u}<\mathbf{G}$ such that $\mathbf{H}_{u}(k)=H_{u}$ and the Borel
map $V\to G/H_{u}:v\mapsto\varphi(u,v)H_{u}$ is
$(\mathscr{V},\alpha_{u})$-equivariant, where $\alpha_{u}:\mathscr{V}\to
G:(v_{1},v_{2})\mapsto\alpha((u,v_{1}),(u,v_{2}))$. Observe that the trivial
pair $(\varphi,\mathfrak{H})$ defined by $\varphi:W\to G:(u,v)\mapsto e$ and
$\mathfrak{H}:U\to\operatorname{Sub}(G):u\mapsto G$ belongs to
$\mathfrak{A}(\alpha)$. Thus, $\mathfrak{A}(\alpha)$ is not the empty set.

As explained in [Zi81], the set $\mathfrak{A}(\alpha)$ is stable under the
following cutting and pasting procedures. Firstly, let
$(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ and $\lambda\in[\mathscr{U}]$.
Define the pair $(\psi,\mathfrak{J})=\lambda_{\ast}(\varphi,\mathfrak{H})$ by
$\psi:W\to G:(u,v)\mapsto\alpha((u,v),(\lambda u,v))\varphi(\lambda u,v)$ and
$\mathfrak{J}:U\to\operatorname{Sub}(G):u\mapsto H_{\lambda u}$. Then a
straightforward computation shows that
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$. Secondly, let
$(\varphi^{i},\mathfrak{H}^{i})\in\mathfrak{A}(\alpha)$ for every
$i\in\\{1,2\\}$. Let $U=U_{1}\sqcup U_{2}$ be a measurable partition. Define
the pair $(\varphi,\mathfrak{H})$ by $\varphi|_{U_{i}\times
V}=\varphi^{i}|_{U_{i}\times V}$ and $\mathfrak{H}|_{U_{i}\times
V}=\mathfrak{H}^{i}|_{U_{i}\times V}$ for every $i\in\\{1,2\\}$. Then it is
plain to see that $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$. We record
the following useful result.

######  Lemma 3.2.

There exists a pair $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ such that
the $\nu$-almost everywhere defined map
$U\mapsto\mathbb{N}:u\mapsto\dim(\mathbf{H}_{u})$ is $\nu$-almost everywhere
constant and for any other pair $(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$,
we have $\dim(\mathbf{J}_{u})\geq\dim(\mathbf{H}_{u})$ for $\nu$-almost every
$u\in U$.

###### Proof.

Consider the nonempty finite subset $\mathscr{F}\subset\mathbb{N}$ that
consists of all integers $j\in\mathbb{N}$ for which there exists
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$ such that $\nu(\\{u\in
U\mid\dim(\mathbf{J}_{u})=j\\})>0$. Set $d=\min(\mathscr{F})$ and choose a
pair $(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$ such that $\nu(\\{u\in
U\mid\dim(\mathbf{J}_{u})=d\\})>0$. Choose $n\geq 1$ and a measurable subset
$U_{1}\subset\\{u\in U\mid\dim(\mathbf{J}_{u})=d\\}$ such that
$\nu(U_{1})=\frac{1}{n}$. Choose measurable subsets $U_{2},\dots,U_{n}\subset
X$ so that $(U_{i})_{1\leq i\leq n}$ forms a measurable partition of $X$.
Since $\mathscr{U}$ is ergodic on $(U,\nu)$, for every $1\leq i\leq n$, we may
choose $\lambda_{i}\in[\mathscr{U}]$ such that $\lambda_{i}(U_{i})=U_{1}$.
Then for every $1\leq i\leq n$, we have
$(\psi^{i},\mathfrak{J}^{i})={\lambda_{i}}_{\ast}(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$.
Define the pair $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ by
$\varphi|_{U_{i}\times V}=\psi^{i}|_{U_{i}\times V}$ and
$\mathfrak{H}|_{U_{i}\times V}=\mathfrak{J}^{i}|_{U_{i}\times V}$ for every
$1\leq i\leq n$. Then we have $d=\dim(\mathbf{H}_{u})$ for $\nu$-almost every
$u\in U$. ∎

We then simply say that $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ is a
minimal pair and we write $d=\dim(\mathbf{H}_{u})$ for $\nu$-almost every
$u\in U$.

The following theorem is an extension of [Zi81, Lemma 2.1] to the case of
algebraic groups defined over a local field $k$ of characteristic zero. For
the sake of completeness, we give a proof following the one of [Zi81, Lemma
2.1] and we add a few details regarding the measurability of certain maps.

######  Theorem 3.3.

Keep the same notation as above. The following assertions hold:

  * $(\rm i)$

Let $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ and
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$ be two minimal pairs. Then there
exists a measurable map $U\to G:u\mapsto b_{u}$ such that
$\mathbf{H}_{u}^{0}=b_{u}\mathbf{J}_{u}^{0}b_{u}^{-1}$ for $\nu$-almost every
$u\in U$.

  * $(\rm ii)$

Let $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ be a minimal pair. Then
there exists a Zariski connected $k$-subgroup $\mathbf{H}<\mathbf{G}$ and a
measurable map $U\to G:u\mapsto g_{u}$ such that
$\mathbf{H}_{u}^{0}=g_{u}\mathbf{H}g_{u}^{-1}$ for $\nu$-almost every $u\in
U$.

  * $(\rm iii)$

Let $(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ and
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$ be two minimal pairs for which
there exists a Zariski connected $k$-subgroup $\mathbf{H}<\mathbf{G}$ such
that $\mathbf{H}_{u}^{0}=\mathbf{H}=\mathbf{J}_{u}^{0}$ for $\nu$-almost every
$u\in U$. Set $\mathbf{L}=\mathscr{N}_{\mathbf{G}}(\mathbf{H})$,
$H=\mathbf{H}(k)$, $L=\mathbf{L}(k)$ and consider the factor map $p:G\to G/L$.
Then we have $p\circ\varphi=p\circ\psi$ $\zeta$-almost everywhere.

###### Proof.

$(\rm i)$ After discarding a $\nu$-null Borel subset, we may assume that the
Borel maps $\mathfrak{H}:U\to\operatorname{Sub}(G):u\mapsto H_{u}$ and
$\mathfrak{J}:U\to\operatorname{Sub}(G):u\mapsto J_{u}$ satisfy that for every
$u\in U$, there exist a $k$-subgroup $\mathbf{H}_{u}<\mathbf{G}$ such that
$\mathbf{H}_{u}(k)=H_{u}$ and a $k$-subgroup $\mathbf{J}_{u}<\mathbf{G}$ such
that $\mathbf{J}_{u}(k)=J_{u}$, and the Borel maps $V\to
G/H_{u}:v\mapsto\varphi(u,v)H_{u}$ and $V\to G/J_{u}:v\mapsto\psi(u,v)J_{u}$
are $(\mathscr{V},\alpha_{u})$-equivariant. Denote by
$\mathscr{G}=\operatorname{L}^{0}(V,\eta,G)$ the space of all
$\eta$-equivalence classes of measurable functions $f:V\to G$. Endowed with
the topology of convergence in measure, $\mathscr{G}$ is a Polish group. For
every $u\in U$, consider the $k$-$\mathbf{G}$-variety
$\mathbf{M}_{u}=\mathbf{G}/\mathbf{H}_{u}\times\mathbf{G}/\mathbf{J}_{u}$,
where $\mathbf{G}\curvearrowright\mathbf{M}_{u}$ acts diagonally. Set
$M_{u}=\mathbf{M}_{u}(k)$ and regard $N_{u}=G/H_{u}\times G/J_{u}\subset
M_{u}$ as a closed subset. Define the Borel map

