Amenability of Bounded Automata Groups on Infinite Alphabets

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire functions.


Introduction
Self-similar groups provide many examples of "exotic" amenable groups.The Grigorchuk group [Gri83] was the first example of a group of intermediate growth.Groups of intermediate growth are always amenable, but not elementary amenable (see [Cho80]).The basilica group is amenable [BV05], but not elementary subexponentially amenable [G Ż02].
Both the Grigorchuk group and the basilica group are examples of automata groups on a two-letter alphabet of bounded activity growth.They fit into the hierarchy of polynomial activity growth introduced in [Sid00], where both finite and infinite alphabets are considered.Under certain assumptions (which are always satisfied for finite alphabets), these groups do not contain free subgroups (see [Sid04]).For finite alphabets, it is shown in [BKN10] that the group generated by bounded activity automata is amenable.A large family of such groups are iterated monodromy groups of post-critically finite polynomials [Nek09].Furthermore, in [AAV13] it is shown that automata groups on finite of linear activity growths are amenable.The techniques of [BKN10] and [AAV13] have been conceptualized in [JNdlS16].
In [Rei], we show that iterated monodromy groups of post-singularly finite entire functions are given by bounded activity automata on infinite alphabets.We expect many similarities of these groups to their polynomial counterparts, so one question in particular is amenability.We can not expect all iterated monodromy groups of post-singularly finite entire functions to be amenable, as there are entire functions with monodromy group In the forthcoming paper [Rei], we show the following: Theorem A (Main application).Let f be a post-singularly finite entire function.Then the iterated monodromy group of f is amenable if and only if the monodromy group of f is amenable.
In this paper we provide the main group theoretic part of the proof of this theorem.We show the following: Theorem B. Let P be an amenable subgroup of Sym(X).Suppose that the action of P on X is recurrent.Then Aut f.s.B (X * ; P ) is amenable.
See Section 2 for a precise definition of Aut f.s.B (X * ; P ), it is roughly the groups of bounded activity automata were every first level action is in P .
We note that Theorem A is our main motivation for Theorem B, but this paper does not logically depend on [Rei].
In Section 2, we start by introducing self-similar groups on infinite alphabets and related concepts, such as the space of ends.We continue in Section 3 with a discussion of recurrent random walks and how to pass from a recurrent action on the alphabet to a recurrent action of a bounded activity group on the space of ends.This will be a key ingredient to invoke the amenability criterion of [JNdlS16] in Section 4 to prove Theorem B. In Section 5, we briefly discuss the forthcoming paper and further related open questions.
Acknowledgements.We gratefully acknowledge support by the Advanced Grant HOLOGRAM by the European Research Council.Part of this research was done during visits at Texas A&M University and at UCLA.We would like to thank our hosts, Volodymyr Nekrashevych and Mario Bonk, as well as the HOLOGRAM team, in particular Kostiantyn Drach, Dzmitry Dudko, Mikhail Hlushchanka, David Pfrang and Dierk Schleicher, for helpful discussions and comments.

