Co-spectral radius of intersections

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.


Introduction
Let Γ be a countable group and let S be a finite symmetric subset of Γ. The co-spectral radius of a subgroup H ⊂ Γ (with respect to S) is defined as the norm of the operator M : L 2 (Γ/H) → L 2 (Γ/H): The groups with co-spectral radius 1 for every choice of S are called coamenable. If the group Γ 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 ⊂ Γ there is not much that can be said about ρ(Γ/(H 1 ∩ H 2 )) other than the trivial inequality ρ(Γ/(H 1 ∩ H 2 )) ≤ min{ρ(Γ/H 1 ), ρ(Γ/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 Γ 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 Γ be a finitely generated group and let S be a finite symmetric generating set. Let H 1 , H 2 be subgroups of Γ and assume that H 1 is co-amenable. Then  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 Γ-actions on suitable spaces X 1 and X 2 . Since H 1 is a (deterministic) subgroup, we can take X 1 = Γ/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 Γ. The intersection H 1 ∩ H 2 is then the stabilizer of a point in X 1 × 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 \Γ. The product of E with X 1 will locally approximate the coset space of (H 1 ∩ H 2 )\Γ. The co-amenability of H 2 means that the set E contains a subset P that is nearly Γ-invariant. The product of such a set with a Følner set F in X 1 should be a nearly Γ-invariant set in X 1 × E, and hence witnesses the amenability of the coset graph (H 1 ∩ H 2 )\Γ. The latter implies that H 1 ∩ 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 ρ(Γ/H 2 ) = λ 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. showing us a related problem for intersections of percolations on 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.
co-spectral radius of intersections 4 2. Background 2.1. Co-amenability. Let Γ be a finitely generated group and let S be a finite symmetric set of generators. A subgroup H of Γ is called coamenable if the Schreier graph Sch(H\Γ, S) is amenable, i.e for any ε > 0 and any S there exists a set F ⊂ H\Γ such that |F ∆F S| ≤ ε|F |. Such sets will be called ε-Følner sets. Alternatively, a subgroup H is co-amenable if and only if the representation ℓ 2 (Γ/H) has almost invariant vectors, or that M L 2 (H\Γ) = 1.

2.2.
Invariant random subgroups. Let Sub Γ be the space of subgroups of Γ, equipped with the topology induced from {0, 1} Γ . An invariant random subgroup is a probability measure µ ∈ P(Sub Γ ) which is invariant under conjugation of Γ. An IRS is called co-amenable if Similarly, we say that an IRS H has co-spectral radius at least λ if ρ(H\Γ) ≥ λ almost surely.
For any action Γ X and x ∈ X write Γ x for the stabilizer of x. Every IRS can be realized as a stabilizer of a random point in a probability measure preserving system: [2]). For every IRS µ, there exists a standard Borel probability space (X, ν) and a Borel p.m.p. Γ-action on (X, ν) such that µ = X δ Γx dν(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 Γ be a countable group and let Γ (X, Σ X , ν) be a nonsingular Borel action on a standard Borel probability space. Then there exist a standard Borel probability space (Z, Σ Z , τ ) and a family of quasi-invariant, ergodic, pairwise mutually singular probability measures {ν z } z∈Z with the same Radon-Nikodym cocycle as ν, and such that for every B ∈ Σ X , we have (2.1) As an application we have: Let Γ be a countable group and let Γ (X 1 , Σ X 1 , ν 1 ) and Γ (X 2 , Σ X 2 , ν 2 ) be measure-preserving Borel actions on standard Borel spaces. Suppose that (X 1 , ν 1 ) is ergodic and that ν 2 (X 2 ) = 1. Then there exists a standard Borel probability space (Z, Σ Z , τ ) and a family of Γ-invariant, ergodic, pairwise mutually singular measures {ν z } z∈Z on X 1 × X 2 such that for every B ∈ Σ X 1 ×X 2 , we have Moreover, for every measurable set F ⊂ X 1 and z ∈ Z we have Proof. Fix a countably-valued Borel function w : X 1 → R >0 , such that Since {(wν) z } z are ergodic and pairwise mutually singular, the same is true 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 ⊂ X 1 and renormalize ν z and τ as follows: By ergodicity of ν 1 , this normalization does not depend on the choice of F .

