Pattern closure of groups of tree automorphisms

It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its level s stabilizer. As an application, it is shown that there are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size two.


Introduction
The group of automorphisms of a regular rooted tree carries three structures, namely a self-similarity structure (related to symbolic dynamics on the tree), a metric structure (with Cantor set topology), and a group theoretic structure (of an iterated wreath product). Each of the tree structurers comes with a naturaly associated closure oerator. Namely, given a set S of tree automorphisms, we may consider the self-similar closure of S (the smallest self-similar set containing S), the topological closure of S, and the group closure of S (the group generated by S). The study of the interaction of these three closures naturally leads to the study of patterns in tree automorphisms.
The main results proved here are as follows.
Theorem 1. Let G be a finitely constrained group of tree automorphisms of X * defined by allowing all patterns of size s + 1, s ≥ 0, that appear in some self-similar group K (and forbidding those that do not). Then G is the topological closure (in Aut(X * )) of a self-similar, countable, regular branch group H, branching over its level s stabilizer H s . Moreover, if K is contracting, H may be chosen to be contracting as well.
As an application of Theorem 1, we prove the following.
Theorem 2. There are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size at most 2.
Note that the closure of the first Grigorchuk group is an infinite, finitely constrained, topologically finitely generated group of binary tree automorphisms defined by patterns of size 4 [Gri05]. The closures of the groups defined by polynomials in [Šun07] provide examples of infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by patterns of size s (but not size s − 1), for any s ≥ 4. Thus, by Theorem 2, the question of existence of infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by patterns of size s remains open only for s = 3.
The necessary background on groups of automorphisms of rooted regular trees is provided in the next two sections, which are followed by a section in which the main results are proved. More extensive background infromation may be found in [Gri05,Nek05,GŠ07,BGŠ03].
1. Background on symbolic dynamics on rooted trees 1.1. Rooted trees. Let X be a finite alphabet of cardinality k (our standard choice is X = {0, 1, . . . , k − 1}). The rooted tree over X is the k-ary rooted tree in which the vertices are the finite words over X, the empty word ∅ is the root and every vertex u is connected by k directed edges to its k children ux, for x in X. The edge connecting u to ux is labeled by x. Level n of the tree X * is the set X n of words of length n over X. We use X * to denote both the rooted tree over X and the set of all words over X. Note also that the rooted tree X * is the right Cayley graph of the free monoid X * over X.
1.2. Portrait space. Let A be a finite alphabet (in order to avoid confusion, this alphabet is usually disjoint from X). The portrait space on the tree X * over the alphabet A is the space A X * of all maps from X * to A. This space is also called the shift space or the full shift space on X * over the alphabet A. The elements of A X * are called portraits (X-tree portraits over A). For a portrait g in the portrait space, denote by g (u) the symbol from A at vertex u in the tree (note that (u) is in the subscript position with respect to g). The symbol g (u) is sometimes called the decoration at u in the portrait g and the alphabet A is called the decoration alphabet.
The portrait space A X * is a metric space in which, for distinct portraits g and h, the distance is given by The topology on A X * is just the product topology on A X * induced by the discrete finite space A. Thus, as long as |A| ≥ 2, A X * is a Cantor set (in particular, it is compact).
For u in X * , the section map σ u : A X * → A X * at u (also known as the shift map) on the portrait space is defined by The section maps provide a right action of X * on the portrait space by continuous maps. Note that, more generally, portrait spaces may be defined over any semigroup (not only over the free monoid X * as defined here; see for instance [CP93]).
Definition 1. A set of portraits is self-similar if it is invariant under the section maps. Other terms used for self-similar sets are X * -invariant or shift invariant sets.
Note that the case k = 1 is not excluded from our considerations. In this case the tree X * has the structure of a ray (one-way inifinite path), the monoid X * is isomorphic to the monoid of natural numbers N and the shift space A N is just the standard one-dimensional one-way shift (see [LM95] or [Kit98]).
1.3. Forbidden patterns. Let s ≥ 1. Rooted tree of size s over X, is the subtree of X * consisting of the vertices in X [s] = ∪ s−1 i=0 X i (we denote this subtree also by X [s] ). An X-tree pattern of size s over A is a map in A X [s] . All 8 X-tree patterns, where X = {0, 1}, of size 2 over A = { , } are presented in Figure 1. A tree Let F be any set of X-tree patterns over A. Denote by G(F ) the set of all portraits in the portrait space A X * that do not contain any pattern from F at any vertex. A subset G of the portrait space is defined by a set of forbidden patterns if G = G(F ) for some set of tree patterns F . The set F is called the set of forbidden tree patterns defining G.
