# Co-spectral radius of intersections

Mikolaj Fraczyk  Department of Mathematics
University of Chicago
Chicago, IL 60637
USA  and  Wouter van Limbeek  Department of Mathematics, Statistics, and
Computer Science
University of Illinois at Chicago
Chicago, IL 60647
USA

(Date: August 27, 2024)

###### Abstract.

We study the behavior of the co-spectral radius of a subgroup $H$ of a
discrete group $\Gamma$ under taking intersections. Our main result is that
the co-spectral radius of an invariant random subgroup does not drop upon
intersecting with a deterministic co-amenable subgroup. As an application, we
find that the intersection of independent co-amenable invariant random
subgroups is co-amenable.

##  1\. Introduction

Let $\Gamma$ be a countable group and let $S$ be a finite symmetric subset of
$\Gamma$. The co-spectral radius of a subgroup $H\subset\Gamma$ (with respect
to $S$) is defined as the norm of the operator $M\colon L^{2}(\Gamma/H)\to
L^{2}(\Gamma/H)$:

$M\phi(\gamma H)=\frac{1}{|S|}\sum_{s\in S}\phi(s\gamma
H),\quad\rho(\Gamma/H):=\|M\|.$

The groups with co-spectral radius $1$ for every choice of $S$ are called co-
amenable. If the group $\Gamma$ is finitely generated, one needs only to
verify that the co-spectral radius is $1$ for some generating set $S$.

In this paper we investigate the behavior of co-spectral radius under
intersections. For general subgroups $H_{1},H_{2}\subset\Gamma$ there is not
much that can be said about $\rho(\Gamma/(H_{1}\cap H_{2}))$ other than the
trivial inequality

$\rho(\Gamma/(H_{1}\cap
H_{2}))\leq\min\\{\rho(\Gamma/H_{1}),\rho(\Gamma/H_{2})\\}.$

The problem of finding lower bounds on the co-spectral radius of an
intersection is even more dire, as there are examples of non-amenable $\Gamma$
with two co-amenable subgroups $H_{1},H_{2}$ with trivial intersection (see
Example 3.1). However, when considering all conjugates simultaneously, we have
the following elementary lower bound on the co-spectral radius of an
intersection. Here and for the remainder of the paper, we write
$H^{g}:=g^{-1}Hg$.

######  Theorem 1.1.

Let $\Gamma$ be a finitely generated group and let $S$ be a finite symmetric
generating set. Let $H_{1},H_{2}$ be subgroups of $\Gamma$ and assume that
$H_{1}$ is co-amenable. Then

$\sup_{g\in\Gamma}\rho(\Gamma/(H_{1}\cap H_{2}^{g}))=\rho(\Gamma/H_{2}).$

The supremum over all conjugates in the statement of the theorem is in fact
necessary, as shown by Example 3.1. However, in the presence of invariance,
this can be improved upon: E.g. if $H_{2}=N$ is normal, Theorem 1.1
immediately implies that $\rho(\Gamma/(H_{1}\cap N))=\rho(\Gamma/N)$. Our aim
is to generalize this to invariant random subgroups.

An invariant random subgroup of $\Gamma$ (see [2]) is a random subgroup of
$\Gamma$ whose distribution is invariant under conjugation. Invariant random
subgroups simultaneously generalize the notion of finite index subgroups and
normal subgroups. They have proven to be very useful tools in measured group
theory (see for example [1, 5, 4]). Many results on invariant random subgroups
are obtained as generalizations of statements previously known for normal
subgroups. We follow this tradition and show that one can remove the supremum
in Theorem 1.1 when $H_{2}$ is an IRS. The co-spectral radius of a subgroup
$H$ is invariant under conjugation of $H$ by elements of $\Gamma$ and defines
a measurable function111The measurability follows from the fact that
$\rho(H\backslash\Gamma)=\lim_{R\to\infty}\sup_{\operatorname{supp}f\subset
B(R)}\frac{\langle Mf,f\rangle}{\|f\|^{2}}$ where $B(R)$ is the $R$-ball
around $H$ in $H\backslash\Gamma$ and $f$ runs over non-zero functions
supported in $B(R)$. The ball $B(R)$ depends measurably on $H$, so the co-
spectral radius must be measurable as well. on the space of subgroups of
$\Gamma$. It follows that the co-spectral radius of an ergodic IRS $H$ is
constant a.s. and therefore $\rho(\Gamma/H)$ is well-defined. We say an IRS is
co-amenable if it is co-amenable a.s. Our main result is:

######  Theorem 1.2.

Let $\Gamma$ be a countable with a finite symmetric subset $S$. Let
$H_{1}\subset\Gamma$ be a deterministic co-amenable subgroup and let $H_{2}$
be an ergodic invariant random subgroup of $\Gamma$. Then

$\rho(\Gamma/(H_{1}\cap H_{2}))=\rho(\Gamma/H_{2})$

almost surely.

This result was inspired by a question of Alex Furman, asking whether the
intersection of a co-amenable IRS’ses remains co-amenable. A positive answer
follows from Theorem 1.2 applied in the case when both $H_{1},H_{2}$ are co-
amenable IRS’es.

######  Corollary 1.3.

Let $\Gamma$ be a countable group and let $H_{1},H_{2}$ be independent co-
amenable invariant random subgroups. Then the intersection $H_{1}\cap H_{2}$
is co-amenable.

######  Remark 1.4.

The independence assumption in the above corollary is necessary (see Example
3.2).

Combined with Cohen-Grigorchuk’s co-growth formula [6, 9] and Gekhtman-Levit’s
lower bound on the critical exponent of an IRS of a free group [7, Thm 1.1],
Theorem 1.2 yields the following corollary on the critical exponents of
subgroups of the free group.

######  Corollary 1.5.

Let $F_{d}$ be a free group on $d\geq 2$ generators and let $S$ be the
standard symmetric generating set. Write $\delta(H)$ for the critical exponent
of a subgroup $H$. Suppose $H_{1}$ is co-amenable and $H_{2}$ is an ergodic
IRS. Then

$\delta(H_{1}\cap H_{2})=\delta(H_{2}),$

almost surely.

###  1.1. Outline of the proof

We outline the proof of Theorem 1.2. For the sake of simplicity we restrict to
the case when both $H_{1},H_{2}$ are co-amenable. We realize $H_{1}$ and
$H_{2}$ as stabilizers of $\Gamma$-actions on suitable spaces $X_{1}$ and
$X_{2}$. Since $H_{1}$ is a (deterministic) subgroup, we can take
$X_{1}=\Gamma/H_{1}$. On the other hand, $H_{2}$ is an ergodic IRS so $X_{2}$
is a probability space with an ergodic measure-preserving action of $\Gamma$.
The intersection $H_{1}\cap H_{2}$ is then the stabilizer of a point in
$X_{1}\times X_{2}$ that is deterministic in the first variable and random in
the second. Using an analogue of the Rokhlin lemma, we can find a positive
measure subset $E$ of $X_{2}$ that locally approximates the coset space
$H_{2}\backslash\Gamma$. The product of $E$ with $X_{1}$ will locally
approximate the coset space of $(H_{1}\cap H_{2})\backslash\Gamma$. The co-
amenability of $H_{2}$ means that the set $E$ contains a subset $P$ that is
nearly $\Gamma$-invariant. The product of such a set with a Følner set $F$ in
$X_{1}$ should be a nearly $\Gamma$-invariant set in $X_{1}\times E$, and
hence witnesses the amenability of the coset graph $(H_{1}\cap
H_{2})\backslash\Gamma$. The latter implies that $H_{1}\cap H_{2}$ is co-
amenable.