Regular trees
In this section we introduce of self-similar groups and other relevant concepts and fix the notation.
Definition 2.1.Let X be a countable infinite set.The standard X-regular tree has as vertex set X * , the set of finite words in X.Its root is the empty word ∅.Its edges are all pairs (v, vx) for v ∈ X * , x ∈ X.By abuse of notion, we denote the standard X-regular tree also as X * , and we denote by Aut(X * ) the group of rooted tree automorphisms of X * .We denote the identity of Aut(X * ) by 1.
For v ∈ X * , let vX * be the subtree of all descendants of v.If g ∈ Aut(X * ), v ∈ X * , there is a unique g |v ∈ Aut(X * ) given by g(vw) = g(v)g |v (w).This is called the section of g along v.
A set S ⊂ Aut(X * ) is called self-similar if it is closed under taking sections, i.e. g |v ∈ S for all g ∈ S, v ∈ X * .We are mainly interested in self-similar groups, i.e. subgroups G ⊂ Aut(X * ) that are self-similar as sets.
For g ∈ Aut(X * ), the activity α n (g) ∈ N ∪ ∞ of g on level n is the number of words v of length n for which the section g| v is not trivial.We denote by Aut fin.(X * ) the set of automorphisms with finite activity on every level.If g ∈ Aut fin.(X * ) has a n so that g |v = 1 for all v ∈ X n , we say that that g is finitary.If g ∈ Aut fin.(X * ) has a c ∈ N so that α n (g) ≤ c for all n, we say that g has bounded activity.
We denote by Aut B (X * ) the set of automorphisms with bounded activity, and by Aut F (X * ) the set of finitary automorphisms.
We also have maps ρ n : Aut(X * ) → Sym(X n ), which are induced by the action of Aut(X * ) on the n-th level.Let P be a subgroup of Sym(X).Let Aut(X * ; P ) denote the set of automorphisms such that ρ 1 (g |v ) ∈ P for all v ∈ X * .We denote by Aut fin.(X * ; P ), Aut B (X * ; P ), Aut F (X * ; P ) the intersections of Aut fin.(X * ), Aut B (X * ), Aut F (X * ) with Aut(X * ; P ) respectively.
Since we consider infinite alphabets, let us fix notations for the two versions of wreath products.
Notation 2.2.Let A and B be groups, L be a set with an A-left action.The unrestricted wreath product l∈L B ⋊ A is denoted B Wr L A, the restricted wreath product l∈L B ⋊ A is denoted B ≀ L A. We will mainly work with the restricted wreath product.We denote the right factor embedding A → B ≀ L A by ι, and by b@l the image of b under the embedding of B into the component indexed by l.
For a subgroup P of Sym(X), we denote the n-th iterated restricted wreath product (along X) by P n .So P 1 = P and P n+1 = P n ≀ X P .Note that if P is amenable, then all P n are amenable.With this in mind we have the following: is an isomorphism of groups.It restricts to isomorphisms Aut fin.(X * ; P ) ∼ = Aut fin.(X * ; P ) ≀ X P Aut B (X * ; P ) ∼ = Aut B (X * ; P ) ≀ X P Aut F (X * ; P ) ∼ = Aut F (X * ; P ) ≀ X P △ For the first line, see for example [Sid00].By iteration, we also get isomorphisms Aut fin.(X * ; P ) → Aut fin.(X * ; P ) and Aut fin.(X * ; P ) ∼ = Aut fin.(X n * ; P n ).
2.1.Action on space of ends X ω .We will also use the action of Aut(X * ) on the space of ends of X * .The set of ends of X * can be identified with X ω , the set of right infinite words in X.The open cylinder sets C(v) = {vw : w ∈ X ω } form a basis of the end topology on X ω .Since X is countable infinite, X ω is homeomorphic to the Baire space N N , in particular X ω is Hausdorff, but not locally compact.The action of Aut(X * ) on X ω is faithful, so we can also think of elements of Aut(X * ) as homeomorphisms on X ω .We will use the language of germs: these are equivalence classes of pairs (g, w) ∈ Aut(X * ) × X ω , where (g, w) ∼ (h, w ′ ) if w = w ′ and g and h agree on a neighborhood of w.Since we only consider germs of Aut(X * ), and {C(v) : v is a prefix of w} forms a neighborhood basis, (g, w) ∼ (h, w ′ ) is equivalent to w = w ′ , g(w) = h(w) and g |v = h |v for some v prefix of w.We denote by T the groupoid of germs of tail equivalences, that is germs of the form (g, w) with g |v trivial for some prefix v of w.Given a groupoid of germs H, we denote by [[H]] the set of global homeomorphisms, such that all their germs belong to H.
We have Aut F (X * ) ⊂ [[T ]].In contrast to the case when X is finite, we do not have equality, as we can easily produce elements in [[T ]] which are not even in Aut fin.(X * ).
If w, w ′ ∈ X ω can be factored as w = vu, w ′ = v ′ u with v, v ′ ∈ X n , u ∈ X ω , we say that w and w ′ are n-tail equivalent.We say that w and w ′ are tail equivalent (or cofinal) if they are n-tail equivalent for some n.The n-tail equivalence class of w is denoted by T n (w) and T (w) = n∈N T n (w) is the cofinality class of w.
Lemma 2.4.Let g ∈ Aut B (X * ).There are only finitely many w such that (g, w) is not in T .If (g, w) is in T , then w and g(w) are cofinal.
Proof.The w where the germ of g is not in T are those where the sections along all prefixes are nontrivial.So they can be identified with the projective limit lim ← − v ∈ X n : g |v = 1 .Since g ∈ Aut B (X * ), the sets in the limit are uniformly bounded.Hence the projective limit is also finite.This proves the first claim.For the second claim, if (g, w) is in T then w factors as vu with g |v trivial, so g(w) = g(vu) = g(v)g |v (u) = g(v)u, so w and g(w) are cofinal.