Theorem 3. Let G be a set of X-tree portraits over A. The following are equivalent.
(i) G is closed, self-similar subset of the full tree portrait space A X * .
(ii) G is defined by a set of forbidden X-tree patterns.
Closed self-similar sets of portraits are called portrait subspaces (or sometimes portrait spaces, shifts, or subshifts). A portrait space defined by finitely many forbidden patters is called a portrait space of finite type.
Example 1. Let X = {0, 1}, A = { , } and consider the X-tree patterns of size 2 over A provided in Figure 1.
If we forbid the patterns in the bottom row, i.e., we define the set of forbidden patterns B = {t, t 3 , at, at 3 }, the automorphisms in the corresponding portrait space of finite type G(B) can be characterized as follows. A portrait g belongs to G(B) if and only if, for every vertex u in X * , Similarly, we may forbid the patterns in the right half of Figure 1, i.e., we define the set of forbidden patterns R = {a, at 2 , at, at 3 }. A portrait g belongs to the portrait space of finite type G(R) if and only if, for every vertex u in X * , where we interpret the addition and the equality modulo 2, and we interpret as 0 and as 1.

Background on groups of tree automorphisms
Let X * be a rooted k-ary tree tree. We consider the special case when the alphabet A is the finite symmetric group S(X). i.e., the case when the decoration at each vertex of the tree is a permutation of the alphabet X. Every portrait g of this type defines a rooted tree automorphism of X * , also denoted by g, defined by Conversely, if g is a tree automorphism, it defines a portrait on X * , also denoted by g, where the permutation of X at the vertex u is uniquely determined by The group Aut(X * ) of rooted tree automorphisms of X * inherits the self-similarity and the metric structure from the X-tree portrait space S(X) X * . In particular, Aut(X * ) is compact and so is each of its closed subgroups.
Note that Aut(X * ) has the structure of an iterated permutational wreath product where the isomorphism Aut(X * ) ∼ = S(X) ⋉ (Aut(X * )) X is given by and, for x ∈ X, the automorphism g| x is just the section σ x (g) of g at x. If we identify Aut(X * ) and S(X) ⋉ (Aut(X * )) X under this isomorphism then, for any two automorphims g and h, We will make use of the equalities expressing the sections of products and inverses as products and inverses of appropriate sections. For a set of tree automorphisms S we may define the group S generated by S, the (topological) closure S of S and the smallest self-similar setS of tree automorphisms containing S (it consist of all sections of all elements in S). Further, we can combine these closure operators. For instance, the closure of a group of tree automorphisms is a group and therefore S is the smallest closed group containing S. Similarly, the closure of a self-similar set is self-similar and thereforẽ S is the smallest closed self-similar set containing S. Finally, a group of tree automorphisms generated by a self-similar set is self-similar and therefore S is the smallest self-similar group containing S. The smallest closed self-similar group of tree automorphisms containing S is S .
A closed group G of tree automorphisms is topologically finitely generated if G = S for some finite set S. A tree automorphism g is a finite-state automorphism (we also say that g is defined by a finite automaton) ifg (the set of sections of g) is finite. A group G of tree automorphisms is called an automaton group if it is generated by a finite self-similar set, i.e., G = S , whereS is finite. A selfsimilar group G of tree automorphisms is contracting if there exist a finite set N of automorphisms such that, for every g in G, there exists a level n (depending on g) such that, for all m ≥ n, sections of g at level m are elements of N . Note that finitely generated contracting groups are automaton groups (each element of a contracting group has only finitely many distinct sections, so it is a finite-state automorphism).
Proposition 1. Let G = G(F ) be a closed, self-similar subset of the tree portrait space S(X) X * defined by a set of forbidden patterns F . The set G is a subgroup of Aut(X * ) if and only if, for every s ≥ 1, the set of essential patterns E s of size s (patterns of size s that actually appear in some element of G) forms a subgroup of Aut(X s ) (the automorphism group of the finite regular tree over X of depth s).
In case F is a finite set of patterns of size s, G is a group if and only if E s is a subgroup of Aut(X s ).
Theorem 4. Let G be a group of tree automorphisms of X * . The following are equivalent.
(i) The group G is closed, self-similar subgroup of Aut(X * ).