The actual proof is more complicated because if the product system is not
ergodic, one has to show the product set is nearly invariant after restriction
to each ergodic component and not just on average. Otherwise we can only
deduce the bound from Theorem 1.1 using a supremum over all conjugates.
Obtaining control on each ergodic component is a key part of the proof where
we actually use the invariance of $H_{2}$.

To prove Theorem 1.2 for the co-spectral radius, one should replace the Følner
set in $X_{2}$ with a function $f_{2}$ that (nearly) witnesses the fact that
$\rho(\Gamma/H_{2})=\lambda_{2}$, and adapt the remainder of the proof
accordingly.

### Outline of the paper

Section 2 contains background material. In Section 3 we prove the
deterministic bound on spectral radius given by Theorem 1.1. Next, in Section
4, we rephrase co-spectral radius of the (discrete) orbits in terms of
_embedded spectral radius_ on a (continuous) measure space.

### Acknowledgments

The authors thank Gabor Pete and Tianyi Zheng for showing us a related problem
for intersections of percolations on $\mathbb{Z}$. We thank Alex Furman for
suggesting the problem as well as for helpful discussions. MF thanks Miklos
Abert for useful discussions. We thank the anonymous referee for valuable
comments. We thank the University of Illinois at Chicago for providing support
for a visit by MF. MF was partly supported by ERC Consolidator Grant 648017.
WvL is supported by NSF DMS-1855371.

##  2\. Background

###  2.1. Co-amenability

Let $\Gamma$ be a finitely generated group and let $S$ be a finite symmetric
set of generators. A subgroup $H$ of $\Gamma$ is called  co-amenable if the
Schreier graph $\operatorname{Sch}(H\backslash\Gamma,S)$ is amenable, i.e for
any $\varepsilon>0$ and any $S$ there exists a set $F\subset
H\backslash\Gamma$ such that $|F\Delta FS|\leq\varepsilon|F|$. Such sets will
be called $\varepsilon$-Følner sets. Alternatively, a subgroup $H$ is co-
amenable if and only if the representation $\ell^{2}(\Gamma/H)$ has almost
invariant vectors, or that $\|M\|_{L^{2}(H\backslash\Gamma)}=1.$

###  2.2. Invariant random subgroups

Let $\operatorname{Sub}_{\Gamma}$ be the space of subgroups of $\Gamma$,
equipped with the topology induced from $\\{0,1\\}^{\Gamma}$. An invariant
random subgroup is a probability measure
$\mu\in\mathcal{P}(\operatorname{Sub}_{\Gamma})$ which is invariant under
conjugation of $\Gamma$. An IRS is called co-amenable if

$\mu(\\{H\in\operatorname{Sub}_{\Gamma}|H\textrm{ is co-amenable}\\})=1.$

Similarly, we say that an IRS $H$ has co-spectral radius at least $\lambda$ if
$\rho(H\backslash\Gamma)\geq\lambda$ almost surely.

For any action $\Gamma\curvearrowright X$ and $x\in X$ write $\Gamma_{x}$ for
the stabilizer of $x$. Every IRS can be realized as a stabilizer of a random
point in a probability measure preserving system:

######  Theorem 2.1 (Abert-Glasner-Virag [2]).

For every IRS $\mu$, there exists a standard Borel probability space $(X,\nu)$
and a Borel p.m.p. $\Gamma$-action on $(X,\nu)$ such that
$\mu=\int_{X}\delta_{\Gamma_{x}}\,d\nu(x).$

###  2.3. Ergodic decomposition of infinite measures

The material in this subsection is well-known to experts but difficult to
locate in the literature. Our goal is to construct an ergodic decomposition
for measure-preserving actions of countable groups on spaces with an infinite
measure. We deduce this from the corresponding result for nonsingular actions
on probability spaces:

######  Theorem 2.2 (Greschonig-Schmidt [8, Thm 1]).

Let $\Gamma$ be a countable group and let
$\Gamma\curvearrowright(X,\Sigma_{X},\nu)$ be a nonsingular Borel action on a
standard Borel probability space. Then there exist a standard Borel
probability space $(Z,\Sigma_{Z},\tau)$ and a family of quasi-invariant,
ergodic, pairwise mutually singular probability measures $\\{\nu_{z}\\}_{z\in
Z}$ with the same Radon-Nikodym cocycle as $\nu$, and such that for every
$B\in\Sigma_{X}$, we have

$\nu(B)=\int_{Z}\nu_{z}(B)d\tau(z).$ (2.1)

As an application we have:

######  Corollary 2.3.

Let $\Gamma$ be a countable group and let
$\Gamma\curvearrowright(X_{1},\Sigma_{X_{1}},\nu_{1})$ and
$\Gamma\curvearrowright(X_{2},\Sigma_{X_{2}},\nu_{2})$ be measure-preserving
Borel actions on standard Borel spaces. Suppose that $(X_{1},\nu_{1})$ is
ergodic and that $\nu_{2}(X_{2})=1.$ Then there exists a standard Borel
probability space $(Z,\Sigma_{Z},\tau)$ and a family of $\Gamma$-invariant,
ergodic, pairwise mutually singular measures $\\{\nu_{z}\\}_{z\in Z}$ on
$X_{1}\times X_{2}$ such that for every $B\in\Sigma_{X_{1}\times X_{2}}$, we
have

$\nu(B)=\int_{Z}\nu_{z}(B)d\tau(z).$ (2.2)

Moreover, for every measurable set $F\subset X_{1}$ and $z\in Z$ we have

$\nu_{z}(F\times X_{2})=\nu_{1}(F).$

###### Proof.

Fix a countably-valued Borel function $w\colon X_{1}\to\mathbb{R}_{>0}$, such
that $\int_{X_{1}}w\,d\nu_{1}=1$. Write $w=c_{i}$ on the set $A_{i}$, where
$\\{A_{i}\\}$ is a Borel partition of $X_{1}$.