Bounded Automata.
Definition 2.5.An automorphism g ∈ Aut(X * ) is called a finite state automorphism if the set of sections {g |v : v ∈ X * } is finite.
We denote by Aut f.s.B (X * ; P ) the subgroups of finite state automorphisms in Aut B (X * ; P ).Note that every g ∈ Aut F (X * ) is a finite state automorphism.
An automorphism g ∈ Aut fin.(X * ) is called directed if there is a word v ∈ X n with g |v = g and g |u ∈ Aut F (X * ) for all u ∈ X n , u = v.
Every finitary automorphism is a finite state automorphism.A directed automorphism has bounded activity growth.We will use the following structural result about finite state automata of bounded activity growth, see [Sid00].
Lemma 2.6.Let g ∈ Aut B (X * ) be a finite state automorphism.Then there exists a n such that for all v ∈ X n , g |v is either directed or finitary.△ 3. Random walks 3.1.Potential theoretic background.We will use the potential theoretic setting as in [Woe00]: Let N = (X, E, r) be a network, i.e. (X, E) is a connected locally finite graph, and r : E → (0, ∞) is a function.We think of r(e) as the resistance of e and denote by a(e) = 1/r(e) the conductivity of e.If Y is a subset of X, we denote by χ Y : X → {0, 1} the characteristic function of Y .
Our main examples will be Schreier graphs: if G is a group generated by a finite set S and G has a left action on X, then Γ(G, S, X) is the graph with vertex set X and edges x → s(x) for every x ∈ X, s ∈ S, all of unit resistance.We allow parallel edges and loops.
We are mostly interested in the space D(N ) of functions f : X → R with finite Dirichlet energy D(f ) = e∈E a(e)(f (e + ) − f (e − )) 2 .For any choice of base point All choices of o give equivalent norms, so there is a well-defined topology on D(N ), so that f n converges to f if and only if lim n D(f n − f ) = 0 and f n converges to f point-wise.Let D 0 (N ) be the closure of functions with finite support in D(N ).By [Woe00, Theorem I.2.12], the random walk on N is recurrent if and only if χ X ∈ D 0 (N ).We also use D 0 (N ) to get the following shorting criterion.
Lemma 3.1 ([Woe00, I.2.19]).Let X = i∈I X i be a partition of X such that χ Xi ∈ D 0 (N ) for all i ∈ I. Consider the shorted network N ′ with vertex set I and conductivity a ′ (i, j) = x∈Xi,y∈Xj (a(x, y)) As a special case we want to mention the Nash-Williams criterion [NW59]: We will also use the following lemma.
Lemma 3.3.Let N = (X, E, r) be a network, Y ⊂ X with ∂Y finite.Suppose N ′ = (Y, E ′ , r ′ ) is a network on Y obtained from N by restricting to Y and adding and removing finitely many edges and changing finitely many resistances. Suppose We extend f n to X by 0. Then ) differ in only finitely many summands, and these go to 0 by point-wise convergence of the f n .So we have lim n D N (f n − χ Y ) = 0 and thus χ Y ∈ D 0 (N ).

