Topological full groups of minimal subshifts with subgroups of intermediate growth

We show that every Grigorchuk group $G_\omega$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementary amenable, answering questions of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_\omega$ can be embedded in a finitely generated simple group that has trivial Poisson boundary for every simple random walk.


Introduction
Let (X, ϕ) be a Cantor minimal system, i.e. a minimal dynamical system where X is a compact space homeomorphic to the Cantor set, and ϕ is a homeomorphism of X. Recall that a dynamical system is said to be minimal if every orbit is dense. The topological full group of (X, ϕ) is the group [[ϕ]] of all homeomorphisms of X that locally coincide with a power of ϕ; in other words the homeomorphisms g of X so that there exists a continuous function n : X → Z that verifies g(x) = ϕ n(x) (x) for every x ∈ X. Giordano, Putnam and Skau show in [GPS99] that this group characterizes completely the dynamics of (X, ϕ), up to replacing ϕ by ϕ −1 .
In this note we are mainly interested in the possible behaviors for the growth of finitely generated subgroups of [[ϕ]]. The growth function of a finitely generated group G equipped with a finite symmetric generating set S is the function b G,S : N → N that counts the number of group elements that can be obtained as a product of at most n generators in S. The growth of G is said to be polynomial if there exist C, D > 0 such that b G,S (n) ≤ Cn D for every n ∈ N, exponential if there exists c > 1 such that b G,S (n) ≥ c n for every n ∈ N, and intermediate otherwise. These properties do not depend on the choice of S.
The susbshift (X ω , ϕ ω ) in Theorem 1.1 can be defined as a space of labeled graph structures on the set of integers, that have the same finite patterns as the orbital Schreier graphs for the action of G ω on the boundary of the binary rooted tree. Similar constructions of group actions based on graph colourings have been used, in this context, by Elek and Monod [EM13] and by van Douwen [vD90].
A minor modification of the construction allows to embed the groups G ω in the commutator subgroup of the topological full group of a minimal subshift, this is shown in Corollary 4.4. The latter is a finitely generated simple group by results of Matui [Mat06,Theorem 4.9, Theorem 5.4]. Relying on a result from [MB14] we deduce the following statement in Theorem 4.1: every Grigorchuk group G ω can be embedded in a finitely generated simple group with the Liouville property (in particular, amenable). Note that the topological full group of any Cantor minimal system is amenable by a result of Juschenko and Monod [JM13].
Related constructions appear in Vorobets' articles [Vor10,Vor12]. In [Vor10] he studies dynamical properties of a minimal one-sided substitutional subshift, which can be shown to be a one-sided version of our (X ω , ϕ ω ) for the periodic sequence ω = 012 · · · (the group G ω corresponding to this ω is known as the first Grigorchuk group). In [Vor12] he studies the dynamics of the action of the first Grigorchuk group on the space of marked Schreier graphs.
The paper is structured as follows. Section 2 recalls the definition of Grigorchuk groups and the construction of their orbital Schreier graphs. Proofs concerning this construction are postponed to the Appendix at the end. Section 3 contains the definition of the subshift (X ω , ϕ ω ) and to the proof of Theorem 1.1; this section can be read without reading the Appendix, but it requires notations from Section 2. In Section 4 we modify the construction to get an embedding in the commutator subgroup; then we study the complexity of the subshift (X ω , ϕ ω ) and deduce the above-mentioned embedding result for Grigorchuk groups.
b ω permutes the letter at position j + 1 in v, while if ω(j) = 2 then b ω (v) = v. An analogous description holds for c ω , d ω after replacing 2 by 1, 0, respectively.
Note that for every ω ∈ Ω we have the relations Grigorchuk proved in [Gri84, Theorem 2.1, Corollary 3.2] that 1. the group G ω is residually finite and amenable for every sequence ω; 2. if ω eventually constant the group G ω is virtually abelian, otherwise it has intermediate growth and it is not elementary amenable (recall that the class of elementary amenable groups is the smallest class of groups containing finite and abelian groups and which is closed by taking extensions, inductive limits, subgroups and quotients); 3. the group G ω is 2-torsion if ω has infinitely many appearances of each of the symbols 0, 1, 2, otherwise G ω admits elements of infinite order.

