On the topological full group of a minimal Cantor Z^2-system

Grigorchuk and Medynets recently announced that the topological full group of a minimal Cantor Z-action is amenable. They asked whether the statement holds for all minimal Cantor actions of general amenable groups as well. We answer in the negative by producing a minimal Cantor Z^2-action for which the topological full group contains a non-abelian free group.


Introduction
Let G be a group acting on a compact space Σ by homeomorphisms. The topological full group associated to this action is the group of all homeomorphisms of Σ that are piecewise given by elements of G, each piece being open. Thus there are finitely many pieces at a time, all are clopen, and this construction is most interesting when Σ is a Cantor space. The importance of the topological full group has come to the fore in the classification results of Giordano-Putnam-Skau [2,3]. Grigorchuk and Medynets announced that the topological full group of a minimal Cantor Z-action is amenable [6]. This is particularly interesting in combination with the work of Matui [8], who showed that the derived subgroup is often a finitely generated simple group. Grigorchuk-Medynets further asked in [6] whether their result holds for actions of general amenable groups as well. We shall prove that it fails already for the group Z 2 : Theorem 1. There exists a free minimal Cantor Z 2 -action whose topological full group contains a non-abelian free group.
Three comments are in order, see the end of this note: 1. There also exist free minimal Cantor Z 2 -actions whose topological full group is amenable, indeed locally virtually abelian. 2. Minimality is fundamental for the study of topological full groups. Even for Z, it is easy to construct Cantor systems whose topological full group contains a non-abelian free group (using e.g. ideas from [9] or [4]). 3. Our example will be a minimal subshift and in this situation the topological full group is sofic by a result of [1].

Proof of the Theorem
We realize the Cantor space as the space Σ of all proper edge-colourings of the "quadrille paper" two-dimensional Euclidean lattice by the letters A, B, C, D, E, F (with the topology of pointwise convergence relative to the discrete topology on the finite set of letters). Recall here that an edge-colouring is called proper if the edges adjacent to a given vertex are coloured differently. There is a natural Z 2 -action on Σ by homeomorphisms defined by translations.
To each letter x ∈ {A, . . . , F } corresponds a continuous involution of Σ, which we still denote by the same letter. It is defined as follows on σ ∈ Σ: if the vertex zero is connected to one of its four neighbours v by an edge labelled by x, then v is uniquely determined and xσ will be the colouring σ translated towards v (i.e. the origin is now where v was). Otherwise, xσ = σ. This involution is contained in the topological full group of the Z 2 -action.
We have thus a homomorphism from the free product A * · · · * F to the topological full group. Notice that this free product preserves any Z 2 -invariant subset of Σ. We shall establish Theorem 1 by proving that Σ contains a minimal non-empty closed Z 2 -invariant subset M on which the Z 2 -action is free and on which the action of ∆ := A * B * C is faithful. This implies the theorem indeed, for ∆ has a (finite index) non-abelian free subgroup.
A pattern of a colouring σ ∈ Σ is the isomorphism class of a finite labelled subgraph of σ. We call σ homogeneous if for any pattern P of σ there is a number f (P ) such that the f (P )-neighbourhood of any vertex in the lattice contains the pattern P . The following facts are well-known and elementary (see e.g. [5]).
Lemma 2. The orbit closure of σ ∈ Σ is minimal if and only if σ is homogeneous. In that case, any τ in the orbit closure has the same patterns as σ and is homogenous with the same function f . Now, we first enumerate the non-trivial elements of the free product ∆. Then, we label the integers with the natural numbers in such a way that the following property holds: for each i ∈ N there is g(i) ≥ 1 such that any subinterval of length g(i) in Z contains at least one element labelled by i. Such a labelling exists: for instance, label an integer by the exponent of 2 in its prime factorization (with an arbitrary adjustment for 0).
We use the labelling above to construct a specific proper edge-colouring λ ∈ Σ. Let w be a word in ∆ that is the i-th in the enumeration. Consider the vertical vertex-lines (v, ·) in the lattice such that v is labelled by i. Colour those vertical lines the following way. Starting at the point (v, 0), copy the string w onto the half-line above, beginning from the right end of w (i.e. write w −1 upwards). Then colour the following edge by D, then copy the string w again and repeat the process ad infinitum. Also, continue the process below (v, 0) so as to obtain a periodic colouring of the whole vertical line. Repeating the process for all non-trivial words w, we have coloured all vertical lines. Finally, colour all horizontal lines periodically with E and F .
The resulting colouring λ has the following property. For any non-trivial w ∈ ∆ there is a number h(w) such that the h(w)-neighbourhood of any vertex of the lattice contains a vertical string of the form w −1 D. Let Ω(λ) ⊆ Σ be the Z 2 -orbit closure of λ. Then all the elements of Ω(λ) have the same property. Now, let M be an arbitrary minimal subsystem of Ω(λ) (in fact it is easy to see that λ is homogeneous and hence Ω(λ) is already minimal). Notice that the Z 2 -action on M is free because λ has no period. In order to prove the theorem, it is enough to show that for any σ in M and any non-trivial w ∈ ∆ there exists a Z 2 -translate of σ which is not fixed by w.
Pick thus any σ ∈ M . Then, by the above property of the orbit closure, there exists a translate τ of σ such that the vertical half-line pointing upwards from the origin starts with the string w −1 D. Hence if we apply w to the translate we reach a point τ such that the colour of the edge pointing upwards from the the origin is coloured by D. Thus τ is not fixed by w, finishing the proof.

Comments
Some Z 2 -systems have a completely opposite behaviour to the ones constructed for Theorem 1. We shall see this by extending the method of Proposition 2.1 in [7].
Recall that the p-adic odometer is the minimal Cantor system given by adding 1 in the ring Z p of p-adic integers. Taking the direct product, we obtain a minimal Cantor Z 2 -action on Σ := Z p × Z p . The proposition below and its proof can be immediately extended to products of more general odometers.
Proposition 3. The full group of this minimal Cantor Z 2 -system is an increasing union of virtually abelian groups.
Proof (compare [7]). Consider Z p as the space of Z/pZ-valued (infinite) sequences. Given a pair of finite sequences of length n, we obtain an n-cylinder set in Σ as the space of pairs of sequences starting with the given prefixes. Thus, n-cylinders determine a partition P n of Σ into p 2n clopen subsets. Moreover, the clopen partition associated to any given element g of the topological full group can be refined to P n when n is large enough. It remains only to observe that the collection of all such g, when n is fixed, is a subgroup of the semi-direct product (Z 2 ) Pn ⋊ Sym(P n ), where Sym(P n ) is the permutation group of the coördinates indexed by P n .
Regarding the second comment of the introduction, suffice it to say that a generic proper colouring of the linear graph by three letters A, B, C gives a faithful non-minimal representation of the free product A * B * C into the topological full group of the associated Z-subshift (compare [9] or [4] for generic constructions).
As for the last comment, Proposition 5.1(1) in [1] implies that the topological full group of any minimal subshift of any amenable group is a sofic group (in the notations of [1], the kernel N Γ is trivial by an application of Lemma 2). In combination with Matui's results [8], this already shows the existence of a sofic finitely generated infinite simple group without appealing to [6].