tree-wreathing applied to generation of groups by finite automata

December 23, 2005 14:23 WSPC/132-IJAC

00255

International Journal of Algebra and Computation Vol. 15, Nos. 5 & 6 (2005) 1205–1212 c World Scientific Publishing Company

Int. J. Algebra Comput. 2005.15:1205-1212. Downloaded from www.worldscientific.com by UNIVERSIDADE DE BRASILIA on 07/02/15. For personal use only.

TREE-WREATHING APPLIED TO GENERATION OF GROUPS BY FINITE AUTOMATA

SAID SIDKI Departamento de Matem´ atica, Universidade de Bras´ılia 70910-900 Bras´ılia-DF, Brazil [email protected] Received 19 June 2003 Communicated by the Guest Editors To Rostislav Grigorchuk on his 50th birthday We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n, Z), a canonical subgroup of finite index in GL(n, Z), then the restricted wreath product HwrB can be generated by finite synchronous automata on 0, 1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0’s is trivial. The uni-triangular group U = U(n, Z) is a subgroup of B(n, Z) and so, HwrU also can be generated by finite synchronous automata on 0, 1. Keywords: Wreath product; binary tree; synchronous automata; pro-2 group. Mathematics Subject Classification 1991: Primary 20E08, 20E22; Secondary 20F10, 22C05.

1. Introduction Which groups can be generated by finite synchronous automata is a challenging question in contemporary group theory. Synchronous automata are finite-state automata with the same input and output finite alphabet. This question, even when the automata are restricted to a special type, has received formal attention only recently (see [4–6]). Synchronous automata on a finite alphabet of size n can be viewed as endomorphisms of a regular one-rooted n-ary tree. The problem then translates to investigating the structure of the group of finite-state automorphisms of such trees. The restricted wreath product W = HwrK of residually-finite groups is intimately related to the group of automorphisms of one-rooted n-ary trees. As W is required to be residually finite, a well-known theorem of Gruenberg tells us that 1205



1206

00255

S. Sidki

H must be abelian or K finite (see [1]). The present paper is a sequel to [3] which addressed the problem of identifying the pairs of groups H, K whose restricted wreath product W = HwrK admits a faithful representation as a finite-state group of automorphisms of the binary tree. Denote by A the group of automorphisms of the binary tree T and F the subgroup of A of finite-state automorphisms. Let H, K be subgroups of F . When K is finite, W is easily realized as a subgroup of F ; this is indeed visible from the geometry of the tree. Thus, when H is abelian, our interest is in finding conditions on K such that W = HwrK is realizable as a subgroup of F . We introduced in [3] a construction called tree-wreathing which produced in particular a finite state group W = HwrK for H abelian and K free abelian of finite rank. The construction seems to depend on stabK (0∞ ) — the stabilizer in K of the 0∞ ray — being trivial. Thus, we extend further this construction from this point of view. We prove in Sec. 1: Theorem 1. Let H, K be finite state groups of automorphisms of the binary tree ˜ K˙ of and suppose that stabK (0∞ ) = 1. T hen, there exist f inite-state copies H, ∞ ˜ ˙ H, K, respectively, with stabK˙ (0 ) = 1 such that G = H, K contains a normal subgroup ν(H ) of G which is a direct sum of countable copies of the derived subgroup H and furthermore, the quotient group G/ν(H ) is isomorphic to the restricted wreath product HwrK. We call G the tree-wreath product of H by K and denote it by G = HwtK. In order to produce finite-state groups K such that stabK (0∞ ) we resort to B(n, Z), the subgroup of finite index in GL(n, Z) formed by matrices B = (Bij ) where Bij is even for all pairs (i, j) with j > i. A faithful representation of B(n, Z) into F , was given in [2]. We prove in Sec. 2: Theorem 2. Let n ≥ 2. Then there exists an isomorphic copy K˙ of K = B(n, Z) in F such that stabK˙ (0∞ ) = 1. Two noteworthy subgroups of B(n, Z) are the group of lower-triangular matrices and the free group of rank 2. In particular, if H is a countable abelian group which is residually a finite 2-group and K is one of these two groups then HwrK admits a faithful representation as a subgroup of F . 1.1. Two copies of the group A ˜ A˙ of the The purpose of this section is to construct two noncommuting copies A, ˜ automorphism group A of the binary tree such that A commutes with as many as ˙ possible of its A-conjugates. The binary tree T is indexed by finite sequences on 0, 1 (in other words, elements from the free monoid M = 0, 1∗ ) with the order relation v ≤ u if and only if u is a prefix of v. Note that for any u ∈ M , the set uM indexes a subtree of T isomorphic to T ; the isomorphism is uv → v, the cancellation of the prefix u. Let σ be the transposition which interchanges 0, 1 and let σ also denote its extension to the automorphism of T defined by 0u ↔ 1u. Then on using the