Schreier graphs of Grigorchuk groups
The action of Grigorchuk groups on the binary tree extends to the boundary at infinity of the tree, identified with the set of right-infinite strings {0, 1} ∞ . Hereinafter let ρ be the constant 1-string ρ = 111 · · · ∈ {0, 1} ∞ , and let G ω ρ be its orbit. We denote by Γ ω the Schreier graph for the action of G ω on the orbit of ρ, with respect to the generating set S = {a, b ω , c ω , d ω }. This is the labelled graph whose vertex set is G ω ρ and such that γ, η ∈ G ω ρ are connected by an edge if there exists s ∈ S such that sγ = η. Such an edge is labelled by s. Loops and multiple edges are allowed. Since all generators are involutions, the orientation of edges is unimportant and we shall work with non-oriented graphs.
The Schreier graphs Γ ω can be described elementary, see Bartholdi and Grigorchuk [BG00, Section 5] where the graph Γ ω is constructed for the sequence ω = 012012 · · · (the corresponding group G ω is known as the first Grigorchuk group). The construction of the graph Γ ω for other choices of ω is also well-known and can easily be adapted from the case of the first Grigorchuk group. We now recall this construction and some related facts; since we were not able to locate a reference for a generic ω, we refer to the Appendix for proofs. Figure 1: The beginning of the graph Γ ω for ω = 012012 · · ·.
Proposition 2.2. As an unlabelled graph, Γ ω does not depend on the sequence ω and is isomorphic to the half-line with additional loops and double edges shown in Figure 1 (apart from labels). The endpoint of the half-line is the vertex ρ. Moreover for every sequence ω • the three loops at ρ are labelled by "b ω ", "c ω ", "d ω "; • every simple edge is labelled "a".
The remaining edges (loops and double edges) are labelled by b ω , c ω , d ω . The labelling of these depends on ω and is described in Lemma 2.4.
Before describing the labelling of Γ ω we fix some notation.
Definition 2.3. 1. We denote Θ, Λ 0 , Λ 1 , Λ 2 the four finite labelled graphs shown in Figure 2, and Ξ the graph with one vertex and three loops, labelled by b ω , c ω , d ω . 2. We denote * the operation of gluing two graphs by identifying two vertices. For well-definiteness, we assume that all finite graphs that we consider have a distinguished "leftmost vertex" and a "rightmost vertex" (that will be clear from the context); the operation * corresponds to identifying the rightmost vertex of the first graph with the leftmost vertex of the second. Figure 2: The "building blocks" of Γ ω .
By Proposition 2.2, with this notation Γ ω has the form It will be convenient to denoteΓ ω the modification of Γ ω obtained by erasing the three loops at ρ, i.e.Γ We now state a lemma, that allows to constructΓ ω recursively (part (i) is sufficient for this purpose). We refer to the Appendix for a proof.

Construction of the subshift
We now turn to the proof of Theorem 1.1.
We first address separately the degenerate case of an eventually constant ω.