2.4.
Ergodic theory of equivalence relations. Let (X, ν) be a probability measure space and let ϕ i : U i → X be a finite family of non-singular measurable maps defined on subsets U i of X. The triple (X, ν, (ϕ i ) i∈I ) is called a graphing. We assume that (ϕ i ) i∈I is symmetric, i.e. for each 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 Γ, equipped with the graphing corresponding to a finite symmetric generating set S of Γ, and ν will be the restricted measure or the restriction of an ergodic component.
Let R be the orbit equivalence relation generated by the maps (ϕ i ) i∈I . A measured graphing yields a random graph in the following way: For every x ∈ X, let G x be the graph with vertex set given by the equivalence class co-spectral radius of intersections 6 [x] R and place an edge between y, z ∈ [x] R whenever z = ϕ i (y) for some i ∈ I (multiple edges are allowed). The graphs G x have degrees bounded by |I| and are undirected since (ϕ i ) i∈I is symmetric. If we choose a ν-random point x, the resulting graph G x is a random rooted graph. The properties of 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 : R → R, we have

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 := 1 |S| s∈S s ∈ C[Γ]. We have the following identity between the unitary representations of Γ: Write π 1 , π 2 for the unitary representations corresponding to L 2 (H 1 \Γ) and L 2 (H 2 \Γ). The above identity implies that To prove the theorem, it is enough to verify that Letting ε → 0 we conclude that (π 1 ⊗ π 2 )(M ) ≥ π 2 (M ) .
The supremum in the inequality seems to be necessary. Below we construct an example of a non-amenable finitely generated group Γ with two co-amenable subgroups H 1 , H 2 such that the intersection 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 Z in Γ. The triple {s, a, b} generates Γ.
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 ⊆ Z contains arbitrarily long segments. Then for any g ∈ Γ, the Schreier graphs for H C and H g C are isomorphic, so Then H C−n k converges to H Z in Sub(Γ) as k → ∞. Since the spectral radius is lower semi-continuous on the space of subgroups, we get 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 Γ be as in the previous example and let A be an invariant percolation on 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.