$\theta_{u}:V\to N_{u}:v\mapsto(\varphi(u,v)H_{u},\psi(u,v)J_{u}).$

Since $\mathbf{M}_{u}$ is an algebraic $k$-$\mathbf{G}$-variety, the
continuous action $G\curvearrowright M_{u}$ has locally closed orbits (see
Facts 2.5$(\rm i)$). Thus, the action $G\curvearrowright N_{u}$ has locally
closed orbits and so the quotient space $G\backslash N_{u}$ is a standard
Borel space. Since the measurable map

$V\to G\backslash N_{u}:v\mapsto G\cdot\theta_{u}(v)$

is $\mathscr{V}$-invariant and since $\mathscr{V}$ is ergodic, $V\to
G\backslash N_{u}:v\mapsto G\cdot\theta_{u}(v)$ is $\eta$-almost everywhere
constant. We may choose $b\in G$ such that $\theta_{u}(v)\in
G\cdot(H_{u},bJ_{u})$ for $\eta$-almost every $v\in V$. Letting
$L=\operatorname{Stab}_{G}(H_{u},bJ_{u})<G$, the map

$G/L\to G\cdot(H_{u},bJ_{u}):gL\mapsto g(H_{u},bJ_{u})$

is a homeomorphism. By considering a Borel section $G/L\to G$ of the
projection map $G\to G/L$, we may then find $F\in\mathscr{G}$ such that
$\theta_{u}(v)=(F(v)H_{u},F(v)bJ_{u})$ for $\eta$-almost every $v\in V$.

For every $u\in U$, define $\varphi_{u}:V\to G:v\mapsto\varphi(u,v)$,
$\psi_{u}:V\to G:v\mapsto\psi(u,v)$ and regard $\varphi_{u}\in\mathscr{G}$,
$\psi_{u}\in\mathscr{G}$. Then the maps $U\to\mathscr{G}:u\mapsto\varphi_{u}$
and $U\to\mathscr{G}:u\mapsto\psi_{u}$ are Borel. For every $u\in U$, set
$\mathscr{H}_{u}=\operatorname{L}^{0}(V,\eta,H_{u})$ and
$\mathscr{J}_{u}=\operatorname{L}^{0}(V,\eta,J_{u})$. By Proposition 2.9, the
maps $U\to\operatorname{Sub}(\mathscr{G}):u\mapsto\mathscr{H}_{u}$ and
$U\to\operatorname{Sub}(\mathscr{G}):u\mapsto\mathscr{J}_{u}$ are Borel.
Denote by $\pi_{U}:U\times G\times\mathscr{G}\to U$ the projection map.
Consider the subset

$B=\left\\{(u,b,F)\in U\times G\times\mathscr{G}\mid
F^{-1}\varphi_{u}\in\mathscr{H}_{u}\;\text{ and
}\;b^{-1}F^{-1}\psi_{u}\in\mathscr{J}_{u}\right\\}.$

The reasoning in the first paragraph shows that for every $u\in U$, there
exist $F\in\mathscr{G}$ and $b\in G$ such that
$(\varphi(u,v)H_{u},\psi(u,v)J_{u})=(F(v)H_{u},F(v)bJ_{u})$ for $\eta$-almost
every $v\in V$. This means exactly that $(u,b,F)\in B$. Therefore, we have
$\pi_{U}(B)=U$. Using (2.7), it follows that $B\subset U\times
G\times\mathscr{G}$ is a Borel subset. By the measurable selection theorem
(see Theorem 3.1), there exists a Borel map $U\to
G\times\mathscr{G}:u\mapsto(b_{u},F_{u})$ such that $(u,b_{u},F_{u})\in B$ for
$\nu$-almost every $u\in U$. Thus, we may choose Borel maps $\theta:W\to G$
and $\mathfrak{L}:U\mapsto\operatorname{Sub}(G):u\mapsto L_{u}$ such that
$\theta(u,v)=F_{u}(v)$ for $\zeta$-almost every $(u,v)\in W$ and
$L_{u}=H_{u}\cap b_{u}J_{u}b_{u}^{-1}$ for $\nu$-almost every $u\in U$. Note
that for $\nu$-almost every $u\in U$, letting
$\mathbf{L}_{u}=\mathbf{H}_{u}\cap b_{u}\mathbf{J}_{u}b_{u}^{-1}<\mathbf{G}$,
which is a $k$-subgroup, we have $L_{u}=\mathbf{L}_{u}(k)$. For $\nu$-almost
every $u\in U$, using the $G$-equivariant homeomorphism

$G/L_{u}\to G\cdot(H_{u},b_{u}J_{u}):gL_{u}\mapsto(gH_{u},gb_{u}J_{u}),$

we may identify

$\displaystyle\theta(u,v)L_{u}$
$\displaystyle=(\theta(u,v)H_{u},\theta(u,v)b_{u}J_{u})$
$\displaystyle=(F_{u}(v)H_{u},F_{u}(v)b_{u}J_{u})$
$\displaystyle=(\varphi(u,v)H_{u},\psi(u,v)J_{u}).$

for $\zeta$-almost every $(u,v)\in W$. Since for $\nu$-almost every $u\in U$,
the Borel maps $V\to G/H_{u}:v\mapsto\varphi(u,v)H_{u}$ and $V\to
G/J_{u}:v\mapsto\psi(u,v)J_{u}$ are $(\mathscr{V},\alpha_{u})$-equivariant, it
follows that the Borel map $V\to G/L_{u}:v\mapsto\theta(u,v)L_{u}$ is
$(\mathscr{V},\alpha_{u})$-equivariant. Therefore, we infer that
$(\theta,\mathfrak{L})\in\mathfrak{A}(\alpha)$. By minimality of the pairs
$(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ and
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha)$, we have that
$\dim(\mathbf{H}_{u})=\dim(\mathbf{H}_{u}\cap
b_{u}\mathbf{J}_{u}b_{u}^{-1})=\dim(\mathbf{J}_{u})$ for $\nu$-almost every
$u\in U$. This further implies that $\mathbf{H}_{u}^{0}=(\mathbf{H}_{u}\cap
b_{u}\mathbf{J}_{u}b_{u}^{-1})^{0}=b_{u}\mathbf{J}_{u}^{0}b_{u}^{-1}$ for
$\nu$-almost every $u\in U$.

