Schreier graphs of invariant random subgroups

Let \( \Gamma \) be a group generated by a finite subset \( S \). To a subgroup \( H \le \Gamma \) one associates the Schreier coset graph \( \mathrm{Sch}(H \backslash \Gamma, S) \): this is the graph with vertex set \( H \backslash \Gamma \), where two vertices \( Hg, Hg’ \) are joined by an edge if and only if \( g’ \in HgS \); in other words it is the quotient of the Cayley graph of \( \Gamma \) with respect to \( S \) by the action of \( H \) by left-translations.

A random subgroup in \( \Gamma \) thus yields a random rooted graph: the Schreier graph of the random subgroup, rooted at the identity coset. Let \( \mathcal G^S \) be the set of rooted \( S \)-labeled graphs (i.e. every vertex has exactly one incoming and one outcoming edge labeled by every element of \( S \)). This set has an action of the free group \( F_S \) on \( S \), where a generator moves the root to the head of the outcoming edge lebeled by it. The following result is essentially immediate as the Schreier graph \( \mathrm{Sch}(sHs^{-1} \backslash \Gamma, S) \) is the same as \( \mathrm{Sch}(H \backslash \Gamma, S) \) but rooted at the coset \( sHs^{-1}s \).

Proposition: A random subgroup is conjugacy-invariant if and only if the corresponding random rooted graph is invariant under this action.

We can reformulate this property as follows: let \( \mathcal G_2 \) be the space of doubly pointed, \( S \)-labeled graphs, up to isomorphism.

Proposition (Mass Transport Principle for unimodular graphs): Let \( \lambda \) be a Borel measure on \( \mathcal G \). It is \( F_S \)-invariant if and only if, for any positive Borel function \( f \) on \( \mathcal G_2 \) we have:
\[
(\mathrm{MTP}) \hspace{1cm} \int_{\mathcal G} \sum_{o_2 \in X} f(X, o_1, o_2) d\lambda(X, o_1) = \int_{\mathcal G} \sum_{o_1 \in X} f(X, o_1, o_2) d\lambda(X, o_2).
\]

The proof of this is not very long. Assume first that \( \lambda \) satisfies (MTP). We want to prove that \( \lambda(sE) = \lambda(E) \) for all \( s \in S \) and all Borel sets \( E \subset G \). For this we define the following function on \( \mathcal G_2 \) to which we’ll apply (MTP):
\[
f(X, v, w) =
\begin{cases}
1 & \text{ if } (X, v) \in E \text{ and } (X, v) = s\cdot (X, w) \\
0 & \text{ otherwise}
\end{cases}
\]
(in other words it indicates whether the two roots are joined by an \( s \)-labeled edge). We compute:
\[
\int_{\mathcal G} \sum_w f(X, v, w) d\lambda(X, v) = \int_{\mathcal G} 1_{(X,v) \in E} d\lambda(X, v) = \lambda(E)
\]
as there is exactly one nonzero value of \( f \) in the sum over \( w \) when \( (X, v) \in E \) and none otherwise; and similarly
\[
\int_{\mathcal G} \sum_{v \in X} f(X, v, w) d\lambda(X, w) = \lambda(sE).
\]
The mass transport principle implies that both integral are equal, hence \( \lambda(sE) = \lambda(E) \).

The converse follows from a similar computation by reducing to the case where \( f \) is supported on a set where the two roots are adjacent.

Unimodularity for unlabeled graphs

From now on we use \( \mathcal G \) (resp. \( \mathcal G_2 \)) to denote the space of pointed (resp. doubly pointed) unlabeled locally finite (not necessary regular?) graphs. We say that a random graph is unimodular if its law satisfies the mass transport principle (MTP) (which makes sense in this generality). Some easy examples are:

1. A finite fixed graph with uniformly chosen root;
2. A de-labeled Schreier graph of an invariant random subgroup in a discrete group;
3. If a graph is vertex transitive with unimodular isomorphism group then it is (rooted at an arbitrary vertex) a unimodular graph (note that this is not true without the unimodularity hypothesis on the group of isomorphisms).

The no-core principle

This is a statement which formalises the idea that a unimodular random graph looks everywhere the same, more precisely that it is impossible to distinguish a finite region (a « core ») in an infinite unimodular random graph in a measurable manner. It goes as follows.

Theorem: Let \( \lambda \) be a unimodular random graph. Let \( A \) be a Borel subset of the space \( \mathcal G \) of pointed graphs. If
\[
0 < |\{ w \in X :\: (X, w) \in A \}| < +\infty \quad (\ast)
\]
with positive probability then \( X \) is finite with positive probability.