(ii) The group G is defined by a set of forbidden patterns.
A group of tree automorphisms defined by a finite set of forbidden patterns is called a finitely constrained group (a more appropriate term would probably be a group of finite type, as in [Gri05], where this kind of groups were introduced, but this term seems to be already overused and so we will avoid it; the term finitely constrained group was used for the first time in [GNŠ06]).
Example 2. Consider again the patterns of size 2 given in Figure 1 and interpret as the trivial permutation of X = {0, 1} and as the non-trivial permutation (01). The group Aut(X 2 ) of tree automorphisms of the X-tree of depth 2 is isomorphic to the dihedral group D 4 and is generated by the automorphism t of order 4 and the automorphism a of order 2 (subject to the relation ata = t 3 ).
Note that, for self-similar groups of binary tree automorphisms, being infinite and being transitive on each level of the tree are equivalent properties (see [BGK + 07] or [BGK + 08, Lemma 3, page 112]).
The only proper transitive subgroups of Aut(X 2 ) are the groups {1, t, t 2 , t 3 }, with complement is R, and the group {1, a, t 2 , at 2 }, with complement is B. Thus, G(R) and G(B) are the only infinite, finitely constrained groups of binary tree automorphisms defined by forbidden patterns of size 2 (in addition to the full group Aut(X * ), which is defined by declaring the empty set to be the set of forbidden patterns). The group G(B) appears explicitly in [Gri05, page 174] as one of the simplest nontrivial examples of finitely constrained groups (nontrivial in the sense that the group is neither finite nor Aut(X * )).
2.1. Pattern closure construction. Given a self-similar group K and we may construct the finitely constrained group G(F s (K)) defined by the set of forbidden patterns F s (K) of size s, which is simply the set of patterns of size s that do not appear in any element of K. Theorem 1 describes the finitely constrained groups that can be obtained by this pattern closure construction.
Theorem 1 may be seen as a refinement of the direction (ii) implies (i) of Theorem 5 below, since its proof provides an explicit way to construct the group H (and since H is countable). In addition, Theorem 1 shows that the contraction property is, in a sense, compatible with the pattern closure construction.
Theorem 5. Let G be a group of tree automorphisms of X * and s ≥ 0. The following are equivalent.
(i) The group G is the closure of some self-similar, regular branch group H, branching over its level s stabilizer H s .
(ii) The group G is finitely constrained group defined by patterns of size s + 1.
The direction (ii) implies (i) is proved in [Gri05, Proposition 7.5] and the other direction in [Šun07, Theorem 3]. Recall that a group H is regular branch group over its level s stabilizer H s if and only if for all h 0 , . . . , h k−1 ∈ H s the tree automorphism (h 0 , h 1 , . . . , h k−1 ) is also an element of H s .
The following example of the pattern closure construction plays a role in the proof of Theorem 2.
Example 3. The group G(R) from Example 1 is just one example in the family of finitely constrained groups defined by the pattern closure construction with respect to various sizes applied to the, so called, odometer group (G(R)) corresponds to size 2).
The k-ary odometer automorphism t of X * is defined by where ρ = (0 1 . . . k − 1) is the standard cycle on the alphabet X = {0, . . . , k − 1}. The group T = t is self-similar, contracting, level transitive group. For a fixed size s + 1, s ≥ 0, define G(k, s + 1) = G(F s+1 (T )) as the finitely constrained group of k-ary rooted tree automorphisms for which the forbidden patters are precisely the patterns of size s + 1 that do not appear in any element of t = T ∼ = Z.

Proofs of Theorem 1 and Theorem 2
For a word u over X and a tree automorphism f , denote by δ u (f ) the unique tree automorphism that stabilizes level |u| and has trivial section at each vertex at level |u| except at u where its section is equal to f . Lemma 1. Let h and g be automorphisms of the tree X * . For any vertex u, Proof. Let |u| = n and v be arbitrary vertex at level n. Since δ u (h) stabilizes level n of X * , we have g −1 δ u (h)g(v) = g −1 g(v) = v. Thus (δ u (h)) g stabilizes level n. Further, showing that (δ u (h)) g = δ v (h g|v ), where v = g −1 (u).