00255


Tree-Wreathing Applied to Generation of Groups by Finite Automata

1207

identification of the subtrees 0M and 1M with M , every element of A can be written as a = (a0 , a1 )σ iφ (a) where a0 , a1 ∈ A and iφ (a) = 0, 1. If iφ (a) = 1, we say that a is active; otherwise, it is inactive. The form of a establishes the factorization of A as (A × A)σ. We note that the group A has pro-2 topology inherited from the structure of the tree as the inverse limit of its truncations at its different levels. Given γ ∈ A, we define γ (0) = γ and inductively, γ (i+1) = (γ (i) , γ (i) ) ∈ A for all i ≥ 0. Also, given a subgroup R of A and u ∈ M , we define u ∗ R to be the subgroup of A which acts on the subtree uM as R and fixes all the vertices outside this subtree. Furthermore, we define ν(R) to be the group generated by u ∗ R for all u ∈ U = ({02 , 10}∗ ){01, 11}; thus, ν(R) = (ν(R) × R) × (ν(R) × R). We note that elements of ν(R) have finite support with respect to the index set U . The automorphism τ = (1, τ )σ of T is called the binary adding machine. It is transitive on all levels of the tree. The topological closure of τ in A is ¯ τ = {τ ξ | ∞ ξ ∈ Z2 } where Z2 is the ring of 2-adic integers. The stabilizer of 0 in A is A0∞ and τ ∩ A0∞ = 1 is formed by a ∈ A where a0i is inactive for all i ≥ 0. We note that ¯ ∞ τ . and A = A0 ¯ The first copy of A is simply constructed as follows: given a ∈ A, let a ˜ ∈ A be defined recursively as a ˜ = ((˜ a, a), (1, 1)). Let A˜ = {˜ a | a ∈ A}. Then, A˜ is clearly isomorphic to A. Now, let T be the subtree of T formed by vertices indexed by elements from the submonoid 00, 10∗ and let A(T ) be the subgroup of A formed by those automorphisms which leave T invariant. Then A(T ) = (A(T ) × A × A(T ) × A)σ. Define the following subgroup of A(T ), A˙ = (A˙ × {1} × A˙ × {1})σ. Furthermore, define the map λ : A → A˙ recursively by λ : a = (a0 , a1 )σ iφ (a) → aλ = (((a0 )λ , 1), ((a1 )λ , 1)) σ iφ (a) . It is straightforward to prove that λ is an isomorphism from A onto A˙ which preserves the finite-state property. It also maps A0∞ onto A˙ 0∞ and τ to α = τ λ = τ onto A˙ = (A˙ 0∞ )¯ α. ((1, 1), (α, 1))σ; thus, it maps the decomposition A = (A0∞ )¯ ˙ It follows that any d ∈ A can be expressed uniquely as d = κ(d)αξ(d) where κ(d) ∈ A˙ 0∞ , ξ(d) ∈ Z2 . Elements d of A˙ 0∞ have the form d = ((d00 , 1), (d10 , 1)), d00 ∈ ˙ A˙ 0∞ , d10 ∈ A. Given the following elements of A˙ d = ((d00 , 1), (d10 , 1))σ i , f = ((f00 , 1), (f10 , 1))σ j , we note that A˙ 0∞ d = A˙ 0∞ f holds if and only if i = j and A˙ 0∞ d00 = A˙ 0∞ f00 .


1208

00255

S. Sidki

1.2. The tree-wreath product ˜ and K. ˙ Given H, K subgroups of A we define G to be the group generated by H We call this group the tree-wreath product of H by K and denote it by HwtK. Let ˜ in G. B(H, K) be the normal closure of H We reprove the following fact from [3].


Proposition 1. Let Υ = ¯ τ , H ≤ A, and let G = HwtΥ. Then, G/ν(H ) ∼ = Hwr Υ. ˙ = ¯ α where α = (1, (α, 1))σ. Then, Proof. Let ξ = ε + 2ξ , ε = 0, 1. Recall Υ αξ = (αξ , 1)(1) , ξ

if ε = 0, 1+ξ

= ((α , 1), (α

, 1))σ,

if ε = 1.

˜ = ((h, ˜ h), 1), ˜b = ((˜b, b), 1) ∈ H, ˜ we have Furthermore, for γ = αξ and h αξ ˜γ = h ˜ αξ = h ˜ , h , 1 if ε0 = 0, h αξ ˜ ,h if ε0 = 1, = 1, h and ˜ ˜bγ ] = 1, if ε0 = 1, [h, αξ ˜ ˜b = h, , [h, b] , 1 , if ε0 = 0. ˜ ˜bγ ], seen as a partial map from the monoid M into H , has a finite Then, [h, number of nontrivial entries in places with indices from U = ({02 , 10}∗ ){01, 11} and ˜ H ˜ γ ] | γ ∈ K that all of these entries are equal to [h, b]. The normal closure of [H, in HwtK is ν(H ). For a dyadic integer ξ, let l(ξ) be its 2-valuation and |ξ| = 2−l(ξ) . The coset representatives of ν(H ) in G can be chosen as expressions of form γs γs+1 γs+2 γs+t 1 γ1 h 2 γ2 · · · w = (h hs )(b1 · · · bt )αξ , b2