Recurrence on orbital Schreier graphs.
Definition 3.4.Let A be a group, L a left A-set.We say that the action of A on L is recurrent if for all finitely supported symmetric measures λ on A, the random walk on L induced by λ is recurrent for all starting points l 0 ∈ L. Remark 3.5.If A is finitely generated, it is enough to show this for one finitely supported symmetric measure whose support generates A. If S is a finite generating set of A, it is enough to consider the simple random walk on the Schreier graph Γ(G, S, X).See for example [Woe00].With this definition it is also clear that recurrent actions are closed under taking subgroups.
Lemma 3.6.Let A, B are groups, L a left A-set, M a left B-set such that the actions are both recurrent.Then the action of B ≀ L A on L × M is also recurrent.
Proof.Let us first reduce to the case where A and B are both finitely generated and both actions are transitive: Let (l, m) ∈ L × M, λ a symmetric finitely supported measure on B ≀ L A. Then there are finitely generated subgroups A ′ ⊂ A, B ′ ⊂ B such that supp(λ) ⊂ B ′ ≀ L A ′ .So wlog.let A and B be finitely generated.Let L ′ be the orbit of l.Then we have a quotient map π : B ≀ L A → B ≀ L ′ A and we can replace λ by π * (λ) to assume wlog.that the action of A on L is transitive.We can easily replace M with the orbit of m.
We can now assume that S and T are finite generating sets of A and B respectively, and both actions are transitive.Instead of showing recurrence for arbitrary λ, we can now fix a preferred generating set of B ≀ L A and show recurrence of the simple random walk on the Schreier graph.
Fix any base point l 0 ∈ L. We take as our generating set of B ≀ L A the set ι(S) ∪ T @l 0 , let N be the resulting network on the Schreier graph.We use Nash-Williamson criterion by partitioning L × M = m∈M L × m.Now ∂(L × m) is a finite collection of edges at (l 0 , m), and the random walk on Γ(A, S, L) is recurrent.So by Lemma 3.3, obtain χ L×m ∈ D 0 (N ).The shorted network is the Schreier graph of M with respect to T , so it is also recurrent.By Lemma 3.1, the network N is also recurrent.
Lemma 3.7.Let G be a finitely generated subgroup of Aut B (X * ).Assume that the action of G on every finite level is recurrent.Then the action of G on every component of the orbital Schreier graph is recurrent.
Proof.Let S be a finite symmetric generating set of G. Let K > 0 be a uniform bound on α n (s) for all n ∈ N, s ∈ S. Let Ω be a component of the orbital Schreier graph.Let N the network on associated with the simple random walk on Ω.
Let E be the set of edges in N which go between different cofinality classes.By Lemma 2.4, E is finite.Since Ω is connected, its vertex set must by contained in finitely many cofinality classes C 1 , . . .C n .Choose representatives w i ∈ C i ∩ Ω.
We claim that ∂T m (w i ) is uniformly bounded by Since Ω is connected, ∂(T m (w i ) ∩ Ω) is also uniformly bounded by K |S| and T m (w i ) ∩ Ω has only finitely many components.Each such component is a subnetwork of the (recurrent) random walk of G on level m, so by Lemma 3.3, their characteristic functions are in D 0 (N ).
Let X m := 1≤i≤n T m (i) ∩ Ω.Then χ Xm is the finite sum of characteristic functions of components of T m (w i ) ∩ Ω, so we obtain χ Xm ∈ D 0 (N ).Also, ∂X m ⊂ 1≤i≤n ∂(T m (w i ) ∩ Ω), so ∂X m is uniformly bounded by nK |S|.
We can now take a subsequence X mi such that ∂X mi is properly contained in X mi+1 .By applying Lemma 3.2 to the sequence X mi , the random walk on N is recurrent.

Amenability of groups generated by bounded activity automata
In this section we will prove the following theorem: Theorem B. Let P be an amenable subgroup of Sym(X).Suppose that the action of P on X is recurrent.Then Aut f.s.B (X * ; P ) is amenable.We will use the following criterion: Theorem C (Theorem 3.1 in [JNdlS16]).Let G be a finitely generated group of homeomorphisms of a topological space Y , and G be its groupoid of germs.Let H be a groupoid of germs of homeomorphisms of Y .Suppose that the following conditions hold: (1) The group [[H]] ∩ G is amenable.
(2) For every g ∈ G the germ of g at y belongs to H for all but finitely many y ∈ Y .We say that y ∈ Y is singular if there exists g ∈ G such that (g, y) / ∈ H.
Here P x is the stabilizer of x ∈ X of the action of P on X.The group homomorphism is injective, and the codomain is amenable, so G w is amenable.

Outlook
Our main application of the main theorem are iterated monodromy groups of post-singularly finite entire functions, see [Rei].We use the version of Theorem C from [JNdlS16], which impose a recurrence condition on the random walk on the orbital Schreier graphs.This recurrence condition was generalized to an extensive amenability condition in [JBMdlS16].It is shown in [JBMdlS16] that every recurrent action is also extensive amenable.In our Theorem B, it would be interesting to see whether we could weaken the recurrence condition to an condition about extensive amenability.Another direction to generalize is to step up in the hierarchy of automata with polynomial activity growth.In [AAV13,JNdlS16], it is shown that the group of automata of linear activity growth acting on a finite alphabet is amenable.Again, crucial step here is the recurrence of the random walk of the orbital Schreier graphs.It is not clear how this generalizes to infinite alphabets, as it seems that the estimates to show recurrence used finiteness of the alphabet at an important point.