$(\rm ii)$ Consider the adjoint $k$-representation
$\operatorname{Ad}:\mathbf{G}\to\operatorname{GL}(\operatorname{Lie}(\mathbf{G}))$.
Denote by $M=\operatorname{Gr}_{d}(\operatorname{Lie}(G))$ (resp.
$\mathbf{M}=\operatorname{Gr}(\operatorname{Lie}(\mathbf{G}))$) the
$k$-analytic Grassmannian $G$-manifold (resp. algebraic
$k$-$\mathbf{G}$-variety) of all $d$-dimensional subspaces of
$\operatorname{Lie}(G)$ (resp. $\operatorname{Lie}(\mathbf{G})$). We
naturally have $M=\mathbf{M}(k)$ as $k$-analytic manifolds. The map $U\to
M:u\mapsto\operatorname{Lie}(H_{u})$ is measurable (see [NZ00, Section 3]).
Since $\mathbf{M}$ is an algebraic $k$-$\mathbf{G}$-variety, the continuous
action $G\curvearrowright M$ has locally closed orbits and so the quotient
space $G\backslash M$ is a standard Borel space (see Facts 2.5$(\rm i)$).
Consider the measurable map $\beta:U\to G\backslash M:u\mapsto
G\cdot\operatorname{Lie}(H_{u})$. Let $\lambda\in[\mathscr{U}]$. Since
$\lambda_{\ast}(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha)$ is a minimal
pair, item $(\rm i)$ implies that there exists a measurable map $U\to
G:u\mapsto b_{u}$ such that $\mathbf{H}_{u}^{0}=b_{u}\mathbf{H}_{\lambda
u}^{0}b_{u}^{-1}$ for $\nu$-almost every $u\in U$. Then we have $\beta(\lambda
u)=\beta(u)$ for $\nu$-almost every $u\in U$. This implies that the measurable
map $\beta:U\to G\backslash M$ is $\mathscr{U}$-invariant. Since $\mathscr{U}$
is ergodic, $\beta:U\to G\backslash M$ is $\nu$-almost everywhere constant and
so there exists a point $w\in M$ such that $\operatorname{Lie}(H_{u})\in
G\cdot w$ for $\nu$-almost every $u\in U$. Then there exists a measurable map
$U\to G:u\mapsto g_{u}$ such that $\operatorname{Lie}(H_{u})=g_{u}\cdot w$ for
$\nu$-almost every $u\in U$. By Fubini’s theorem, there exists $u_{0}\in U$
such that letting $\mathbf{H}=g_{u_{0}}^{-1}\mathbf{H}_{u_{0}}^{0}g_{u_{0}}$,
we have
$\operatorname{Lie}(\mathbf{H}_{u}^{0})=\operatorname{Lie}(g_{u}\mathbf{H}g_{u}^{-1})$
for $\nu$-almost every $u\in U$. Since the characteristic of $k$ is zero, it
follows that $\mathbf{H}_{u}^{0}=g_{u}\mathbf{H}g_{u}^{-1}$ for $\nu$-almost
every $u\in U$.

$(\rm iii)$ By the proof of item $(\rm i)$, there exist measurable maps
$\theta:W\to G$ and $U\to G:u\mapsto b_{u}$ such that

$(\varphi(u,v)H_{u},\psi(u,v)J_{u})=(\theta(u,v)H_{u},\theta(u,v)b_{u}J_{u})$

for $\zeta$-almost every $(u,v)\in W$. For $\nu$-almost every $u\in U$, since
$\dim(\mathbf{H}_{u})=\dim(\mathbf{H}_{u}\cap
b_{u}\mathbf{J}_{u}b_{u}^{-1})=\dim(\mathbf{J}_{u})$ and
$\mathbf{H}_{u}^{0}=\mathbf{H}=\mathbf{J}_{u}^{0}$, we have
$\mathbf{H}=\mathbf{H}_{u}^{0}=b_{u}\mathbf{J}_{u}^{0}b_{u}^{-1}=b_{u}\mathbf{H}b_{u}^{-1}$
and so $b_{u}\in\mathscr{N}_{\mathbf{G}}(\mathbf{H})=\mathbf{L}$. Then for
$\zeta$-almost every $(u,v)\in W$, we have

$\varphi(u,v)L=\theta(u,v)L=\theta(u,v)b_{u}L=\psi(u,v)L.$

This shows that $p\circ\varphi=p\circ\psi$ $\zeta$-almost everywhere. ∎

###  3.2. Proof of Theorem A

Let $k$ be a local field of characteristic zero, $\mathbf{G}$ a $k$-isotropic
almost $k$-simple algebraic $k$-group and $\Gamma<\mathbf{G}(k)$ a Zariski
dense discrete subgroup. Let $\Gamma\curvearrowright(X,\nu)$ be a free ergodic
pmp action and set $\mathscr{R}=\mathscr{R}(\Gamma\curvearrowright X)$. Upon
discarding a null Borel subset, we may assume that the Borel action
$\Gamma\curvearrowright X$ is free. Consider the orbit Borel $1$-cocycle
$\beta:\mathscr{R}\to\Gamma:(\gamma x,x)\mapsto\gamma$. Let $\mathscr{S}$ be
an ergodic equivalence relation on $(Y,\eta)$ and consider the Borel
$1$-cocycle $\alpha=\beta\circ q:\mathscr{R}\times\mathscr{S}\to\Gamma$, where
$q:\mathscr{R}\times\mathscr{S}\to\mathscr{R}$ is the canonical factor map.
Then $\ker(\alpha)=\operatorname{id}_{X}\times\mathscr{S}$.

For every $i\in\\{1,2\\}$, let $\mathscr{T}_{i}$ be an ergodic equivalence
relation on $(Z_{i},\zeta_{i})$. Set
$\mathscr{T}=\mathscr{T}_{1}\times\mathscr{T}_{2}$ and $(Z,\zeta)=(Z_{1}\times
Z_{2},\zeta_{1}\otimes\zeta_{2})$. Assume that
$\mathscr{R}\times\mathscr{S}\cong\mathscr{T}$. Then there exists a conull
Borel subset $Z_{0}\subset Z$ such that we may identify
$(\mathscr{R}\times\mathscr{S})|_{Z_{0}}$ with $\mathscr{T}|_{Z_{0}}$
everywhere and regard $Z_{0}\subset X\times Y$ as a conull Borel subset.

We prove Theorem A by contradiction. Assume that for every $i\in\\{1,2\\}$, we
have $\mathscr{T}_{i}\npreceq_{\mathscr{T}}\mathscr{S}$ as subequivalences
relations of $\mathscr{T}$.

By [Zi84, Proposition 9.3.2], for every $i\in\\{1,2\\}$, there exists an
amenable ergodic subequivalence relation $\mathscr{U}_{i}\leq\mathscr{T}_{i}$.
Set $\mathscr{U}=\mathscr{U}_{1}\times\mathscr{U}_{2}$ and
$\mathscr{U}_{0}=\mathscr{U}|_{Z_{0}}$ which is an amenable ergodic
subequivalence relation of $\mathscr{T}|_{Z_{0}}$.

######  Claim 3.4.

There exist a proper $k$-subgroup $\mathbf{H}<\mathbf{G}$ and a
$(\mathscr{U},\alpha|_{\mathscr{U}})$-equivariant measurable map $f:Z\to G/H$,
where we regard $G/H\subset(\mathbf{G}/\mathbf{H})(k)$ with $H=\mathbf{H}(k)$.

By contradiction, assume that the assertion on Claim 3.4 does not hold. By
applying Theorem 2.6 to $\alpha|_{\mathscr{U}_{0}}:\mathscr{U}_{0}\to G$,
there exists a compact subgroup $L<G$ and a
$(\mathscr{U}_{0},\alpha|_{\mathscr{U}_{0}})$-equivariant measurable map
$f:Z_{0}\to G/L$. Upon discarding a null Borel subset, we may assume that the
map $f:Z_{0}\to G/L$ is Borel and strictly
$(\mathscr{U}_{0},\alpha|_{\mathscr{U}_{0}})$-equivariant. Choose a Borel
section $\sigma:G/L\to G$ and define the Borel map $\psi=\sigma\circ
f:Z_{0}\to G$. Then for every $(z_{1},z_{2})\in\mathscr{U}|_{Z_{0}}$, we have
$\psi(z_{1})L=\alpha(z_{1},z_{2})\psi(z_{2})L$. Since $G$ is $\sigma$-compact,
there exists a compact subset $C\subset G$ such that the Borel subset
$V=\psi^{-1}(C)\subset Z_{0}$ satisfies $\zeta(V)>0$. Then for every
$(z_{1},z_{2})\in\mathscr{U}|_{V}$, we have
$\alpha(z_{1},z_{2})\in\psi(z_{1})L\psi(z_{2})^{-1}\subset
CLC^{-1}\cap\Gamma$. Since $CLC^{-1}\subset G$ is compact and $\Gamma<G$ is
discrete, $F=CLC^{-1}\cap\Gamma$ is a finite subset of $\Gamma$.