Then $w(x_{1})d\nu(x_{1})d\nu_{2}(x_{2})$ is a $\Gamma$-quasi-invariant
probability measure on $X_{1}\times X_{2}$ with Radon-Nikodym cocycle
$dw(x_{1},x_{2},\gamma)=\frac{w(\gamma x_{1})}{w(x_{1})}$. Let
$(Z,\Sigma_{Z},\tau)$, $z\mapsto(w\nu)_{z}$, be its ergodic decomposition as
provided by Theorem 2.1. Now pass back from
$w(x_{1})d\nu_{1}(x_{1})d\nu_{2}(x_{2})$ to $\nu_{1}\times\nu_{2}$ by setting

$d\nu_{z}(x_{1},x_{2}):=w(x_{1})^{-1}d(w\nu)_{z}.$

Since $\\{(w\nu)_{z}\\}_{z}$ are ergodic and pairwise mutually singular, the
same is true for $\\{\nu_{z}\\}_{z}$. Since $(w\nu)_{z}$ have Radon-Nikodym
cocycle $dw$, the measures $\nu_{z}$ are $\Gamma$-invariant.

It is easy to verify that Equation (2.1) implies the corresponding Equation
(2.2).

Finally, to satisfy the last identity, choose a positive measure subset
$F\subset X_{1}$ and renormalize $\nu_{z}$ and $\tau$ as follows:

$\nu_{z}\mapsto\frac{\nu_{1}(F)}{\nu_{z}(F\times X_{2})}\nu_{z},\quad
d\tau(z)\to\frac{\nu_{z}(F\times X_{2})}{\nu_{1}(F)}d\tau(z).$

By ergodicity of $\nu_{1}$, this normalization does not depend on the choice
of $F$. ∎

###  2.4. Ergodic theory of equivalence relations

Let $(X,\nu)$ be a probability measure space and let $\varphi_{i}:U_{i}\to X$
be a finite family of non-singular measurable maps defined on subsets $U_{i}$
of $X$. The triple $(X,\nu,(\varphi_{i})_{i\in I})$ is called a graphing. We
assume that $(\varphi_{i})_{i\in I}$ is symmetric, i.e. for each $i\in I$ the
map $\varphi_{i}^{-1}\colon\varphi_{i}(U_{i})\to U_{i}$ is also in the set
$(\varphi_{i})_{i\in I}$. A graphing is finite if the index set $I$ is finite.

######  Remark 2.4.

In our applications of this theory, $X$ will be a finite measure subset (not
necessarily invariant) of a measure-preserving action of $\Gamma$, equipped
with the graphing corresponding to a finite symmetric generating set $S$ of
$\Gamma$, and $\nu$ will be the restricted measure or the restriction of an
ergodic component.

Let $\mathcal{R}$ be the orbit equivalence relation generated by the maps
$(\varphi_{i})_{i\in I}$. A measured graphing yields a random graph in the
following way: For every $x\in X$, let $\mathcal{G}_{x}$ be the graph with
vertex set given by the equivalence class $[x]_{\mathcal{R}}$ and place an
edge between $y,z\in[x]_{\mathcal{R}}$ whenever $z=\varphi_{i}(y)$ for some
$i\in I$ (multiple edges are allowed). The graphs $\mathcal{G}_{x}$ have
degrees bounded by $|I|$ and are undirected since $(\varphi_{i})_{i\in I}$ is
symmetric. If we choose a $\nu$-random point $x$, the resulting graph
$\mathcal{G}_{x}$ is a random rooted graph. The properties of
$\mathcal{G}_{x}$ will depend on the graphing. For example, if the graphing
consists of measure preserving maps then the resulting random graph is
unimodular (see [3]).

Suppose from now on the graphing is measure-preserving. Then the mass
transport principle [3] asserts that for any measurable function
$K:\mathcal{R}\to\mathbb{R}$, we have

$\int_{X}\left(\sum_{x^{\prime}\in[x]_{\mathcal{R}}}K(x,x^{\prime})\right)\,d\nu(x)=\int_{X}\left(\sum_{x\in[x^{\prime}]_{\mathcal{R}}}K(x,x^{\prime})\right)\,d\nu(x^{\prime}).$
(2.3)

##  3\. Co-spectral radius for deterministic intersections

In this section, we prove Theorem 1.1 that gives the elementary deterministic
lower bound on the supremum of co-spectral radii over all conjugates. Then we
show an example that consideration of all conjugates is necessary. This
example will also show the necessity of the independence assumption in
Corollary 1.3 on the co-amenability of the intersection of a pair of
independent co-amenable IRS’es.

###### Proof of Theorem 1.1.

As in the introduction, we let $M:=\frac{1}{|S|}\sum_{s\in
S}s\in\mathbb{C}[\Gamma]$. We have the following identity between the unitary
representations of $\Gamma$:

$L^{2}(H_{1}\backslash\Gamma)\otimes
L^{2}(H_{2}\backslash\Gamma)\simeq\bigoplus_{g\in
H_{1}\backslash\Gamma/H_{2}}L^{2}((H_{1}\cap H_{2}^{g})\backslash\Gamma).$

Write $\pi_{1},\pi_{2}$ for the unitary representations corresponding to
$L^{2}(H_{1}\backslash\Gamma)$ and $L^{2}(H_{2}\backslash\Gamma)$. The above
identity implies that

$\sup_{g\in H_{1}\backslash\Gamma/H_{2}}\rho((H_{1}\cap
H_{2}^{g})\backslash\Gamma)=\|(\pi_{1}\otimes\pi_{2})(M)\|.$

To prove the theorem, it is enough to verify that

$\|(\pi_{1}\otimes\pi_{2})(M)\|\geq\|\pi_{2}(M)\|=\rho(H_{2}\backslash\Gamma).$

Let $\varepsilon>0$. Choose unit vectors $u_{1}\in
L^{2}(H_{1}\backslash\Gamma)$ and $u_{2}\in L^{2}(H_{2}\backslash\Gamma)$ such
that $\langle\pi_{1}(s)u_{1},u_{1}\rangle\geq 1-\varepsilon$ for all $s\in S$
and $\langle\pi_{2}(M)u_{2},u_{2}\rangle\geq\|\pi_{2}(M)\|-\varepsilon$. Then

$\displaystyle\langle(\pi_{1}\otimes\pi_{2})(M)u_{1}\otimes u_{2},u_{1}\otimes
u_{2}\rangle=$ $\displaystyle\frac{1}{|S|}\sum_{s\in
s}\langle\pi_{1}(s)u_{1},u_{1}\rangle\langle\pi_{2}(s)u_{2},u_{2}\rangle$
$\displaystyle\geq$ $\displaystyle\frac{1}{|S|}\sum_{s\in
S}(1-\varepsilon)\langle\pi_{2}(s)u_{2},u_{2}\rangle$ $\displaystyle\geq$
$\displaystyle(1-\varepsilon)(\|\pi_{2}(M)\|-\varepsilon).$

Letting $\varepsilon\to 0$ we conclude that
$\|(\pi_{1}\otimes\pi_{2})(M)\|\geq\|\pi_{2}(M)\|.$ ∎

The supremum in the inequality seems to be necessary. Below we construct an
example of a non-amenable finitely generated group $\Gamma$ with two co-
amenable subgroups $H_{1},H_{2}$ such that the intersection $H_{1}\cap H_{2}$
is trivial. In particular

$\rho(\Gamma)=\rho((H_{1}\cap
H_{2})\backslash\Gamma)<\rho(H_{2}\backslash\Gamma)=1.$

######  Example 3.1.

Let $\Gamma:=F_{2}^{\oplus\mathbb{Z}}\rtimes\mathbb{Z}$, where $F_{2}$ stands
for the free group on two generators. The group is obviously non-amenable. Let
$a,b$ be the standard generators of $F_{2}$ and let $s$ be the generator of
the copy of $\mathbb{Z}$ in $\Gamma$. The triple $\\{s,a,b\\}$ generates
$\Gamma$. Put $S:=\\{s,a,b,s^{-1},a^{-1},b^{-1}\\}.$ For any subset
$E\subset\mathbb{Z}$ let $H_{E}:=F_{2}^{\oplus E}\subset\Gamma$.