Proposition 3.1. Suppose that the sequence ω is eventually constant. Then for every Proof. If ω is eventually constant, the group G ω is virtually abelian, see [Gri84, Theorem 2.1 (3)]. In fact it can be embedded in a semi-direct product of the form Z n S n for sufficiently large n, where the symmetric group S n acts on Z n by permutation of the coordinates (strictly speaking, it is shown in [Gri84] that it can be embedded in D n ∞ S n , where D ∞ is the infinite dihedral group; note for instance that D n ∞ S n embeds in Z 2n S 2n by identifying D ∞ with the subgroup of Z 2 S 2 generated by (1, −1) ∈ Z 2 and by the nontrivial ∈ S 2 ). It is thus sufficient to show that for every minimal Cantor system (X, ϕ) and every n ∈ N, the group Z n S n embeds in , and is the identity on the complement of ϕ i (U ) ∪ ϕ j (U ). The group generated by the involutions σ ij is isomorphic to the symmetric group S n . Let r 0 ∈ [[ϕ]] be any element of infinite order with support contained in U (for instance take the first return map of U extended to the identity on the complement of U ). For i = 0, . . . , n − 1 set r i = ϕ i r 0 ϕ −i . The support of r i is contained in ϕ i (U ), thus the choice of U implies that r i and r j commute if i = j. It follows that the group generated by r 0 , . . . , r n−1 is free abelian of rank n. The group Hereinafter we fix ω ∈ Ω and assume that it is not eventually constant.
By a pattern in a labeled graph we mean the isomorphism class of a finite, connected labelled subgraph.
1. Let X be the the space of edge-labeled, connected graphs with vertex set Z such that any two consecutive integers n, n + 1 ∈ Z are connected by one of the four labelled graphs in Figure 2, and there is no edge between n, m ∈ Z if |n − m| > 1. Endow X with the natural product topology and with the shift map ϕ induced by translations of Z, which makes (X, ϕ) conjugate to the full shift over the alphabet A = {Θ, Λ 0 , Λ 1 , Λ 2 }.
2. Let X ω ⊂ X be the set of all x ∈ X such that every pattern of x appears iñ Γ ω (recall that this is the graph obtained from Γ ω after erasing the three initial loops). This defines a closed, shift-invariant subset of X and we denote (X ω , ϕ ω ) the subshift obtained in this way.
Remark 3.3. Equivalently X ω could be defined as the orbit-closure of any x ∈ X isomorphic to the Schreier graph of the orbit of a point γ ∈ {0, 1} * lying outside the orbit of ρ (these graphs are isomorphic to bi-infinite lines; their labellings admit similar and only slightly more complicated recursive construction).
A labeled graph is said to be uniformly recurrent if for every finite pattern in the graph exists a constant R > 0 such that every ball of radius R in the graph contains a copy of the pattern.
Lemma 3.4. The graphΓ ω is uniformly recurrent. Moreover its labelling is not eventually periodic along the half-line.
Proof. We take notations from Definition 2.3 and Lemma 2.4.
By part (i) of Lemma 2.4, for every pattern ofΓ ω there exists n such that the pattern appears inΓ n,ω . Using this, part (ii) of Lemma 2.4 implies thatΓ ω is uniformly recurrent.
Suppose that the labelling ofΓ ω is eventually periodic, i.e. that the sequence of graphs (∆ j,ω ) j≥1 is eventually periodic with period T . SinceΓ 1,ω = Θ * Λ ω(1) * Θ, part (ii) of Lemma 2.4 for n = 1 implies that all odd terms in the sequence (∆ j,ω ) j≥1 are equal to Λ ω(1) , while the subsequence of even terms is equal to (∆ j,σω ) j≥1 . Since we assume that ω is not eventually constant, part (i) of Lemma 2.4 easily implies that the sequence (∆ j,ω ) j≥1 is also not eventually constant, hence there are infinitely many is for which ∆ i,σω = Λ ω(1) . We conclude that T is even, and that the sequence(∆ j,σω ) j≥1 is also eventually periodic with period T /2. Since ω is not eventually constant, neither is σω, thus we may repeat the same reasoning forΓ σω to conclude that T /2 is even and (∆ j,σ 2 ω ) j≥1 is eventually periodic with period T /4. By induction T is a multiple of 2 n for every n, thus T = 0.
Previous results by Vorobets in [Vor10] imply minimality of a one-sided subshift given by a certain substitution, which is conjugate to the one-sided version of (X ω , ϕ ω ) for the periodic ω = 012 · · ·.