Define the subequivalence relation
$\mathscr{V}=\mathscr{U}|_{V}\cap\ker(\alpha)|_{V}$. Let $z\in V$. Then we can
write $[z]_{\mathscr{U}|_{V}}=\bigcup_{\gamma\in F}C_{\gamma}(z)$, where
$C_{\gamma}(z)=\left\\{t\in[z]_{\mathscr{U}|_{V}}\mid\alpha(z,t)=\gamma\right\\}$
for $\gamma\in F$. Then for every $\gamma\in F$ and all $t_{1},t_{2}\in
C_{\gamma}(z)$, we have
$\alpha(t_{1},t_{2})=\alpha(t_{1},z)\alpha(z,t_{2})=\gamma^{-1}\gamma=e$ and
so $(t_{1},t_{2})\in\mathscr{V}$. Therefore,
$\mathscr{V}\subset\mathscr{U}|_{V}$ has bounded index. Since
$\mathscr{V}\leq\ker(\alpha)|_{V}=(\operatorname{id}_{X}\times\mathscr{S})|_{V}$,
it follows that
$\mathscr{U}\preceq_{\mathscr{T}}\operatorname{id}_{X}\times\mathscr{S}$.
Since $\mathscr{U}_{1}\times\operatorname{id}_{Z_{2}}\leq\mathscr{U}$, this
further implies that
$\mathscr{U}_{1}\times\operatorname{id}_{Z_{2}}\preceq_{\mathscr{T}}\operatorname{id}_{X}\times\mathscr{S}$.
Since $\mathscr{U}_{1}$ is ergodic, Facts 2.2$(\rm v)$ implies that
$\mathscr{T}_{1}\times\operatorname{id}_{Z_{2}}\preceq_{\mathscr{T}}\operatorname{id}_{X}\times\mathscr{S}$.
This contradicts our assumption and finishes the proof of Claim 3.4.

######  Claim 3.5.

There exist a proper $k$-subgroup $\mathbf{L}<\mathbf{G}$ and a
$(\mathscr{T}_{1}\times\mathscr{T}_{2},\alpha)$-equivariant measurable map
$F:Z\to G/L$, where we regard $G/L\subset(\mathbf{G}/\mathbf{L})(k)$ with
$L=\mathbf{L}(k)$.

The proof of Claim 3.5 consists of two steps. Firstly, we consider the product
equivalence $\mathscr{T}_{1}\times\mathscr{U}_{2}$ together with the
subequivalence relation
$\mathscr{U}_{1}\times\mathscr{U}_{2}\leq\mathscr{T}_{1}\times\mathscr{U}_{2}$
and the
$(\mathscr{U}_{1}\times\mathscr{U}_{2},\alpha|_{\mathscr{U}_{1}\times\mathscr{U}_{2}})$-equivariant
measurable map $f:Z\to G/H$. By Theorem 3.3$(\rm ii)$, we may choose a minimal
pair
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$
for which there exists a Zariski connected $k$-subgroup
$\mathbf{J}<\mathbf{G}$ such that $\mathbf{J}=\mathbf{J}_{z_{1}}^{0}$ for
$\zeta_{1}$-almost every $z_{1}\in Z_{1}$. By Claim 3.4, there exist a proper
$k$-subgroup $\mathbf{H}<\mathbf{G}$ and a
$(\mathscr{U},\alpha|_{\mathscr{U}})$-equivariant measurable map $f:Z\to G/H$.
Choose a Borel section $\sigma:G/H\to G$. Consider the measurable map
$\varphi=\sigma\circ f:Z\to G$ and the constant map
$\mathfrak{H}:Z_{1}\to\operatorname{Sub}(G):z_{1}\mapsto H$. Then
$(\varphi,\mathfrak{H})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$.
By minimality of
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$
and since $\mathbf{H}<\mathbf{G}$ is a proper $k$-subgroup, it follows that
$\mathbf{J}<\mathbf{G}$ is a proper $k$-subgroup.

Next, we claim that $\mathbf{J}\neq\\{e\\}$. Upon discarding a null Borel
subset, we may assume that the map
$\mathfrak{J}:Z_{1}\to\operatorname{Sub}(G):z_{1}\mapsto J_{z_{1}}$ is Borel
and for every $z_{1}\in Z_{1}$, there exists an algebraic $k$-subgroup
$\mathbf{J}_{z_{1}}<\mathbf{G}$ such that $\mathbf{J}_{z_{1}}(k)=J_{z_{1}}$
and $\mathbf{J}_{z_{1}}^{0}=\mathbf{J}$. By contradiction, assume that
$\mathbf{J}=\\{e\\}$. Then we have that $J_{z_{1}}<G$ is a finite subgroup for
every $z_{1}\in Z_{1}$. It is well-known that there are only finitely many
conjugacy classes of maximal compact subgroups of $G$ (see e.g. [BT71,
Corollaire 3.3.3]). Therefore, there exists a nonnull Borel subset $W\subset
Z_{1}$ and a maximal compact subgroup $K<G$ such that the subset

$B=\left\\{(z_{1},b)\in W\times G\mid bJ_{z_{1}}b^{-1}<K\right\\}\subset
W\times G$

satisfies $\pi_{W}(B)=W$. Using (2.6), it follows that $B\subset W\times G$ is
a Borel subset. By the measurable selection theorem (see Theorem 3.1), there
exists a Borel map $W\to G:z_{1}\mapsto b_{z_{1}}$ such that
$b_{z_{1}}J_{z_{1}}b_{z_{1}}^{-1}<K$ for $\zeta_{1}$-almost every $z_{1}\in
W$. Upon discarding a null Borel subset, we may assume that the Borel map
$W\to G:z_{1}\mapsto b_{z_{1}}$ satisfies $b_{z_{1}}J_{z_{1}}b_{z_{1}}^{-1}<K$
for every $z_{1}\in W$. We may extend the Borel map $Z_{1}\to G:z_{1}\mapsto
b_{z_{1}}$ by declaring that $b_{z_{1}}=e$ for every $z_{1}\in Z_{1}\setminus
W$. Upon replacing the pair
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$
by the pair
$(\rho,\mathfrak{K})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$,
where $\rho:Z\to G:(z_{1},z_{2})\mapsto\psi(z_{1},z_{2})b_{z_{1}}^{-1}$ and
$\mathfrak{K}:Z_{1}\to\operatorname{Sub}(G):z_{1}\mapsto
b_{z_{1}}J_{z_{1}}b_{z_{1}}^{-1}$, we may assume that $J_{z_{1}}<K$ for every
$z_{1}\in W$.