Now let $A,B$ be disjoint subsets of $\mathbb{Z}$ containing arbitrary long
segments. Since $A\cap B=\varnothing$, the intersection $H_{A}\cap H_{B}=1$ is
not co-amenable. On the other hand, we claim that for any subset $C$
containing arbitrarily long segments, $H_{C}$ is co-amenable, so that in
particular $H_{A}$ and $H_{B}$ are co-amenable: Indeed, suppose
$C\subseteq\mathbb{Z}$ contains arbitrarily long segments. Then for any
$g\in\Gamma$, the Schreier graphs for $H_{C}$ and $H_{C}^{g}$ are isomorphic,
so $\rho(H_{C}\backslash\Gamma)=\rho(H_{C}^{g}\backslash\Gamma)$. For every
$n\in\mathbb{N}$

$\rho(H_{C}\backslash\Gamma)=\rho(H_{C}^{s^{n}}\backslash\Gamma)=\rho(H_{C-n}\backslash\Gamma).$

Let $(n_{k})_{k\in\mathbb{N}}$ be a sequence such that
$\\{-k,-k+1,\ldots,k-1,k\\}\subset C-{n_{k}}$. Then $H_{C-n_{k}}$ converges to
$H_{\mathbb{Z}}$ in ${\rm Sub}(\Gamma)$ as $k\to\infty$. Since the spectral
radius is lower semi-continuous on the space of subgroups, we get

$\rho(H_{C}\backslash\Gamma)=\liminf_{k\to\infty}\rho(H_{C-n_{k}}\backslash\Gamma)\geq\rho(H_{\mathbb{Z}}\backslash\Gamma)=\rho(\mathbb{Z})=1.$

######  Example 3.2.

The above example also shows that for the intersection of two co-amenable
IRS’es to be co-amenable (Corollary 1.3), the independence assumption is
necessary. Indeed, let $\Gamma$ be as in the previous example and let $A$ be
an invariant percolation on $\mathbb{Z}$ such that both $A$ and its complement
contain arbitrarily long segments (e.g. Bernoulli percolation). Then $H_{A}$
and $H_{A^{c}}$ are co-amenable but their intersection is trivial.

##  4\. Embedded spectral radius

Let us introduce some terminology. Let $(X,\nu)$ be a measure preserving
$\Gamma$-action, and write $\mathcal{R}$ for the corresponding orbit
equivalence relation. We shall assume that $\nu$ is $\sigma$-finite but not
necessarily finite. We recall that for every $x\in X$, $\mathcal{G}_{x}$ is
the labeled graph with vertex set $[x]_{\mathcal{R}}$ and edge set
$(y,sy),y\in\Gamma x$, labeled by $s\in S$.

######  Definition 4.1.

A set $P\subset X$ is called a finite connected component if for almost all
$x\in P$ the connected component of $x$ in the graph $P\cap\mathcal{G}_{x}$ is
finite. In other words, the graphing restricted to $P$ generates a finite
equivalence relation.

For any subset $P\subset X$ write $\partial P:=SP\setminus P$ for the (outer)
boundary and ${\rm int}(P):=P\setminus\partial(X\setminus P)$ for the interior
of $P$.

######  Definition 4.2.

Let $(X,\nu)$ be a measure-preserving $\Gamma$-action. We say that $(X,\nu)$
has embedded spectral radius $\lambda$ if for every finite measure finite
connected component $P\subset X$ and every $f\in L^{2}(X,\nu)$ supported on
the interior ${\rm int}(P)$, we have

$\langle(I-M)f,f\rangle\geq(1-\lambda)\|f\|^{2},$

and $\lambda$ is minimal with this property.

######  Remark 4.3.

Using the monotone convergence theorem, we may assume that $f$ in the above
definition is bounded. Further, taking the absolute value of $f$ leaves the
right-hand side unchanged, and decreases the left-hand side. Therefore it
suffices to consider nonnegative functions $f\geq 0$.

Our goal in this section is to prove that the embedded spectral radius of a
measure preserving system $\Gamma\curvearrowright(X,\nu)$ is detected by the
co-spectral radius along orbits:

######  Proposition 4.4.

Let $(X,\nu)$ be a $\sigma$-finite measure-preserving $\Gamma$-system. Then
the stabilizer of almost every point has co-spectral radius at least $\lambda$
if and only if almost every ergodic component of $\nu$ has embedded spectral
radius at least $\lambda$.

######  Remark 4.5.

This result can be used to give examples whose embedded spectral radius is
strictly less than the spectral radius of $M$ on $L^{2}_{0}(X,\nu)$. This
happens for example when $\Gamma$ is a non-abelian free group and
$X=X_{1}\times X_{2}$ is a product of an essentially free action $X_{1}$ with
an action $X_{2}$ that has no spectral gap. In this case, the graphs
$\mathcal{G}_{x}$ are just copies of the Cayley graph of $\Gamma$ so their
spectral radius is bounded away from $1$. On the other hand, the spectral
radius of $M$ on $L^{2}_{0}(X,\nu)$ is $1$, because it contains
$L^{2}_{0}(X_{2},\nu_{2})$.

###### Proof.

By passing to ergodic components we can assume without loss of generality that
$(X,\nu)$ is ergodic. If $(X,\nu)$ is periodic then there is nothing to do, so
henceforth we will assume that $(X,\nu)$ is an aperiodic measure preserving
ergodic system.