Embedded spectral radius
Let us introduce some terminology. Let (X, ν) be a measure preserving Γ-action, and write R for the corresponding orbit equivalence relation. We shall assume that ν is σ-finite but not necessarily finite. We recall that for every x ∈ X, G x is the labeled graph with vertex set [x] R and edge set (y, sy), y ∈ Γx, labeled by s ∈ S. Definition 4.1. A set P ⊂ X is called a finite connected component if for almost all x ∈ P the connected component of x in the graph P ∩ G x is finite. In other words, the graphing restricted to P generates a finite equivalence relation.
For any subset P ⊂ X write ∂P := SP \ P for the (outer) boundary and int(P ) := P \ ∂(X \ P ) for the interior of P . Definition 4.2. Let (X, ν) be a measure-preserving Γ-action. We say that (X, ν) has embedded spectral radius λ if for every finite measure finite connected component P ⊂ X and every f ∈ L 2 (X, ν) supported on the interior int(P ), we have and λ 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 ≥ 0.
Our goal in this section is to prove that the embedded spectral radius of a measure preserving system Γ (X, ν) is detected by the co-spectral radius along orbits: Proposition 4.4. Let (X, ν) be a σ-finite measure-preserving Γ-system. Then the stabilizer of almost every point has co-spectral radius at least λ if and only if almost every ergodic component of ν has embedded spectral radius at least λ.
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, ν). This happens for example when Γ is a non-abelian free group and X = X 1 × 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 G x are just copies of the Cayley graph of Γ so their spectral radius is bounded away from 1. On the other hand, the spectral radius of M on L 2 0 (X, ν) is 1, because it contains L 2 0 (X 2 , ν 2 ). Proof. By passing to ergodic components we can assume without loss of generality that (X, ν) is ergodic. If (X, ν) is periodic then there is nothing to do, so henceforth we will assume that (X, ν) is an aperiodic measure preserving ergodic system.
First let us prove that if the co-spectral radius of the stabilizer Γ x is at least λ, then X has embedded spectral radius at least λ. Let ε > 0 be arbitrary. Then ν-almost every orbit G x supports a function f x : (4.1) Since 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 Gx (x, R x ) of radius R x around x supports a function ψ x satisfying (4.1). Since balls of fixed radius depend measurably on x, the map x → R x is measurable, so we can choose R 0 > 0 such that and put X 1 := {x ∈ X | R x ≤ 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 ⊂ X 1 such that for all x ∈ X 2 , the rooted graphs (B Gx (x, R 0 ), x) are all isomorphic to some (G, o) as rooted S-labeled graphs. By restricting to a smaller subset, we can assume that ν(X 2 ) is finite. Fix ψ : G → R satisfying and for x ∈ X 2 , let B x ⊂ X be the image of G via the unique labeled Let S ′ 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 ′ X 2 , ν, S ′ ) we find a partition S ′ X 2 = B ⊔ N j=1 A j with ν(B) < ν(X 2 )/2, such that A j ∩ sA j = {x ∈ A j | sx = x} for every s ∈ S ′ . This translates to the condition that that B x and B x ′ are disjoint for every distinct pair of points x, y ∈ A j . Since the sets A j cover a subset of X 2 of measure at least ν(X 2 )/2, there exists j such that X 3 := X 2 ∩ A j has positive measure. The set P := x∈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 ψ : G → R naturally induces a function f : Then we easily verify ( and similarly Taking ε → 0, we see that X has embedded spectral radius at least λ. 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 (G x , x) where x ∈ X is ν| P -random. We argue by contradiction, so assume that (X, ν) has embedded spectral radius at least λ but at the same time stabilizers have co-spectral radius ρ < λ with positive probability. By ergodicity, there exists an h > 0 such that Since X has spectral radius at least λ, there exists f ∈ L 2 (X, ν) nonzero and supported on the interior of a finite connected component P ⊆ X with ν(P ) < ∞ such that As in Section 2.4, let R P be the equivalence relation generated by the graphing on P . We write P o x := [x] R P for the connected component of By the mass transport principle, we equate We start by computing the integrand on the right-hand side. Rewriting we find (using S is symmetric) for every x ∈ X that Using ρ(G x ) ≤ λ − h and f ≥ 0, we can estimate Therefore we have the following estimate for the right-hand side in the Mass Transport Equation (4.4) Next, we compute the integrand on the left-hand side of the Mass Transport Equation (4.4), namely for y ∈ X, we have 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 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, ν, (ϕ i ) i∈I ) be a finite measure preserving symmetric graphing on a finite measure space. Then, for every δ > 0 there exists a measur- Proof. We start by proving the lemma for a single measure preserving invertible map ϕ : U → ϕ(U ). Since we do not assume that ϕ is defined on all of X we need to treat separately the subset of elements where ϕ can be applied only finitely many times. For any n ∈ N define This reduces the problem to the subset Y := X \ ∞ n=0 E n . By definition, ϕ(Y ) = Y . We further decompose Y into the periodic and aperiodic parts Y p , Y ap . The periodic part can be partitioned into a fixed point set A 2 = {x ∈ Y p | ϕ(x) = x}, finitely many sets A 3 , . . . , A M permuted by ϕ and a remainder B 1 of measure ν(B 1 ) < δ/2 coming from large odd periods. By the usual Rokhlin lemma, the aperiodic part Y ap can be decomposed as  Let Γ be a countable group with a co-amenable subgroup H 1 and an IRS H 2 with co-spectral radius λ 2 . We need to show that H 1 ∩ H 2 has co-spectral radius at least λ 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 \Γ and H 2 is realized as the stabilizer of a random point in a p.m.p. action of Γ on (X 2 , ν 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 λ 2 . Next, using a large Fölner set in X 1 , we produce a new finite connected component in the product system X 1 × X 2 and a new function which certifies that the co-spectral radius of stabilizers in X 1 × X 2 is arbitrarily close to λ 2 on almost every ergodic component of the product measure.