To prove this assume that \( (\ast) \) holds for \( (X, v) \in B \) with \( \lambda(B) > 0 \). As it has an enumerable range we can assume that \( |\{ w \in X :\: (X, w) \in A \}| \) takes only the value \( 0 \in < +\infty \) on \( B \). Then applying (MTP) to the function
\[
(X, v, w) \mapsto 1_A(X,v) 1_B(X,w)
\]
we get that :
\[
\int_B \sum_{w \in X} 1_A(X,v) d\lambda(X, v) = \int_{\mathcal G} \sum_{w \in X} 1_A(X, w) d\lambda(X, v) = N.
\]
It follows that the right-hand side must be finite, but as it is equal to \( \int_{\mathcal G} 1_A(X,v) \cdot |X| d\lambda(X,v) \) this implies that \( |X| \ +\infty \) for all \( (X, v) \in B \).

As an application of the no-core principle one can prove the following classification result for unimodular random graphs.

Corollary: Let \( \lambda \) be a unimodular random graph. Then for \( \lambda \)-almost all \( (X, v) \in \mathcal G \) the graph \( X \) is either finite (zero ends) or has exactly one, two or a Cantor set of ends.

We prove first the weaker statement that \( X \) has either \( 0, 1, 2 \) or infinitely many ends. Assume to the contrary that with positive probability \( X \) has \( N \) ends, where \( N \ge 3 \) is an integer. Then there exists \( R > 0 \) such that with positive probability for \( (X, v) \) the ball \( B_X(v, R) \) of radius \( R \) in \( X \) around \( v \) separates those \( N \) ends. We define a function as follows:
\[
f(X, v) =
\begin{cases}
1 & \text{ if } X \setminus B_X(v, R) \text{ has at least } N \text{ unbounded components; } \\
0 & \text{otherwise. }
\end{cases}
\]
This yields a contradiction with the NCP: given \( X \) with \( N \) ends and \( v, w \in X \), if both \( f(X, v), f(X, w) = 1 \) then \( B_X(v, R) \cap B_X(w, R) \neq \emptyset \) (otherwise \( X \) would have \( > N \) ends). THis implies that for such \( X \) the subset \( \{v \in X :\: f(X, v) = 1\} \) is finite. But as \( X \) is infinite this contradicts the theorem above.

Unimodular random manifolds

Let \( \mathcal M^d \) be the space of isometry classes of pointed complete Riemannian manifolds (the topology on this space is not obvious but it exists and is locally compact) and \( \mathcal M_2^d \) the space of doubly pointed such manifolds.

A Borel probability measure \( \lambda \) on \( \mathcal M^d \) is called unimodular if it satisfies a mass transport principle, that is for any Borel function \( f \) on \( _mathcal M_2^d \) the followin gequality holds:
\[
\int_{\mathcal M^d} \int_M f(M, x, y) d\mathrm{vol}_M(y) d\lambda(M, x) = \int_{\mathcal M^d} \int_M f(M, x, y) d\mathrm{vol}_M(x) d\lambda(M, y).
\]

Examples

1. A finite-volume complete manifold, with the root chosen with respect to the normalised volume measure;
2. If \( \mathrm{Isom}(X) \) is transitive, then the random manifold \( (X, x_0) \) (where \( x_0 \in X \) is an arbitrarily chosen point) is unimodular if and only if the locally compact group \( \mathrm{Isom}(X) \) is unimodular;
3. Let \( X \) be a symmetric space without compact or Euclidean factors and \( G = \mathrm{Isom}(X) \). Then any torsion-free and discrete IRS in \( G \) gives a unimodular random manifold locally isometric to \( X \), and vice-versa.

Benjamini–Schramm convergence

As a space of probability measures on a locally compact space the set of nimodular random manifolds has a natural locally topology (weak-star convergence). Convergence in this space is called Benjamini–Schramm convergence (these authors first studied the corresponding notion for unimodular random graphs). This topology has many applications in Riemannian geometry, such as the following result.

Theorem (Abért–Bergeron–Biringer–Gelander): Let \( X \) be a symmetric space without Euclidean or compact factors, and assume that it is not isometric to hyperbolic 3–space \( \mathbb H^3 \). Let \( M_n \) be a sequence of finite-volume, complet \( X \)-manifolds which is convergent in the Benjamini–Schramma topology. Then the sequences of normalised Betti numbers \( b_k(M_n) / \mathrm{vol}(M_n) \) converge for every \( 0 \le k \le \dim(X) \).