First let us prove that if the co-spectral radius of the stabilizer
$\Gamma_{x}$ is at least $\lambda$, then $X$ has embedded spectral radius at
least $\lambda$. Let $\varepsilon>0$ be arbitrary. Then $\nu$-almost every
orbit $\mathcal{G}_{x}$ supports a function
$f_{x}:\mathcal{G}_{x}\to\mathbb{R}$ such that

$\langle(I-M)f_{x},f_{x}\rangle\leq(1-\lambda+\varepsilon)\|f_{x}\|^{2}.$
(4.1)

Since $\mathcal{G}_{x}$ is countable, using the Monotone Convergence Theorem,
we can assume $f_{x}$ is supported on a finite ball. Let $R_{x}>0$ be minimal
such that the interior of the ball $B_{\mathcal{G}_{x}}(x,R_{x})$ of radius
$R_{x}$ around $x$ supports a function $\psi_{x}$ satisfying (4.1). Since
balls of fixed radius depend measurably on $x$, the map $x\mapsto R_{x}$ is
measurable, so we can choose $R_{0}>0$ such that

$\nu\,(\\{x\in X\mid R_{x}\leq R_{0}\\})>0$

and put $X_{1}:=\\{x\in X\mid R_{x}\leq R_{0}\\}.$ Since there are only
finitely many rooted graphs of radius $R_{0}$ labeled by $S$, there exists a
positive measure set $X_{2}\subset X_{1}$ such that for all $x\in X_{2}$, the
rooted graphs $(B_{\mathcal{G}_{x}}(x,R_{0}),x)$ are all isomorphic to some
$(\mathcal{G},o)$ as rooted $S$-labeled graphs. By restricting to a smaller
subset, we can assume that $\nu(X_{2})$ is finite. Fix
$\psi:\mathcal{G}\to\mathbb{R}$ satisfying

$\langle(I-M)\psi,\psi\rangle\leq(1-\lambda+\varepsilon)\|\psi\|^{2},$

and for $x\in X_{2}$, let $B_{x}\subset X$ be the image of $\mathcal{G}$ via
the unique labeled isomorphism
$(\mathcal{G},o)\simeq(B_{\mathcal{G}_{x}}(x,R_{0}),x)$.

Let $S^{\prime}$ be the set of all products of at most $2R_{0}$ elements of
$S$. At this point we need to use a Rokhlin-type lemma, which will be stated
and proved below (see Lemma 4.6). Upon applying this to the graphing
$(S^{\prime}X_{2},\nu,S^{\prime})$ we find a partition
$S^{\prime}X_{2}=B\sqcup\bigsqcup_{j=1}^{N}A_{j}$ with $\nu(B)<\nu(X_{2})/2$,
such that $A_{j}\cap sA_{j}=\\{x\in A_{j}\mid sx=x\\}$ for every $s\in
S^{\prime}$. This translates to the condition that that $B_{x}$ and
$B_{x^{\prime}}$ are disjoint for every distinct pair of points $x,y\in
A_{j}$. Since the sets $A_{j}$ cover a subset of $X_{2}$ of measure at least
$\nu(X_{2})/2$, there exists $j$ such that $X_{3}:=X_{2}\cap A_{j}$ has
positive measure. The set $P:=\bigcup_{x\in X_{3}}B_{x}$ is then a disjoint
union of its finite connected components $B_{x}$, so it is a finite connected
component in the sense of Definition 4.1. The function
$\psi:\mathcal{G}\to\mathbb{R}$ naturally induces a function
$f:P\to\mathbb{R}$ defined by $f|_{B_{x}}:=\psi$ for all $x\in X_{3}$, where
we have identified $B_{x}$ with $\mathcal{G}$ using the isomorphism of rooted
$S$-labeled graphs $(B_{x},x)\simeq(\mathcal{G},o)$.

Then we easily verify
$\langle(I-M)f,f\rangle\leq(1-\lambda+\varepsilon)\|f\|^{2}$, namely

$\|f\|^{2}=\int_{X_{3}}\|\psi\|^{2}d\nu=\nu(X_{3})\|\psi\|^{2},$

and similarly

$\langle Mf,f\rangle=\int_{X_{3}}\langle
M\psi,\psi\rangle\,d\nu=\nu(X_{3})\langle M\psi,\psi\rangle.$

Taking $\varepsilon\to 0$, we see that $X$ has embedded spectral radius at
least $\lambda$.

We prove the other direction. The proof will use the mass transport principle
for unimodular random graphs. In our case the unimodular random graph is given
by $(\mathcal{G}_{x},x)$ where $x\in X$ is $\nu|_{P}$-random. We argue by
contradiction, so assume that $(X,\nu)$ has embedded spectral radius at least
$\lambda$ but at the same time stabilizers have co-spectral radius
$\rho<\lambda$ with positive probability. By ergodicity, there exists an $h>0$
such that $\rho(\mathcal{G}_{x})\leq\lambda-h$ a.s.

Since $X$ has spectral radius at least $\lambda$, there exists $f\in
L^{2}(X,\nu)$ nonzero and supported on the interior of a finite connected
component $P\subseteq X$ with $\nu(P)<\infty$ such that

$\langle(I-M)f,f\rangle\leq\left(1-\lambda+\frac{h}{2}\right)\|f\|^{2}.$ (4.2)

As in Section 2.4, let $\mathcal{R}_{P}$ be the equivalence relation generated
by the graphing on $P$. We write $P_{x}^{o}:=[x]_{\mathcal{R}_{P}}$ for the
connected component of $x\in P$. Define $K:\mathcal{R}_{P}\to\mathbb{R}$ by

$K(x,y):=\frac{f(x)^{2}}{\|f\|_{P_{x}^{o}}^{2}}(2|S|)^{-1}\sum_{s\in
S}(f(ys)-f(y))^{2}$ (4.3)

By the mass transport principle, we equate

$\int_{P}\sum_{x\in P_{y}^{o}}K(x,y)\,d\nu(y)=\int_{P}\sum_{y\in
P_{x}^{o}}K(x,y)\,d\nu(x).$ (4.4)

We start by computing the integrand on the right-hand side. Rewriting

$(f(ys)-f(y))^{2}=f(ys)(f(ys)-f(y))+f(y)(f(y)-f(ys)),$

we find (using $S$ is symmetric) for every $x\in X$ that

$\displaystyle\sum_{y\in P_{x}^{o}}K(x,y)$
$\displaystyle=\frac{f(x)^{2}}{\|f\|_{P_{x}^{o}}^{2}}|S|^{-1}\sum_{y\in
P_{x}^{o}}\sum_{s\in S}f(y)\left(f(y)-f(ys)\right)$
$\displaystyle=\frac{f(x)^{2}}{\|f\|_{P_{x}^{o}}^{2}}\left\langle(I-M)f,f\right\rangle_{P_{x}^{o}}.$

Using $\rho(\mathcal{G}_{x})\leq\lambda-h$ and $f\geq 0$, we can estimate

$\sum_{y\in P_{x}^{o}}K(x,y)\geq(1-\lambda+h)f(x)^{2}.$ (4.5)

Therefore we have the following estimate for the right-hand side in the Mass
Transport Equation (4.4)

$\int_{P}\left(\sum_{y\in
P_{x}^{o}}K(x,y)\right)\,d\nu(x)\geq(1-\lambda+h)\|f\|^{2}.$ (4.6)

Next, we compute the integrand on the left-hand side of the Mass Transport
Equation (4.4), namely for $y\in X$, we have

$\displaystyle\sum_{x\in P_{y}^{o}}K(x,y)$ $\displaystyle=\sum_{x\in
P_{y}^{o}}\frac{f(x)^{2}}{\|f\|^{2}_{P_{y}^{o}}}(2|S|)^{-1}\sum_{s\in
S}(f(ys)-f(y))^{2}$ $\displaystyle=f(y)(I-M)(f)(y),$

where we used $S$ is symmetric and the action is measure-preserving. Hence,
integrating the above equation over $y$ and using the mass transport principle
to estimate this by the right-hand side of Equation (4.6), we find