Choose a conull Borel subset $V\subset(W\times Z_{2})\cap Z_{0}$ such that the
restricted Borel map $\psi|_{V}:V\to G$ satisfies

$\forall((z_{1},s),(z_{1},t))\in(\operatorname{id}_{Z_{1}}\times\mathscr{U}_{2})|_{V},\quad\psi(z_{1},s)J_{z_{1}}=\alpha((z_{1},s),(z_{1},t))\psi(z_{1},t)J_{z_{1}}.$

Since $G$ is $\sigma$-compact, there exists a compact subset $C\subset G$ such
that $V_{0}=(\psi|_{V})^{-1}(C)\subset V$ satisfies $\zeta(V_{0})>0$. Then for
every
$((z_{1},s),(z_{1},t))\in(\operatorname{id}_{Z_{1}}\times\mathscr{U}_{2})|_{V_{0}}$,
we have

$\alpha((z_{1},s),(z_{1},t))\in\psi(z_{1},s)J_{z_{1}}\psi(z_{1},t)^{-1}\in
CJ_{z_{1}}C^{-1}\cap\Gamma\subset CKC^{-1}\cap\Gamma.$

Since $CKC^{-1}\subset G$ is compact and $\Gamma<G$ is discrete,
$F=CKC^{-1}\cap\Gamma$ is a finite subset of $\Gamma$. Now a reasoning
entirely analogous to the one as in the second paragraph of the proof of Claim
3.4 shows that
$\operatorname{id}_{Z_{1}}\times\mathscr{U}_{2}\preceq_{\mathscr{T}}\operatorname{id}_{X}\times\mathscr{S}$.
This contradicts our assumption and so $\mathbf{J}\neq\\{e\\}$.

Since $\mathbf{G}$ is almost $k$-simple and since the Zariski connected
$k$-subgroup $\mathbf{J}$ is different from $\\{e\\}$ and $\mathbf{G}$, it
follows that the $k$-subgroup
$\mathbf{L}=\mathscr{N}_{\mathbf{G}}(\mathbf{J})<\mathbf{G}$ is a proper
subgroup. Set $L=\mathbf{L}(k)$. Choose a countable subgroup
$\Lambda<[\mathscr{T}_{1}]$ such that
$\mathscr{T}_{1}=\mathscr{R}(\Lambda\curvearrowright Z_{1})$. For every
$\lambda\in\Lambda$, the pair
$\lambda_{\ast}(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$
is minimal and for $\zeta_{1}$-almost every $z_{1}\in Z_{1}$, we have
$(\mathbf{J}_{\lambda z_{1}})^{0}=\mathbf{J}$. Then Theorem 3.3$(\rm iii)$
implies that for $\zeta$-almost every $(z_{1},s)\in Z_{1}\times Z_{2}$, we
have $\psi(z_{1},s)L=\alpha((z_{1},s),(\lambda z_{1},s))\psi(\lambda
z_{1},s)L$. This further implies that for almost every
$(z_{1},s_{1})\in\mathscr{T}_{1}$ and almost every
$(z_{2},s_{2})\in\mathscr{U}_{2}$, we have

$\displaystyle\psi(z_{1},s_{1})L$
$\displaystyle=\alpha((z_{1},s_{1}),(z_{2},s_{1}))\psi(z_{2},s_{1})L$
$\displaystyle=\alpha((z_{1},s_{1}),(z_{2},s_{1}))\alpha((z_{2},s_{1}),(z_{2},s_{2}))\psi(z_{2},s_{2})L$
$\displaystyle=\alpha((z_{1},s_{1}),(z_{2},s_{2}))\psi(z_{2},s_{2})L,$

where in the second line we used the fact that
$(\psi,\mathfrak{J})\in\mathfrak{A}(\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$.
It follows that the measurable map $Z\to G/L:z\mapsto\psi(z)L$ is
$(\mathscr{T}_{1}\times\mathscr{U}_{2},\alpha|_{\mathscr{T}_{1}\times\mathscr{U}_{2}})$-equivariant.

Now, we can consider the product equivalence relation
$\mathscr{T}_{1}\times\mathscr{T}_{2}$ together with the subequivalence
relation
$\mathscr{T}_{1}\times\mathscr{U}_{2}\leq\mathscr{T}_{1}\times\mathscr{T}_{2}$
and the $(\mathscr{T}_{1}\times\mathscr{U}_{2})$-equivariant measurable map
$Z\to G/L:z\mapsto\psi(z)L$. Repeating the same argument as in the first four
paragraphs of the proof of Claim 3.5 and since
$\mathscr{T}_{1}\times\operatorname{id}_{Z_{2}}\npreceq_{\mathscr{T}}\operatorname{id}_{X}\times\mathscr{S}$,
upon modifying the proper $k$-subgroup $\mathbf{L}<\mathbf{G}$, we obtain the
desired conclusion. This finishes the proof of Claim 3.5.

Consider now the pmp action $\Gamma\curvearrowright(Z,\zeta)$ defined by
$\gamma z=(\gamma x,y)$ for every $z=(x,y)\in Z$. Then for every
$\gamma\in\Gamma$ and almost every $z=(x,y)\in Z$, we have

$F(\gamma z)=F(\gamma x,y)=\alpha((\gamma x,y),(x,y))F(x,y)=\gamma F(z).$

Then the measurable map $F:Z\to G/L$ is $\Gamma$-equivariant. Consider the
Borel probability measure
$\mu=F_{\ast}\zeta\in\operatorname{Prob}(G/L)^{\Gamma}$. Since $\Gamma<G$ is
Zariski dense in $\mathbf{G}$, [BDL14, Proposition 1.9] implies that there
exists a normal $k$-subgroup $\mathbf{N}\lhd\mathbf{G}$ such that the image of
$\Gamma$ in $(\mathbf{G}/\mathbf{N})(k)$ is precompact and $\mu$ is supported
on $(\mathbf{G}/\mathbf{L})^{\mathbf{N}}\cap G/L$. Since
$\mathbf{L}<\mathbf{G}$ is a proper $k$-subgroup and since
$(\mathbf{G}/\mathbf{L})^{\mathbf{N}}\neq\emptyset$, we have
$\mathbf{N}\neq\mathbf{G}$. Since $\mathbf{G}$ is almost $k$-simple, it
follows that $\mathbf{N}\lhd\mathbf{G}$ is a finite normal $k$-subgroup.
Letting $N=\mathbf{N}(k)$, the image of $\Gamma$ in
$(\mathbf{G}/\mathbf{N})(k)$ is contained in $G/N$ which is closed in
$(\mathbf{G}/\mathbf{N})(k)$. Then the image of $\Gamma$ in $G/N$ is
precompact. Since $N<G$ is finite, this further implies that $\Gamma<G$ is
compact, which is a contradiction. Therefore, we have showed that the orbit
equivalence relation $\mathscr{R}(\Gamma\curvearrowright X)$ is strongly
prime. This concludes the proof of Theorem A.

The following variation of Theorem A will be useful in the proof of Theorem D.

######  Theorem 3.6.

Let $k$ be a local field of characteristic zero, $\mathbf{G}$ a $k$-isotropic
almost $k$-simple algebraic $k$-group and $\Gamma<\mathbf{G}(k)$ a Zariski
dense discrete subgroup. Let $\Lambda$ be a countable discrete group and
$\Gamma\times\Lambda\curvearrowright(X,\nu)$ an essentially free ergodic pmp
action. Set $\mathscr{R}=\mathscr{R}(\Gamma\times\Lambda\curvearrowright X)$.

For every $i\in\\{1,2\\}$, let $\mathscr{T}_{i}$ be an ergodic equivalence
relation on $(Z_{i},\zeta_{i})$ and assume that
$\mathscr{R}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$. Then there exists
$i\in\\{1,2\\}$ such that
$\mathscr{T}_{i}\preceq_{\mathscr{R}}\mathscr{R}(\Lambda\curvearrowright X)$
as subequivalence relations.