Proof of Theorem 1. Let P be the set of patterns of size s + 1 appearing in the elements of the the self-similar group K. Because K is self-similar, this is the set of patterns of size s + 1 appearing at the root in the elements of K. Let S = {g 1 , . . . , g m } be a set of elements in K such that every pattern in P appears at the root in at least one of the automorphisms in S (note that the set P is finite, so S may be chosen to be finite as well). Let L = S be the smallest self-similar group containing S (this is a subgroup of K) and let L s = S ′ be the stabilizer of level s in L. Note thatS is countable (since S is finite and every tree automorphism has no more than countably many sections). Therefore L is countable and so are L s and S ′ .
Note that H is self-similar. Indeed, H = D ∪ L and all sections of the elements in D ∪ L are trivial or elements in the self-similar group L. Therefore D ∪ L is a self-similar set and H itself is self-similar. We claim that H s = D .
Since every element h ∈ S ′ stabilizes s levels of the tree X * , δ u (h) stabilizes s + |u| levels. Therefore D is a subgroup of H s .
Further, by Lemma 1, for any word u, g ∈S and The last equality shows that the group D is normal subgroup of H.
Since D is normal in H = D ∪S , any element of H s can be written as a product of an element in D and an element in S = L stabilizing s. But the generators of L s are in D , which shows that H s = D .
The group H is a regular branch group, branching over its stabilizer H s of level s. This is clear since, for any words u 0 , . . . , u k−1 in X * , and elements h 0 , . . . , h k−1 in S ′ , (δ u0 (h 0 ), . . . , δ u k−1 (h k−1 )) = (δ u0 (h 0 ), 1, . . . , 1) · · · (1, . . . , δ u k−1 (h k−1 )) = By Theorem 5, the closure H is a finitely constrained group, defined by patterns of size s + 1. Moreover, since H is self-similar, the patterns defining H are the patterns of size s + 1 appearing at the root in the elements of H. Since, for nonempty words u and h ∈ S ′ , δ u (h) stabilizes level s + 1, the patterns of size s + 1 appearing at the root in the elements of H are precisely the patterns of size s + 1 appearing at the root of the elements in L, and these are the patterns defining G. Therefore H = G.
Assume now, in addition, that K is contracting over the finite set N . Redefine S in the above construction so as to include N . Then S is finite and, because of the contraction property, so isS. Thus L is in this case an automaton group that is contracting over N . The group H is also contracting over N since, for h ∈ S ′ each section of δ u (h) at level |u| is an element of L and L is contracting over N .
Note that in the contracting caseS is finite and since L = S is finitely generated so is its finite index subgroup L s . This means that S ′ may be chosen to be finite as well. Further, in some situations the set D = {δ u (h) | h ∈ S ′ , u ∈ X * } in the definition of H may be replaced by some subset such as, for instance, The claim of Theorem 2 follows if we prove than none of the groups Aut(X * ), G(B), and G(R) is topologically finitely generated. This is known for Aut(X * ) (see [Gri05]), the claim for G(R) is proved in more general form in Proposition 2, and the claim for G(B) is proved in Proposition 3.
Proposition 2. The finitely constrained group G(k, s + 1) (defined in Example 3 by allowing the patterns of size s + 1 that appear in the odometer group) is not topologically finitely generated, for k ≥ 2, s ≥ 0.
Proof. We first explicitly determine a self-similar, countable, regular branch group H, branching over its level stabilizer H s , such that G is the closure of H in Aut(X * ). In order to accomplish this we follow the argument in the proof of Theorem 1 (we follow the argument somewhat loosely, since in the concrete situation some of simplifications, as indicated in the remarks after the proof of Theorem 1, apply).
The role of L may be played by T itself. The stabilizer T s of level s in T is generated by t k s . Define t s = t k s and, for n ≥ s, t n+1 = (1, 1, . . . , t n ).
Let H = t, t s+1 , . . . . Then, for the level s stabilizer in H, we have H s = t t i n | n ≥ s, i = 0, . . . , k n−s − 1 The closure H of H in Aut(X * ) is precisely G. This implies that G/G n = H/H n , for n ≥ 0.
Therefore, in order to show that G is not topologically finitely generated, it is sufficient to show that, for n ≥ s + 1, the minimal number of generators of H [n] = H/H n is n − s.
For n ≥ s + 1, let A n = C k s+1 × C k × · · · × C k , where C m denotes the standard cyclic group of order m (the elements are the residue classes modulo m) and the total number of factors is n − s. We claim that, for n ≥ s + 1, there exists a surjective homomorphism from H [n] to A n .