$\langle(I-M)f,f\rangle\geq(1-\lambda+h)\|f\|^{2}.$

This contradicts the choice of $f$ in Equation (4.2). ∎

We end this section with the following technical Rokhlin-type lemma that was
used in the above proof:

######  Lemma 4.6.

$(X,\nu,(\varphi_{i})_{i\in I})$ be a finite measure preserving symmetric
graphing on a finite measure space. Then, for every $\delta>0$ there exists a
measurable partition $X=B\sqcup\bigsqcup_{j=0}^{N}A_{j}$, such that
$\nu(B)\leq\delta$ and

$A_{j}\cap\varphi_{i}(A_{j}\cap U_{i})=\\{x\in A_{j}\cap
U_{i}\mid\varphi_{i}(x)=x\\}.$

###### Proof.

We start by proving the lemma for a single measure preserving invertible map
$\varphi\colon U\to\varphi(U).$ Since we do not assume that $\varphi$ is
defined on all of $X$ we need to treat separately the subset of elements where
$\varphi$ can be applied only finitely many times. For any $n\in\mathbb{N}$
define

$E_{n}:=\\{x\in X\mid\varphi^{n-1}(x)\in U\text{ but }\varphi^{n}(x)\not\in
U\\}.$

Put $A_{\rm 0}=\cup_{n=0}^{\infty}E_{2n}$ and $A_{\rm
1}=\cup_{n=0}^{\infty}E_{2n+1}.$ We obviously have $A_{0}\cap
A_{1}=\emptyset$, $\varphi^{\pm 1}(A_{0})\subset A_{1}$ and $\varphi^{\pm
1}(A_{1})\subset A_{0}$. This reduces the problem to the subset
$Y:=X\setminus\bigcup_{n=0}^{\infty}E_{n}$. By definition, $\varphi(Y)=Y$. We
further decompose $Y$ into the periodic and aperiodic parts $Y^{\rm p},Y^{\rm
ap}$. The periodic part can be partitioned into a fixed point set
$A_{2}=\\{x\in Y^{\rm p}\mid\varphi(x)=x\\}$, finitely many sets
$A_{3},\ldots,A_{M}$ permuted by $\varphi$ and a remainder $B_{1}$ of measure
$\nu(B_{1})<\delta/2$ coming from large odd periods. By the usual Rokhlin
lemma, the aperiodic part $Y^{\rm ap}$ can be decomposed as $Y^{\rm
ap}=B_{2}\sqcup A_{M+1}\sqcup\ldots\sqcup A_{N}$ where $\nu(B_{2})<\delta/2$,
$\varphi(A_{M+k})=A_{M+k+1}$ for all $M+k<N$ and $\varphi(A_{N})\subset
B_{2}.$ Put $B=B_{1}\cup B_{2}$. This ends the construction for a single map.

Suppose now that the graphing consists of $d$ maps
$\varphi_{1},\ldots,\varphi_{d}$ and their inverses. For each $i=1,\ldots d$
there exists a partition $X=B^{i}\sqcup\bigsqcup_{j=1}^{N_{i}}A_{j}^{i}$ such
that $\nu(B^{i})<\frac{\delta}{d}$ and $\varphi_{i}(A_{j}^{i}\cap U_{i})\cap
A_{j}^{i}=\\{x\in(A_{j}^{i}\cap U_{i})\mid\varphi_{i}(x)=x\\}.$ Let
$B:=\cup_{i=1}^{d}B^{i}$ and define the partition $\\{A_{j}\\}$ as the product
partition $\bigwedge_{i=1}^{d}\\{A_{j}^{i}\\}.$ This partition satisfies all
the desired conditions. ∎

##  5\. Proof of the Main Theorem

###  5.1. Preliminary reductions and general strategy

Let $\Gamma$ be a countable group with a co-amenable subgroup $H_{1}$ and an
IRS $H_{2}$ with co-spectral radius $\lambda_{2}$. We need to show that
$H_{1}\cap H_{2}$ has co-spectral radius at least $\lambda_{2}$ as well. First
of all, without loss of generality we can assume $H_{2}$ is ergodic. The group
$H_{1}$ is realized as the stabilizer of a point in
$X_{1}:=H_{1}\backslash\Gamma$ and $H_{2}$ is realized as the stabilizer of a
random point in a p.m.p. action of $\Gamma$ on $(X_{2},\nu_{2})$. We use
Proposition 4.4 to find a finite connected component $P_{2}$ of $X_{2}$ and a
function $f_{2}$ on $P_{2}$ that witnesses the spectral radius $\lambda_{2}$.
Next, using a large Fölner set in $X_{1}$, we produce a new finite connected
component in the product system $X_{1}\times X_{2}$ and a new function which
certifies that the co-spectral radius of stabilizers in $X_{1}\times X_{2}$ is
arbitrarily close to $\lambda_{2}$ on almost every ergodic component of the
product measure.

###  5.2. Reformulation of the problem in measure theoretic terms.

Write $(X_{1},\nu_{1})$ for the set $H_{1}\backslash\Gamma$ endowed with the
counting measure. It is an infinite ergodic measure-preserving action of
$\Gamma$. Let $\Gamma\curvearrowright(X_{2},\nu_{2})$ be a p.m.p. Borel action
on a standard Borel probability space such that $H_{2}=\Gamma_{x}$ for
$\nu_{2}$-random $x$. We will consider the action of $\Gamma$ on the product
system $(X_{1}\times X_{2},\nu_{1}\times\nu_{2}).$ To shorten notation we
write $\nu=\nu_{1}\times\nu_{2}$. The intersection $H_{1}\cap H_{2}$ is
nothing else than the stabilizer of a random point $x\in\\{[H_{1}]\\}\times
X_{2}$. Note that for such $x$, we have
$\rho(\Gamma_{x}\backslash\Gamma)\leq\lambda_{2}:=\rho(H_{2}\backslash\Gamma)$
almost surely. Set

$C_{0}:=\\{x\in X_{1}\times
X_{2}\mid\rho(\Gamma_{x}\backslash\Gamma)=\lambda_{2}\\}.$

Since conjugate subgroups have the same co-spectral radius, the set $C_{0}$ is
invariant under the action of $\Gamma$. Let $(X_{1}\times
X_{2},\nu)\to(Z,\tau)$ be the ergodic decomposition given by Corollary 2.3,
and set

$Z_{0}:=\\{z\in Z\mid\nu_{z}(C_{0})>0\\}.$

By ergodicity and invariance, the set $C_{0}$ has full $\nu_{z}$-measure for
every $z\in Z_{0}$. Theorem 1.2 is equivalent to the identity
$C_{0}=X_{1}\times X_{2}$ modulo a null set, so it will follow once we show
that $\tau(Z_{0})=1$. By Proposition 4.4, $z\in Z_{0}$ if and only if the
following condition holds: For every $\eta>0$ there exists a function $h$
supported on the interior of a finite connected component of $(X_{1}\times
X_{2},\nu_{z},S)$ (according to Definition 4.1), such that