Proof. Let us first check that X ω is not empty. To see this, let y ∈ X be any graph that agrees withΓ ω on the positive half-line. Then any cluster point of (ϕ n (y)) n≥0 belongs to X ω . Let x ∈ X ω be arbitrary. It is a consequence of Lemma 3.4 that every pattern ofΓ ω appears in x infinitely many times, and that x is uniformly recurrent. Namely every pattern ofΓ ω appears in every sufficiently long segment ofΓ ω , thus in every sufficiently long segment of x by the construction of X ω . This easily implies that the orbit of x is dense in X ω , and that (X ω , ϕ ω ) is minimal by a well-known characterization of minimality (see [Got46] or [Que87, Proposition 4.7]). Finally the fact thatΓ ω is not eventually periodic (Lemma 3.4) implies that x is not periodic, sinceΓ ω and x have the same finite patterns and periodicity can be easily characterized in terms of these (for instance by Morse and Hedlund's theorem, see [CN10, Theorem 4.3.1]). Thus X ω is infinite.
Remark 3.6. It follows from the proof that every pattern inΓ ω appears in every x ∈ X ω . Proof. Let x ∈ X ω . By construction, for every f ∈ {a, b ω , c ω , d ω } the vertex 0 ∈ Z is the endpoint of exactly one edge in x which is labelled by f (this edge is possibly a loop at 0). We define four elementsā,b ω ,c ω ,d ω of the topological full group [[ϕ ω ]] as follows: if the label f is on an edge connecting 0 and 1 (respectively on an edge connecting 0 and -1, on the loop at 0). It We claim that the map ι : a →ā, b ω →b ω , c ω →c ω , d ω →d ω extends to an injective group homomorphism ι : To see that ι extends to a well-defined homomorphism, it is sufficient to check that relations are respected, i.e. that for every n ∈ N and h 1 , . . . , h n ∈ {a, b ω , c ω , d ω } such that h 1 · · · h n = e in G ω we haveh 1 · · ·h n = e in [[ϕ ω ]]. Suppose thath 1 · · ·h n = e in [[ϕ ω ]]. Then there exists a point x ∈ X ω such thath 1 · · ·h n x = x. Write x| n for the finite subgraph of x spanned by the interval [−n, n] ⊂ Z. Observe that the fact whether x is fixed or not byh 1 · · ·h n only depends on x through x| n . By the construction of X ω , the graph x| n isomorphic to a labeled subgraph ofΓ ω , hence of Γ ω . Let γ ∈ Γ ω be the midpoint of this subgraph. It follows from the definition ofā,b ω ,c ω ,d ω that h 1 · · · h n γ = γ. We conclude that h 1 · · · h n = e in G ω .
The verification that ι is injective is similar. Let h 1 , · · · , h n ∈ {a, b ω , c ω , d ω } be such that h 1 · · · h n = e in G ω . Then there is a vertex of the rooted tree, say v ∈ {0, 1} * , which is moved by h 1 · · · h n . Recall that the orbit G ω ρ consists exactly of all sequences γ ∈ {0, 1} * that are cofinal with ρ (Proposition 2.1). In particular, v is a prefix of infinitely many such sequences, each of which is moved by h 1 · · · h n . Thus we can find γ ∈ Γ ω lying at a distance greater than n from ρ, and such that h 1 · · · h n γ = γ. Since we have chosen γ at a distance greater than n from ρ, γ is the midpoint of a connected graph of length 2n in Γ ω . This graph is also a subgraph ofΓ ω . By Remark 3.6 and by shift-invariance, there exists x ∈ X ω such that x| n is isomorphic to this graph (recall that x| n denotes the finite subgraph of x spanned by [−n, n]). Again by construction ofā,b ω ,c ω ,d ω , we haveh 1 · · ·h n x = ϕ ±k ω (x), where k > 0 is the distance by which γ is displaced on Γ ω by the action of h 1 · · · h n . But Corollary 3.5 implies that there is no periodic point for ϕ ω , thush 1 · · ·h n (x) = ϕ ±k ω (x) = x, and soh 1 · · ·h n = e in [[ϕ ω ]]. We conclude that ι is injective.