###### Proof.

The proof is identical to the proof of Theorem A, the only difference is that
the pmp action $\Gamma\times\Lambda\curvearrowright(X,\nu)$ is no longer
assumed to be a product action. To circumvent this issue, we exploit Lemma
2.3. Upon discarding a null Borel subset, we may assume that the Borel action
$\Gamma\times\Lambda\curvearrowright X$ is free. Consider the Borel
$1$-cocycle $\alpha:\mathscr{R}\to\Gamma:((\gamma,\lambda)x,x)\mapsto\gamma$.
Then $\ker(\alpha)=\mathscr{R}(\Lambda\curvearrowright X)$.

By contradiction, assume that for every $i\in\\{1,2\\}$, we have
$\mathscr{T}_{i}\npreceq_{\mathscr{R}}\mathscr{R}(\Lambda\curvearrowright X)$
as subequivalence relations. By Lemma 2.3, for every $i\in\\{1,2\\}$, there
exists an amenable ergodic subequivalence relation
$\mathscr{U}_{i}\leq\mathscr{T}_{i}$ such that
$\mathscr{U}_{i}\npreceq_{\mathscr{R}}\mathscr{R}(\Lambda\curvearrowright X)$.
Then we can literally repeat the exact same proof as in Claims 3.4 and 3.5 to
deduce a contradiction. ∎

We use Theorem 3.6 in combination with Connes–Jones’ construction [CJ81] to
provide examples of type ${\rm II_{1}}$ ergodic equivalence relations
$\mathscr{R}$ that are prime but for which the associated type ${\rm II_{1}}$
factor $\operatorname{L}(\mathscr{R})$ is not prime.

######  Proposition 3.7.

Let $\Gamma=\operatorname{SL}_{3}(\mathbb{Z})$ and $H$ be a nonabelian finite
discrete group. Set $\Lambda=\bigoplus_{\mathbb{N}}H$,
$(Y,\eta)=(H^{\mathbb{N}},\operatorname{Haar})$ and
$(X,\nu)=(Y^{\Gamma},\eta^{\otimes\Gamma})$. Consider the free ergodic pmp
action $\Gamma\times\Lambda\curvearrowright(X,\nu)$, where
$\Lambda\curvearrowright(X,\nu)$ acts diagonally and
$\Gamma\curvearrowright(X,\nu)$ acts by Bernoulli shifts. Set
$\mathscr{R}=\mathscr{R}(\Gamma\times\Lambda\curvearrowright X)$.

Then $\mathscr{R}$ is prime but $\operatorname{L}(\mathscr{R})$ is not prime.
In fact, $\operatorname{L}(\mathscr{R})$ is a McDuff factor meaning that
$\operatorname{L}(\mathscr{R})\cong\operatorname{L}(\mathscr{R})\mathbin{\overline{\otimes}}R$,
where $R$ is the unique hyperfinite type ${\rm II_{1}}$ factor.

###### Proof.

By Connes–Jones’ result [CJ81], $\operatorname{L}(\mathscr{R})$ is a McDuff
factor. Since $\Gamma$ has Kazhdan’s property (T) and since
$\Gamma\curvearrowright(X,\nu)$ is ergodic, it follows that $\mathscr{R}$ is
strongly ergodic. Assume that
$\mathscr{R}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, where
$\mathscr{T}_{1}$, $\mathscr{T}_{2}$ are ergodic equivalence relations. By
Theorem 3.6, there exists $i\in\\{1,2\\}$ such that
$\mathscr{T}_{i}\preceq_{\mathscr{R}}\mathscr{R}(\Lambda\curvearrowright X)$
as subequivalence relations. Since $\mathscr{R}(\Lambda\curvearrowright X)$ is
amenable, it follows that $\mathscr{T}_{i}$ is amenable and so
$\mathscr{T}_{i}$ is hyperfinite by [CFW81]. Since $\mathscr{R}$ is strongly
ergodic, it follows that $\mathscr{T}_{i}$ is finite. Therefore, $\mathscr{R}$
is prime. ∎

###  3.3. Proof of Corollary B

Let $n\geq 1$ and $\mathscr{R}_{1},\dots,\mathscr{R}_{n}$ be ergodic
equivalence relations that belong to $\mathfrak{Z}$. For every
$i\in\\{1,\dots,n\\}$, write
$\mathscr{R}_{i}=\mathscr{R}(\Gamma_{i}\curvearrowright X_{i})$.

$(\rm i)$ Assume that
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$,
where $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic
equivalence relations acting on $(Y_{1},\eta_{1})$ and $(Y_{2},\eta_{2})$
respectively. For every $j\in\\{1,2\\}$, denote by
$T_{j}\subset\\{1,\dots,n\\}$ a minimal subset for which
$\mathscr{T}_{j}\preceq_{\mathscr{R}}\mathscr{R}_{T_{j}}$ as subequivalence
relations, where $\mathscr{R}_{T_{j}}=\prod_{i\in T_{j}}\mathscr{R}_{i}$.
Since $\mathscr{T}_{j}$ is a type ${\rm II_{1}}$ ergodic equivalence relation,
we have $T_{j}\neq\emptyset$. Set $T=T_{1}\cup T_{2}$ and
$\mathscr{R}_{T}=\prod_{i\in T}\mathscr{R}_{i}$.

Firstly, we claim that $T=\\{1,\dots,n\\}$. Indeed, we have
$\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{T}$ and
$\mathscr{T}_{2}\preceq_{\mathscr{R}}\mathscr{R}_{T}$ as subequivalence
relations. By Facts 2.2$(\rm i)$, we obtain that
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T^{c}})$
and
$\operatorname{L}^{\infty}(Y_{1})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{T}_{2})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T^{c}})$.
By [DHI16, Lemma 2.4(3)], we moreover have
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T^{c}})$
and
$\operatorname{L}^{\infty}(Y_{1})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{T}_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T^{c}})$.
Then [BV12, Lemma 2.3] implies that
$\operatorname{L}(\mathscr{R})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T^{c}})$.
Since the group $\Gamma_{i}$ is infinite and the pmp action
$\Gamma_{i}\curvearrowright(X_{i},\nu_{i})$ is essentially free ergodic for
every $i\in\\{1,\dots,n\\}$, [HI17, Theorem 4.4] implies that we necessarily
have $T=\\{1,\dots,n\\}$.

Secondly, we claim that $T_{1}\cap T_{2}=\emptyset$. By contradiction, assume
that $T_{1}\cap T_{2}\neq\emptyset$ and let $i\in T_{1}\cap T_{2}$. Write
$\mathscr{R}=\mathscr{R}_{i}\times\mathscr{S}$, where
$\mathscr{S}=\mathscr{R}_{\\{1,\dots,n\\}\setminus\\{i\\}}$. By Theorem A, we
have $\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{S}$ or
$\mathscr{T}_{2}\preceq_{\mathscr{R}}\mathscr{S}$ as subequivalence relations.
Assume that $\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{S}$ as
subequivalence relations. The reasoning in the previous paragraph shows that
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{S})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{i})$
and
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T_{1}})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T_{1}^{c}})$.
Then [DHI16, Lemma 2.8(2)] implies that
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T_{1}\setminus\\{i\\}})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{(T_{1}\setminus\\{i\\})^{c}})$.
Then Fact 2.2$(\rm i)$ implies that
$\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{T_{1}\setminus\\{i\\}}$ as
subequivalence relations. This however contradicts the minimality of the set
$T_{1}$. Likewise, $\mathscr{T}_{2}\preceq_{\mathscr{R}}\mathscr{S}$
contradicts the minimality of the set $T_{2}$. Therefore, we have $T_{1}\cap
T_{2}=\emptyset$.