$\langle(I-M)h,h\rangle_{\nu_{z}}\leq(1-\lambda_{2}+\eta)\|h\|_{\nu_{z}}^{2}.$
(5.1)

We will refer to nonnegative, nonzero functions supported on interiors of
finite connected components of $(X_{1}\times X_{2},\nu,S)$ as test functions.
It is easy to check that a test function for $\nu$ is also a test function for
almost all ergodic components $\nu_{z}$. It will be convenient to name the set
of ergodic components $z$ for which there exist a test function satisfying
(5.1) with specific $\eta$. Let

$Z_{\eta}:=\\{z\in Z\mid\text{ there exists }h\text{ such that
}\langle(I-M)h,h\rangle_{\nu_{z}}\leq(1-\lambda_{2}+\eta)\|h\|_{\nu_{z}}^{2}\\}.$

Obviously we have $Z_{0}=\bigcap_{\eta>0}Z_{\eta}$ and $Z_{\eta}\subset
Z_{\eta^{\prime}}$ for $\eta<\eta^{\prime}$. In the following sections we show
that $\tau(Z_{\eta})\to 1$ as $\eta\to 0$. This will imply that
$\nu(Z_{\eta})=1$ for every $\eta>0$, and consequently that $\nu(Z_{0})=1$,
which is tantamount to Theorem 1.2.

###  5.3. Construction of test functions.

######  Lemma 5.1.

Let $\delta>0$. There exists a test function $f$ and a set $Z^{\prime}\subset
Z$ such that

  1. (1)

$\|f\|_{\nu_{z}}^{2}\geq(1-\delta)\|f\|_{\nu}^{2}$, for every $z\in
Z^{\prime}$.

  2. (2)

$\tau(Z^{\prime})\geq 1-\delta.$

  3. (3)

$\langle(I-M)f,f\rangle_{\nu}\leq(1-\lambda_{2}+\delta)\|f\|_{\nu}^{2}.$

###### Proof.

Let $\varepsilon_{2}>0$. Since $\Gamma\curvearrowright(X_{2},\nu_{2})$ has
embedded spectral radius $\lambda_{2}$, there is a finite measure, finite
connected component $P_{2}\subset X_{2}$ and nonzero $f_{2}\in
L^{2}(X_{2},\nu_{2})$ as is in Definition 4.2, i.e. $f_{2}$ is supported on
the interior ${\rm int}(P_{2})$ and

$\langle(I-M)f_{2},f_{2}\rangle\leq(1-\lambda_{2}+\varepsilon_{2})\|f_{2}\|^{2}.$
(5.2)

By Remark 4.3, we may assume $f_{2}\geq 0$.

We will show that for a good enough Følner set $F\subseteq X_{1}$ and small
enough $\varepsilon_{2}$, the function

$f:=\mathds{1}_{F}\times f_{2}$

satisfies the conditions of the lemma. While (3) is relatively
straightforward, conditions (1) and (2) require some work and strongly use the
fact that $X_{2}$ is a p.m.p. action.

Consider the following probability measures on $\Gamma$:

$\mu:=\frac{1}{|S|}\sum_{s\in S}\delta_{s}\textrm{ and
}\mu^{m}:=\frac{1}{m}\sum_{i=0}^{m-1}\mu^{\ast i}\textrm{ for
}m\in\mathbb{N}.$

By Kakutani’s ergodic theorem [10], there exists $m_{0}\geq 1$ such that

$\left|\int_{\Gamma}f_{2}^{2}(x\gamma^{-1})d\mu^{m_{0}}(\gamma)-\int_{X_{2}}f_{2}^{2}d\nu_{2}\right|\leq\varepsilon_{2}\|f_{2}\|^{2}$
(5.3)

for all $x\in X_{2}^{\prime}$ where $\nu_{2}(X_{2}^{\prime})\geq
1-\varepsilon_{2}$.

Fix $0<\varepsilon_{1}\ll\varepsilon_{2}$ very small. The precise choice only
depends on $\varepsilon_{2}$ and will be specified at the end of the proof.
Let $F\subset X_{1}$ be an $\varepsilon_{1}$-Følner set and write
$Y=(F\cup\partial F)\times X_{2}$.

Write $F^{\prime}$ for the set of points of $F$ which are at distance at least
$m_{0}$ from the boundary $\partial F$ and set $Y^{\prime}:=F^{\prime}\times
X_{2}^{\prime}$, where $X_{2}^{\prime}$ is as in (5.3). We claim that for
$\varepsilon_{1}$ small enough, we will have
$(\nu_{1}\times\nu_{2})(Y^{\prime})\geq|F|(1-2\varepsilon_{2})$. Indeed, using
that $F$ is $\varepsilon_{1}$-Følner, we have

$|F^{\prime}|\geq|F\cup\partial F|-|\partial
F|\sum_{i=1}^{m_{0}-1}|S|^{i}\geq|F\cup\partial
F|(1-|S|^{m_{0}}\varepsilon_{1}).$

Clearly for sufficiently small $\varepsilon_{1}$, we have

$\nu(Y^{\prime})=|F^{\prime}|\nu_{2}(X^{\prime}_{2})\geq(1-\varepsilon_{1}|S|^{m_{0}})(1-\varepsilon_{2})|F|\geq(1-2\varepsilon_{2})|F|.$
(5.4)

Write $P:=(F\cup\partial F)\times P_{2}\subset Y$. By construction, the
support of $f$ is contained in $P\subset Y$. Note that since $P_{2}$ is a
finite connected component of $X_{2}$ and $F\cup\partial F$ is finite, the set
$P$ will be a finite connected component of $(X_{1}\times X_{2},\nu,S)$ in the
sense of the Definition 4.1. Let $\nu=\int_{Z}\nu_{z}d\tau(z)$ be the ergodic
decomposition of $\nu$ as in Section 2.3. The set $P$ is also a finite
connected component of $(X_{1}\times X_{2},\nu_{z},S)$ for almost every $z\in
Z$. For every $z\in Z$, the measure $\nu_{z}$ is invariant under the action of
$\Gamma$, so that

$\displaystyle\int f^{2}(x_{1},x_{2})d\nu_{z}(x_{1},x_{2})$
$\displaystyle=\int_{\Gamma}\int
f^{2}(x_{1}\gamma^{-1},x_{2}\gamma^{-1})d\nu_{z}(x_{1},x_{2})d\mu^{m_{0}}(\gamma)$
$\displaystyle\geq\int_{Y^{\prime}}\int_{\Gamma}f^{2}(x_{1}\gamma^{-1},x_{2}\gamma^{-1})d\mu^{m_{0}}(\gamma)d\nu_{z}(x_{1},x_{2})$

Since $Y^{\prime}=F^{\prime}\times X_{2}^{\prime}$ and
$F^{\prime}\gamma^{-1}\subset F$ for any
$\gamma\in\operatorname{supp}\mu^{m_{0}}$ we can use the identity
$f=\mathds{1}_{F}\times f_{2}$ to rewrite the last integral as