Remark 3.8. The usual action of G ω on the Cantor set identified with the boundary of the tree {0, 1} ∞ does not belong to the topological full group of a minimal homeomorphism of {0, 1} ∞ . To see this, note that b ω , c ω , d ω fix ρ but act non-trivially on every neighbourhood of ρ. This can not happen in the topological full group of a minimal homeomorphism.

Embedding G ω in a simple Liouville group
It is a classical result, due to Hall [Hal74], Gorjuskin [Gor74] and Schupp [Sch76] that every countable group can be embedded in a finitely generated simple group. These constructions always yield "large" ambient groups (e.g. non-amenable) even if the starting group is "small". In this section we deduce the following consequence for the groups G ω from the construction presented in Section 3.
Theorem 4.1. Every Grigorchuk group G ω can be embedded in a finitely generated, simple group that has trivial Poisson-Furstenberg boundary for every symmetric, finitely supported probability measure on it (in other words, it has Liouville property and in particular it is amenable).
The class of groups with the Liouville property contains the class of subexponentially growing groups (Avez [Ave74]) and it is contained in the class of amenable groups (this is due to Furstenberg, see [KV83,Theorem 4.2]).
Recall that the complexity of a subshift (X, ϕ) over a finite alphabet A is the function ρ X : N → N that associates to n the number of finite words in the alphabet A that appear as subwords of sequences in X. The proof of Theorem 4.1 relies on the following result from [MB14].
The system (Y, ψ) is clearly minimal as soon as (X, ϕ) is. Observe that X × {1} and X × {2} are ψ 2 -invariant and that the restrictions of ψ 2 to X × {1} and to X × {2} are conjugate to (X, ϕ). (ii). If (X, ϕ) is a subshift over the finite alphabet A, the above construction is readily seen to be equivalent to the following one. Consider the alphabet B = A {z} where z is a letter not in A, and define Y ⊂ B Z to be the subshift consisting of all sequences of the form y = · · · zx −2 zx −1 .zx 0 zx 1 · · · or y = · · · zx −2 zx −1 z.x 0 zx 1 · · · where x = · · · x −2 x −1 .x 0 x 1 · · · is a sequence in X. There is a natural partition of Y in two sets homeomorphic to X which are given by the parity of the position of appearances of z. These two sets play the roles of X × {1} and X × {2} in the proof of part (i) and the same argument yields an embedding of H in [[ψ]] . The claim on the complexity is also clear from this description.
Since the groups G ω are generated by elements of order 2, we get the following improvement of Theorem 1.1. This already gives an embedding of G ω into a finitely generated, simple amenable group by the result of Juschenko and Monod [JM13]. To prove that the Liouville property holds, we estimate the complexity of the subshift (X ω , ϕ ω ) constructed in Section 3, then apply part (ii) of Lemma 4.3 and Theorem 4.2. The following lemma is enough to conclude the proof of Theorem 4.1.
Remark 4.6. Here we see (X ω , ϕ ω ) as a subshift in the usual sense, over the (formal) Proof. Let x ∈ X ω and consider a finite subword w of x of length n ∈ N. Let m ∈ N be the smallest integer such that n ≤ 2 m . By part (ii) of Lemma 2.4 the word w appears as a subword of a word of the formΓ m−1,ω * ∆ * Γ m−1,ω , where ∆ ∈ {Λ 0 , Λ 1 , Λ 2 }. Such a word is uniquely determined by ∆ (3 possibilities) and by the position of its first letter (2 m possibilities, since we may assume without loss of generality that the first letter of w is inΓ m−1,ω * ∆). Thus ρ Xω (n) ≤ 3 · 2 m ≤ 6n, where we have used that 2 m−1 < n by the choice of m.

Appendix: Schreier graphs and Gray code
In this Appendix we give proofs of the statements in Section 2.2, namely Propositions 2.1, Proposition 2.2 and Lemma 2.4.
We first state a remark that follows from the definitions of a, b ω , c ω , d ω .