5.2.
Reformulation of the problem in measure theoretic terms.
Write (X 1 , ν 1 ) for the set H 1 \Γ endowed with the counting measure. It is an infinite ergodic measure-preserving action of Γ. Let Γ (X 2 , ν 2 ) be a p.m.p. Borel action on a standard Borel probability space such that H 2 = Γ x for ν 2 -random x. We will consider the action of Γ on the product system (X 1 × X 2 , ν 1 × ν 2 ). To shorten notation we write ν = ν 1 × ν 2 . The intersection H 1 ∩ H 2 is nothing else than the stabilizer of a random point Note that for such x, we have ρ(Γ x \Γ) ≤ λ 2 := ρ(H 2 \Γ) almost surely. Set Since conjugate subgroups have the same co-spectral radius, the set C 0 is invariant under the action of Γ. Let (X 1 × X 2 , ν) → (Z, τ ) be the ergodic decomposition given by Corollary 2.3, and set By ergodicity and invariance, the set C 0 has full ν z -measure for every z ∈ Z 0 . Theorem 1.2 is equivalent to the identity C 0 = X 1 × X 2 modulo a null set, so it will follow once we show that τ (Z 0 ) = 1. By Proposition 4.4, z ∈ Z 0 if and only if the following condition holds: For every η > 0 there exists a function h supported on the interior of a finite connected component of (X 1 × X 2 , ν z , S) (according to Definition 4.1), such that We will refer to nonnegative, nonzero functions supported on interiors of finite connected components of (X 1 × X 2 , ν, S) as test functions. It is easy to check that a test function for ν is also a test function for almost all ergodic components ν 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 η. Let Z η := {z ∈ Z | there exists h such that (I−M )h, h νz ≤ (1−λ 2 +η) h 2 νz }. Obviously we have Z 0 = η>0 Z η and Z η ⊂ Z η ′ for η < η ′ . In the following sections we show that τ (Z η ) → 1 as η → 0. This will imply that ν(Z η ) = 1 for every η > 0, and consequently that ν(Z 0 ) = 1, which is tantamount to Theorem 1.2.

Construction of test functions.
Lemma 5.1. Let δ > 0. There exists a test function f and a set Z ′ ⊂ Z such that ( Proof. Let ε 2 > 0. Since Γ (X 2 , ν 2 ) has embedded spectral radius λ 2 , there is a finite measure, finite connected component P 2 ⊂ X 2 and nonzero f 2 ∈ L 2 (X 2 , ν 2 ) as is in Definition 4.2, i.e. f 2 is supported on the interior int(P 2 ) and By Remark 4.3, we may assume f 2 ≥ 0.
We will show that for a good enough Følner set F ⊆ X 1 and small enough ε 2 , the function f := 1 F × 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.
Write F ′ for the set of points of F which are at distance at least m 0 from the boundary ∂F and set Y ′ := F ′ × X ′ 2 , where X ′ 2 is as in (5.3). We claim that for ε 1 small enough, we will have (ν 1 × ν 2 )(Y ′ ) ≥ |F |(1 − 2ε 2 ). Indeed, using that F is ε 1 -Følner, we have Clearly for sufficiently small ε 1 , we have Write P := (F ∪ ∂F ) × P 2 ⊂ Y . By construction, the support of f is contained in P ⊂ Y . Note that since P 2 is a finite connected component of X 2 and F ∪ ∂F is finite, the set P will be a finite connected component of (X 1 × X 2 , ν, S) in the sense of the Definition 4.1. Let ν = Z ν z dτ (z) be the ergodic decomposition of ν as in Section 2.3. The set P is also a finite connected component of (X 1 × X 2 , ν z , S) for almost every z ∈ Z. For every z ∈ Z, the measure ν z is invariant under the action of Γ, so that On the other hand, Proposition 4.4 and the fact that co-spectral radii of stabilizers are all at most λ 2 , yield the inequality (I − M )f, f νz f 2 νz ≥ 1 − λ 2 for almost all z ∈ Z.