Therefore, we may write $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$. We have
$\mathscr{R}=\mathscr{R}_{T_{1}}\times\mathscr{R}_{T_{2}}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$.
Moreover, we have $\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{T_{1}}$
and $\mathscr{T}_{2}\preceq_{\mathscr{R}}\mathscr{R}_{T_{2}}$ as
subequivalence relations.

$(\rm ii)$ We prove the assertion by complete induction over $n\geq 1$.
Theorem A implies that the assertion holds for $n=1$. Assume that the
assertion holds for every $1\leq j\leq n-1$ and let us prove that it holds for
$n$.

Firstly, assume that
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\cdots\times\mathscr{T}_{p}$,
where $\mathscr{T}_{1},\dots,\mathscr{T}_{p}$ are type ${\rm II_{1}}$ ergodic
equivalence relations. We may assume that $p\geq n\geq 2$. Then item $(\rm i)$
implies that there exists a partition $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into
nonempty subsets such that
$\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{T_{1}}$ and
$\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{p}\preceq_{\mathscr{R}}\mathscr{R}_{T_{2}}$
as subequivalence relations. By Facts 2.2$(\rm vi)$, there exists an ergodic
equivalence relation $\mathscr{V}$ such that
$\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{p}\times\mathscr{V}$ and
$\mathscr{R}_{T_{2}}$ are stably isomorphic. Upon replacing $\mathscr{T}_{2}$
by $\mathscr{T}_{2}^{t}$ for $t>0$, we may assume that
$\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{p}\times\mathscr{V}\cong\mathscr{R}_{T_{2}}$.
By induction hypothesis and since $p-1\geq|T_{2}|$ and $|T_{2}|\leq n-1$, we
have that $\mathscr{V}$ is finite, $p-1=|T_{2}|\leq n-1$ and so $p=n$.

Secondly, assume that
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\cdots\times\mathscr{T}_{n}$.
Then item $(\rm i)$ implies that there exists a partition
$\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty subsets such that
$\mathscr{T}_{1}\preceq_{\mathscr{R}}\mathscr{R}_{T_{1}}$ and
$\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{n}\preceq_{\mathscr{R}}\mathscr{R}_{T_{2}}$
as subequivalence relations. By Facts 2.2$(\rm vi)$, there exists an ergodic
equivalence relation $\mathscr{U}$ such that
$\mathscr{T}_{1}\times\mathscr{U}$ and $\mathscr{R}_{T_{1}}$ are stably
isomorphic and there exists an ergodic equivalence relation $\mathscr{V}$ such
that $\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{n}\times\mathscr{V}$ and
$\mathscr{R}_{T_{2}}$ are stably isomorphic. Upon replacing $\mathscr{T}_{2}$
by $\mathscr{T}_{2}^{t}$ for $t>0$, we may assume that
$\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{n}\times\mathscr{V}\cong\mathscr{R}_{T_{2}}$.
Since $|T_{2}|\leq n-1$, the previous paragraph implies that $\mathscr{V}$ is
finite, $|T_{2}|=n-1$ and so $|T_{1}|=1$. Upon replacing $\mathscr{T}_{n}$ by
$\mathscr{T}_{n}\times\mathscr{V}$, we may assume that $\mathscr{V}$ is
trivial. Upon permuting the indices, we may assume that $T_{1}=\\{1\\}$ and so
$\mathscr{T}_{1}\times\mathscr{U}$ and $\mathscr{R}_{1}$ are stably
isomorphic. Since $\mathscr{R}_{1}$ is strongly prime, we have that
$\mathscr{U}$ is finite and so $\mathscr{T}_{1}$ and $\mathscr{R}_{1}$ are
stably isomorphic. Moreover, we have
$\mathscr{R}_{2}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{2}\times\cdots\times\mathscr{T}_{n}$.
By induction hypothesis and upon permuting the indices, we have that
$\mathscr{T}_{i}$ and $\mathscr{R}_{i}$ are stably isomorphic for every
$i\in\\{2,\dots,n\\}$. This concludes the proof of Corollary B.

###  3.4. Proof of Corollary C

Let $n\geq 1$ and $\mathscr{R}_{1},\dots,\mathscr{R}_{n}$ be strongly ergodic
equivalence relations that belong to $\mathfrak{Z}$. For every
$i\in\\{1,\dots,n\\}$, write
$\mathscr{R}_{i}=\mathscr{R}(\Gamma_{i}\curvearrowright X_{i})$. For every
$i\in\\{1,\dots,n\\}$, since $\Gamma_{i}$ is not inner amenable by Proposition
2.7, the type ${\rm II_{1}}$ factor $\operatorname{L}(\mathscr{R}_{i})$ is
necessarily full by [Ch81].

$(\rm i)$ Assume that
$\mathscr{R}=\mathscr{R}_{1}\times\cdots\times\mathscr{R}_{n}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$,
where $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic
equivalence relations acting on $(Y_{1},\eta_{1})$ and $(Y_{2},\eta_{2})$
respectively. We may literally repeat the proof of Corollary B$(\rm i)$. There
exists a partition $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty subsets
such that
$\operatorname{L}(\mathscr{T}_{1})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(Y_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T_{1}})\mathbin{\overline{\otimes}}\operatorname{L}^{\infty}(X_{T_{2}})$
and
$\operatorname{L}^{\infty}(Y_{1})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{T}_{2})\preceq^{s}_{\operatorname{L}(\mathscr{R})}\operatorname{L}^{\infty}(X_{T_{1}})\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{R}_{T_{2}})$.
A combination of [DHI16, Lemma 2.6(3)] and [OP07, Proposition 2.4(3)] implies
that $\operatorname{L}(\mathscr{T}_{j})$ is amenable relative to
$\operatorname{L}(\mathscr{R}_{T_{j}})$ inside $\operatorname{L}(\mathscr{R})$
for every $j\in\\{1,2\\}$. Note that $\operatorname{L}(\mathscr{R})$ is full
by [Co75, Corollary 2.3] and so are $\operatorname{L}(\mathscr{T}_{1})$,
$\operatorname{L}(\mathscr{T}_{2})$, $\operatorname{L}(\mathscr{R}_{T_{1}})$,
$\operatorname{L}(\mathscr{R}_{T_{2}})$. Then [IM19, Lemma 5.2] implies that
$\operatorname{L}(\mathscr{T}_{j})\preceq_{\operatorname{L}(\mathscr{R})}\operatorname{L}(\mathscr{R}_{T_{j}})$
for every $j\in\\{1,2\\}$. Reasoning as in the proof of [Ho15, Corollary
E(1)], we infer that there exists $t>0$ such that
$\mathscr{T}_{1}^{t}\cong\mathscr{R}_{T_{1}}$ and
$\mathscr{T}_{2}^{1/t}\cong\mathscr{R}_{T_{2}}$.

$(\rm ii)$ The proof follows by combining the ones of item $(\rm i)$,
Corollary B$(\rm ii)$ and [Ho15, Corollary E]. This concludes the proof of
Corollary C.

###  3.5. Proof of Theorem D

We prove the following key intermediate result which is analogous to [Dr19,
Proposition 4.1].

######  Proposition 3.8.

Let $n\geq 1$. For every $1\leq i\leq n$, let $k_{i}$ be a local field of
characteristic zero, $\mathbf{G}_{i}$ a $k_{i}$-isotropic $k_{i}$-simple
algebraic $k_{i}$-group and $\Gamma_{i}<\mathbf{G}(k)$ a Zariski dense
discrete subgroup with Kazhdan’s property (T). Set
$\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$. Let
$\Gamma\curvearrowright(X,\nu)$ be an essentially free ergodic pmp action and
set $\mathscr{R}=\mathscr{R}(\Gamma\curvearrowright X)$. Assume that
$\mathscr{R}\cong\mathscr{T}_{1}\times\mathscr{T}_{2}$, where
$\mathscr{T}_{1},\mathscr{T}_{2}$ are type ${\rm II_{1}}$ ergodic equivalence
relations. Set $M=\operatorname{L}(\mathscr{R})$ and for every
$j\in\\{1,2\\}$, set $P_{j}=\operatorname{L}(\mathscr{T}_{j})$.