$\int_{F^{\prime}\times
X_{2}^{\prime}}\left(\int_{\Gamma}f_{2}^{2}(x_{2}\gamma^{-1})d\mu^{m_{0}}(\gamma)\right)d\nu_{z}(x_{1},x_{2}).$

We use (5.3) to estimate the innermost integral and obtain a lower bound on
$\|f\|_{\nu_{z}}^{2}$:

$\displaystyle\int_{P}f^{2}d\nu_{z}\geq$
$\displaystyle(1-\varepsilon_{2})\nu_{z}(F^{\prime}\times
X_{2}^{\prime})\int_{X_{2}}f_{2}^{2}d\nu_{2}=(1-\varepsilon_{2})\nu_{z}(Y^{\prime})\|f_{2}\|_{\nu_{2}}^{2}.$

By (5.4), we have $\nu(Y^{\prime})\geq(1-2\varepsilon_{2})|F|$, so we can
apply Markov’s inequality to get a set $Z^{\prime}\subset Z$ with
$\tau(Z^{\prime})\geq 1-\sqrt{2\varepsilon_{2}}$ such that
$\nu_{z}(Y^{\prime})\geq(1-\sqrt{2\varepsilon_{2}})|F|$ for $z\in Z^{\prime}$.
Finally we get that for $z\in Z^{\prime}$:

$\|f\|_{\nu_{z}}^{2}\geq(1-\varepsilon_{2})(1-\sqrt{2\varepsilon_{2}})|F|\|f_{2}\|_{\nu_{2}}^{2}=(1-\varepsilon_{2})(1-\sqrt{2\varepsilon_{2}})\|f\|_{\nu}^{2}.$
(5.5)

This establishes Properties (1) and (2) of the lemma. It remains to address
(3). Using $f_{2}\geq 0$, we estimate $\langle f,Mf\rangle_{\nu}$ in terms of
$\langle f_{2},Mf_{2}\rangle_{\nu_{2}}$ as follows:

$\displaystyle\langle f,Mf\rangle_{\nu}$
$\displaystyle=|S|^{-1}\int_{X_{1}}\int_{X_{2}}\mathds{1}_{F}(x_{1})f_{2}(x_{2})\sum_{s\in
S}\mathds{1}_{F}(x_{1}s)f_{2}(x_{2}s)d\nu_{2}(x_{2})d\nu_{1}(x_{1})$
$\displaystyle\geq|S|^{-1}\int_{\text{int}(F)}\int_{X_{2}}f_{2}(x_{2})\sum_{s\in
S}f_{2}(x_{2}s)d\nu_{2}(x_{2})d\nu_{1}(x_{1})$
$\displaystyle=|\text{int}(F)|\langle f_{2},Mf_{2}\rangle_{\nu_{2}}.$

Hence

$\displaystyle\langle(I-M)f,f\rangle_{\nu}$ $\displaystyle\leq|F|\langle
f_{2},f_{2}\rangle_{\nu_{2}}-|\text{int}(F)|\langle
f_{2},Mf_{2}\rangle_{\nu_{2}}$ $\displaystyle=|F|\langle
f_{2}-Mf_{2},f_{2}\rangle_{\nu_{2}}+|F|\left(1-\frac{|\text{int}(F)|}{|F|}\right)\langle
Mf_{2},f_{2}\rangle_{\nu_{2}}.$

Since $F$ is $\varepsilon_{1}$-Følner, we have $|\text{int}(F)|/|F|\geq
1-|S|\varepsilon_{1}$, so finally we obtain

$\displaystyle\langle(I-M)f,f\rangle_{\nu}\leq$ $\displaystyle|F|\left(\langle
f_{2}-Mf_{2},f_{2}\rangle_{\nu_{2}}+|S|\varepsilon_{1}\langle
Mf_{2},f_{2}\rangle_{\nu_{2}}\right)$ $\displaystyle\leq$
$\displaystyle|F|\left(\langle
f_{2}-Mf_{2},f_{2}\rangle_{\nu_{2}}+|S|\varepsilon_{1}\|f_{2}\|_{\nu_{2}}^{2}\right).$

By the defining property of $f_{2}$ (see Equation (5.2)), we then find

$\langle(I-M)f,f\rangle_{\nu}\leq|F|(1-\lambda_{2}+\varepsilon_{2}+|S|\varepsilon_{1})\|f_{2}\|_{\nu_{2}}^{2}=(1-\lambda_{2}+\varepsilon_{2}+|S|\varepsilon_{1})\|f\|_{\nu}^{2}.$

To finish the proof, choose $\varepsilon_{2}>0$ such that
$(1-\varepsilon_{2})(1-\sqrt{2\varepsilon_{2}})\geq(1-\delta)$ and then choose
$\varepsilon_{1}>0$ such that $\varepsilon_{2}+|S|\varepsilon_{1}\leq\delta$
and such that (5.4) holds. ∎

###  5.4. End of the proof.

Let $f,\delta,Z^{\prime}$ be as in the Lemma 5.1. Using Property (3) of the
Lemma, we have

$\displaystyle\int_{Z^{\prime}}\langle(I-M)f,f\rangle_{\nu_{z}}d\tau(z)\leq$
$\displaystyle\int_{Z}\langle(I-M)f,f\rangle_{\nu_{z}}d\tau(z)$
$\displaystyle=$ $\displaystyle\langle(I-M)f,f\rangle_{\nu}$
$\displaystyle\leq$ $\displaystyle(1-\lambda_{2}+\delta)\|f\|_{\nu}^{2}.$

By Property (1), we can then estimate

$\int_{Z^{\prime}}\frac{\langle(I-M)f,f\rangle_{\nu_{z}}}{\|f\|_{\nu_{z}}^{2}}d\tau(z)\leq\frac{1-\lambda_{2}+\delta}{1-\delta}.$

On the other hand, Proposition 4.4 and the fact that co-spectral radii of
stabilizers are all at most $\lambda_{2}$, yield the inequality

$\frac{\langle(I-M)f,f\rangle_{\nu_{z}}}{\|f\|_{\nu_{z}}^{2}}\geq
1-\lambda_{2}\text{ for almost all }z\in Z.$

Therefore, by Markov’s inequality there is a positive $\eta=O(\sqrt{\delta})$
and a subset $Z^{\prime\prime}\subset Z$ such that $\tau(Z^{\prime\prime})\geq
1-\eta$ and

$\langle(I-M)f,f\rangle_{\nu_{z}}\leq(1-\lambda_{2}+\eta){\|f\|_{\nu_{z}}^{2}}\text{
for }z\in Z^{\prime\prime}.$

$\eta$ could be made explicit in terms of $\delta$ and $\lambda_{2}$, but we
will only need that $\eta\to 0$ as $\delta\to 0$. We have
$Z^{\prime\prime}\subset Z_{\eta}$, so it follows that $\tau(Z_{\eta})\to 1$
as $\eta\to 0$. This ends the proof per the discussion at the end of Section
5.2. ∎

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