Remark A.1. Given any binary string γ ∈ {0, 1} ∞ , a generator s ∈ {a, b ω , c ω , d ω } that acts non-trivially on γ can act in two possible ways: Moreover for every γ = ρ it is possible to choose s ∈ {b ω , c ω , d ω } that acts on γ by a move of type (M2).
Proof of Proposition 2.1. Since the action of every generator of G ω changes at most one letter of each binary string, every string in the orbit of ρ is cofinal with ρ. Conversely, it is easy to see that starting from ρ and performing only moves of type (M1) and (M2) it is possible to reach every string which is cofinal with ρ.
Proof of Proposition 2.2. Since ρ is fixed by b ω , c ω , d ω , it has three loops and it is the endpoint of a simple edge (given by the action of a). If γ = ρ there exists exactly one generator s ∈ {b ω , c ω , d ω }, such that sγ = γ. The generator s is determined by the value ω(m), where m is the position of the first 0 appearing in γ. The two other generators s , s ∈ {b ω , c ω , d ω } act non-trivially on γ by a move of type (M2), in particular s γ = s γ. The generator a acts non-trivially on γ by a move of type (M1), and thus aγ = s γ. Hence the vertex corresponding to γ in Γ ω is the endpoint of a loop, a double edge and a simple edge. Finally, observe that the only connected unlabelled graph respecting these local rules is the graph shown in Figure 2.
Before giving the proof of Lemma 2.4 we recall some terminology. The Gray code is a classical non-standard way to code natural numbers with binary strings, having the property that two consecutive strings differ only by one bit. It is a well-known fact that the Gray code can be used to describe the orbital Schreier graphs of several groups acting on rooted trees, including all groups G ω (cf. Bartholdi, Grigorchuk anď Sunić [BGŠ03, Section 10.3] where this is mentioned explicitly for the first Grigorchuk group). We recall the construction of the Gray code in the definition below; the reader should be warned that the roles of 0 and 1 are exchanged here with respect to the more usual definition.
In plain words, the first 2 l terms of the sequence (r For example, the Gray code order of binary strings of length 4 is 1111; 0111; 0011; 1011; 1001; 0001; 0101; 1101; 1100; 0100; 0000; 1000; 1010; 0010; 0110; 1110. Definition A.3. The Gray code enumeration of the orbit G ω ρ is the sequence (ρ j ) j∈N , taking values in {0, 1} ∞ , where ρ j is obtained by attaching infinitely many digits 1 to r (l) j , for any l > 0 such that r (l) j is defined (i.e. such that j ≤ 2 l − 1). Note that, for any such l, the string r (l+1) j is obtained from r (l) j by attaching the digit 1 on the right, in particular ρ j does not depend on the choice of l.
Lemma A.4. The sequence (ρ j ) coincides with the sequence of vertices of Γ ω , ordered by increasing distance from ρ.
Proof. It follows from the definitions that ρ 0 = ρ, and that ρ j+1 is obtained from ρ j by performing a move of type (M1) if j is even and a move of type (M2) if j is odd (this can be immediately proven by induction on l from the definition of the Gray code order). Thus the sequence (ρ j ) coincides with the sequence of vertices of Γ ω by Remark A.1.
Proof of Lemma 2.4. For j ≥ 1 let m j be the position of the first 0 digit in ρ j . There is exactly one generator s ∈ {b ω , c ω , d ω } such that sρ j = ρ j and this generator is determined by ω(m j ) (more precisely s = b ω , c ω , d ω if ω(m j ) = 2, 1, 0, respectively). By comparing this observation with Definition 2.3 and with Figure 2, we see that ∆ i,ω = Λ ω(m2i−1) . Set a i = m 2i−1 . The sequence (a i ) only depends on the definition of the Gray code order, and it is characterized by the recursion a 2 n = n + 1 for every n ≥ 0, a 2 n +j = a 2 n −j for every 1 ≤ j ≤ 2 n − 1.