Then there exist a partition $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty
subsets such that
$P_{j}\preceq^{s}_{M}\operatorname{L}(\Gamma_{T_{j}}\curvearrowright X)$ for
every $j\in\\{1,2\\}$.

###### Proof.

For every $j\in\\{1,2\\}$, denote by $T_{j}\subset\\{1,\dots,n\\}$ a minimal
subset for which $P_{j}\preceq^{s}_{M}M_{T_{j}}$, where
$M_{T_{j}}=\operatorname{L}(\Gamma_{T_{j}}\curvearrowright X)$ and
$\Gamma_{T_{j}}=\prod_{i\in T_{j}}\Gamma_{i}$. Since $P_{j}$ is a type ${\rm
II_{1}}$ factor, we have $T_{j}\neq\emptyset$. Set $T=T_{1}\cup T_{2}$,
$\Gamma_{T}=\prod_{i\in T}\Gamma_{i}$ and
$M_{T}=\operatorname{L}(\Gamma_{T}\curvearrowright X)$. Arguing as in the
proof of Corollary B, we deduce that $T=\\{1,\dots,n\\}$.

It remains to prove that $T_{1}\cap T_{2}=\emptyset$. By contradiction, assume
that $T_{1}\cap T_{2}\neq\emptyset$ and let $i\in T_{1}\cap T_{2}$. Set
$\Lambda=\prod_{i\neq j}\Gamma_{i}$ and
$\mathscr{S}=\mathscr{R}(\Lambda\curvearrowright X)$. Arguing as in the proof
of Corollary B, by minimality of $T_{1}$ and $T_{2}$ and [DHI16, Lemma
2.8(2)], we have that
$P_{j}\npreceq_{M}^{s}\operatorname{L}(\Lambda\curvearrowright X)$ for every
$j\in\\{1,2\\}$. By Fact 2.2$(\rm i)$, this further implies that
$\mathscr{T}_{j}\npreceq_{\mathscr{R}}\mathscr{S}$ as subequivalence
relations. This however contradicts Theorem 3.6. Therefore, we have
$\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$. ∎

###### Proof of Theorem D.

By combining Proposition 3.8 and [Dr19, Theorem 3.2], we infer that there
exist a partition $\\{1,\dots,n\\}=T_{1}\sqcup T_{2}$ into nonempty subsets,
$t>0$, an essentially free ergodic pmp action
$\Gamma_{T_{j}}\curvearrowright(X_{j},\nu_{j})$, where
$\Gamma_{T_{j}}=\prod_{i\in T_{j}}\Gamma_{i}$ and
$\operatorname{L}^{\infty}(X_{j},\nu_{j})=\operatorname{L}^{\infty}(X,\nu)^{\Gamma_{T_{j+1}}}$
for every $j\in\mathbb{Z}/2\mathbb{Z}$, a decomposition
$\operatorname{L}(\mathscr{R})=\operatorname{L}(\mathscr{T}_{1})^{t}\mathbin{\overline{\otimes}}\operatorname{L}(\mathscr{T}_{2})^{1/t}$
and $u\in\mathscr{U}(\operatorname{L}(\mathscr{R}))$, such that
$\Gamma\curvearrowright(X,\nu)$ is isomorphic to
$\Gamma_{T_{1}}\times\Gamma_{T_{2}}\curvearrowright(X_{1}\times
X_{2},\nu_{1}\otimes\nu_{2})$ and

$\operatorname{L}(\mathscr{T}_{1})^{t}=u\operatorname{L}(\Gamma_{T_{1}}\curvearrowright
X_{1})u^{*}\quad\text{and}\quad\operatorname{L}(\mathscr{T}_{2})^{1/t}=u\operatorname{L}(\Gamma_{T_{2}}\curvearrowright
X_{2})u^{*}.$

Arguing as in the proof of [Ho15, Corollary E], we infer that

$\mathscr{T}_{1}^{t}\cong\mathscr{R}(\Gamma_{T_{1}}\curvearrowright
X_{1})\quad\text{and}\quad\mathscr{T}_{2}^{1/t}\cong\mathscr{R}(\Gamma_{T_{2}}\curvearrowright
X_{2}).$

This finishes the proof of Theorem D. ∎

###  3.6. Proof of Corollary E

By applying Theorem D finitely many times, we can find a partition
$\\{1,\dots,n\\}=T_{1}\sqcup\cdots\sqcup T_{r}$ into nonempty subsets, an
essentially free ergodic pmp action
$\Gamma_{T_{j}}\curvearrowright(X_{j},\nu_{j})$, where
$\Gamma_{T_{j}}=\prod_{i\in T_{j}}\Gamma_{i}$ for every $j\in\\{1,\dots,r\\}$,
such that $\Gamma\curvearrowright(X,\nu)$ is isomorphic to
$\Gamma_{T_{1}}\times\cdots\times\Gamma_{T_{r}}\curvearrowright(X_{1}\times\cdots\times
X_{r},\nu_{1}\otimes\cdots\otimes\nu_{r})$ and
$\mathscr{R}(\Gamma_{T_{j}}\curvearrowright X_{j})$ is prime for every
$j\in\\{1,\dots,r\\}$.

To prove the uniqueness part, we consider another partition
$\\{1,\dots,n\\}=S_{1}\sqcup\cdots\sqcup S_{p}$ into nonempty subsets, an
essentially free ergodic pmp action
$\Gamma_{S_{j}}\curvearrowright(Y_{j},\nu_{j})$, where
$\Gamma_{S_{j}}=\prod_{i\in S_{j}}\Gamma_{i}$ for every $j\in\\{1,\dots,p\\}$
such that $\Gamma\curvearrowright(X,\nu)$ is isomorphic to
$\Gamma_{S_{1}}\times\cdots\times\Gamma_{S_{p}}\curvearrowright(Y_{1}\times\cdots\times
Y_{p},\eta_{1}\otimes\cdots\otimes\eta_{p})$ and
$\mathscr{R}(\Gamma_{S_{j}}\curvearrowright Y_{j})$ is prime for every
$j\in\\{1,\dots,p\\}$. Without loss of generality, we may assume that $r\geq
p$. We claim that for any $1\leq j\leq r$, there is $1\leq\ell\leq p$ such
that $T_{j}\subset S_{\ell}$. To see this, note that for any $1\leq j\leq r$,
there is $1\leq\ell\leq p$ such that $S_{\ell}\cap T_{j}\neq\emptyset$. Note
that the action $\Gamma_{T_{j}}\curvearrowright(X,\nu)$ is isomorphic to
$\Gamma_{S_{1}\cap T_{j}}\times\cdots\times\Gamma_{S_{p}\cap
T_{j}}\curvearrowright(Y_{1}\times\cdots\times
Y_{p},\eta_{1}\otimes\cdots\otimes\eta_{p})$. If $T_{j}\not\subset S_{\ell}$,
it is easy to see that $\mathscr{R}(\Gamma_{T_{j}}\curvearrowright X_{j})$ is
not prime, contradiction.

The above claim implies that $r=p$, and hence, $T_{j}=S_{j}$, for any $1\leq
j\leq r$. Then the conclusion follows.

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