First, since the generators t n , t n+1 , . . . stabilize level n, the group H [n] is generated by (the cosets of) {t, t s+1 , . . . , t n−1 }. Define a map β n from the set of group words over {t, t s+1 , . . . , t n−1 } to A n by setting β n (W ) = (expn t (W ), expn ts+1 (W ), . . . , expn tn−1 (W )), where expn t * (W ) denotes the total expnonent of the letter t * in W . We claim that the map β n represents a surjective homomorphism from H [n] to A n . The surjectivity and the homomorphism property follow trivially, once we show that β n is well defined (as a map from H [n] ). Therefore, we need to show that, for every group word W over {t, t s+1 , . . . , t n−1 } representing the identity in H [n] (i.e. every group word W representing an element in the stabilizer H n ), β n (W ) = (0, 0, . . . , 0). We do this by induction on n.
For n = s + 1, H [s+1] = t and, since the smallest power of t stabilizing level s + 1 is t k s+1 , any group word over {t} representing the identity in H [s+1] is a power of t k s+1 .
Since, for n ≥ s + 1, the abelian group A n has rank n − s and β n : H [n] → A n is a surjective homomorphism, we conclude that the closure H = G is not topologically finitely generated.
Proposition 3. The finitely constrained group G(B) (defined in Example 1) is not topologically finitely generated.
Proof. The proof follows the general outline of the Proof of Proposition 2.
The role of a self-similar, countable, regular branch group H, branching over its first level stabilizer H 1 , such that G = G(B) is the closure of H in Aut(X * ) is played by H = a, a 1 , a 2 , a 3 , dots (and the role of L by a, a 1 ), where a = (01)(1, 1), a 1 = (a, a) and for n ≥ 2, a n+1 = (1, a n ). Every generator of H has order 2.
Indeed, to show that β n , for n ≥ 1, is a surjective homomorphism it suffices to show that it is well defined, i.e., it suffices to show that for a group word W over {a, a 1 , . . . , a n−1 } representing an element in H n the exponent expn a (W ) and the exponents expn ai (W ), i = 1, . . . , n − 1, are even. This can be accomplished by induction on n. The claim is clear for n = 1, since a is the only generator that does not stabilize level 1. In fact, exp a (W ) must be even for any group word over {a, a 1 , a 2 , . . . } stabilizing at least one level of the tree. Assume that n ≥ 2 and the claim is correct for n − 1. For any group word W over a, a 1 , . . . , a n−1 representing an element in H n , let the words W 0 and W 1 be the group words over {a, a 1 , . . . , a n−1 } obtained by decomposition. These words represent elements in H n−1 and the induction hypothesis applies. Since expn a (W 0 ) = expn a1 (W ), expn a1 (W 0 ) + expn a1 (W 1 ) = expn a2 (W ), . . .
(2) expn an−2 (W 0 ) + expn an−2 (W 1 ) = expn an−1 (W ). and all exponents on the left are even, all exponent on the right are even as well, completing the proof.

A remark on level transitivity
We observed in Example 1 that only the transitive subgroups od Aut(X 2 ) may lead to infinite (and level transitive) finitely constrained subgroups of Aut(X * ). Here we provide an example that shows that not all transitive subgroups of Aut(X s ) yield spherically transitive (or even nontrivial) finitely constrained groups.
Example 4. Let X = {0, 1}. Forbid all patterns of size 3 that contain a pattern from R as the sub-pattern at the root and all patterns that contain a pattern from B in any of the two bottom sub-patterns of size 2. Let F be the set of forbidden patterns of size 3 we just defined. There are 16 allowed patterns (the sub-pattern of size 2 at the top comes from t and the two sub-patterns in the bottom come from a, t 2 ). The group of allowed patterns (isomorphic to C 4 ⋉ (C 2 × C 2 )) acts transitively on X 3 .
A binary tree automorphism g belongs to the finitely constrained group G(F ) if and only if, for every word u over X and every letter x in X, g (u0) + g (u1) + g (u) = 0, and g (ux0) = g (ux1) , where the equalities are considered modulo 2, the trivial permutation on X is regarded as 0 and the non-trivial as 1. It is easy to check that only the trivial automorphism satisfies the above requirements, i.e., G(F ) = 1. Indeed, assume that, for some u ∈ X * , g (u) = 1. Then exactly one of g (u0) or g (u1) must be equal to 1. Without loss of generality, assume g (u0 ) = 1. The conditions g (u0 ) = 1, g (u00) + g (u01) + g (u0) = 0 and g (u00) = g (u01) contradict each other.