where γi = αξi and ξi are distinct 2-adic integers for 1 ≤ i ≤ s + t and l(ξi ) ≥ 1 for 1 ≤ i ≤ s and l(ξi ) = 0 for s + 1 ≤ i ≤ t. Let us call such a form for w having minimal (|ξ| + i≥0 |ξi | , s + t) as semi-normal. Suppose w ∈ ν(H ) is as above, in semi-normal form, and that w = 1. Since ν(H ) stabilizes the first level of the tree, we have l(ξ) > 0. Therefore, w = (u, v) and by noting that the respective coordinates of u, v are also in ν(H ), we conclude from the minimality of w that ˜ = ((h, ˜ h), 1) and h ∈ H . We reach a contradiction by appealing to the fact w=h ˜ have that elements of ν(H ) have finite support whereas nontrivial elements of H infinite support. Hence G/ν(H ) is isomorphic to HwrΥ. Let a ˜ = ((˜ a, a), (1, 1)), ˜b = ((˜b, b), (1, 1)) ∈ A˜ and d = ((d00 , 1), (d10 , 1)) ˙ ˜d = σ ∈ A. Write d = κ(d)αξ(d) where κ(d) ∈ A˙ 0∞ and ξ(d) ∈ Z2 . Since a ξ(d) d00 σi d α ˜ Therefore, a ˜ =a ˜ . ((˜ a , a), (1, 1)) , we have A˙ 0∞ = CA (A). i


00255


1209

Proposition 2. Let H, K be subgroups of A and C = ν(H ) ∩ B(H, K). Then, ˙ B(H, K)/C is isomorphic to a direct sum of copies of H which are permuted by K. ∼ Furthermore, if K ∩ A0∞ = 1 then HwtK/C = HwrK. ˙ Then, Proof. Let I = {ξ(d) | d ∈ K}.


˜ d | d ∈ K ˙ = H ˜ αξ(d) |d ∈ K ˙ ≤ HwtΥ, B(H, K) = H where Υ = ¯ τ . It follows from the above proposition that B(H, K)/C is isomorphic to a direct sum of ˜ αξ(d) | ξ(d) ∈ I}. Also, HwtK/C is a semidirect product of B(H, K)/C by K˙ {H and K˙ simply permutes by conjugation the set ξ(d)

˜α {H

|ξ(d) ∈ I}

˜ αξ(d) )f˙ = ˜hαξ(d) f˙ = h ˜ αρ according to the following rule: if h ∈ H, f ∈ K, then (h where αξ(d) f˙ = gα ˙ ρ , g ∈ A0∞ , ρ ∈ Z2 . ˙ we have d = f if and only if Suppose K ∩ A0∞ = 1. Then, for d, f ∈ K, ξ(d) = ξ(f ). Hence, HwtK/C ∼ = HwrK. 2. The Stabilizer of p = 0∞ in B(n, Z) Consider B(n, Z), the subgroup of GL(n, Z) formed by matrices B = (Bij ) where Bij is even for all pairs (i, j) with j > i. Write Bij = 2bij if j > i and Bij = bij , otherwise. We recall the representation of the affine group Zn B(n, Z) in [2]. The following elements vi of A generate a free abelian group V of rank n: v1 = (1, v2 )σ, v2 = (v3 , v3 ), . . . , vn = (v1 , v1 ). We identify this set of elements monomial n × n matrix  0 1 0 0   . .  . .  0 0 1 0 2

with the canonical basis of Zn . Let A be the 0 1 . . 0 0

. . . . . .

. . . . . .

. . . . . .

 0 0   . . .  1 0

There is a faithful representation φ of B(n, Z) into the normalizer of V in A −1 −1 defined recursively by B φ = ((A(I − B)v1 ) · (B A )φ , (B A )φ ); the 1st coordinate −1 φ (A(I − B)v1 ) · (B A )φ is an element of the semidirect product V B(n, Z) . We identify the group with its image by dropping the φ symbol from the notation; −1 −1 thus, B = (A(I − B)v1 · B A , B A ).


1210

00255

S. Sidki

Note that




−b21 1 − b22  −b −b32 31   . .   A(I − B) =  . .   −b(n−1)1 −b(n−1)2   −bn1 −bn2 1 −b12 2 (1 − b11 )

−2b23 1 − b33 . . −b(n−1)3 −bn3 −b13

Define in V the vectors wk = A(I − B)A

−k+1

. . . . . . .

. . . . . . .

 . −2b2n . −2b3n    . .   . . .  . −2b(n−1)n   . 1 − bnn  . −b1n

v1 , k ≥ 1. Then, for 1 ≤ k ≤ n,

1 wk = −b(k+1)k v1 − b(k+2)k v2 − · · · − (bkk − 1)vn . 2 1 n Since A is the scalar matrix 2 In , we have wn+k = wk for all k ≥ 1. If w = c1 v1 + c2 v2 + · · · + cn vn is an integral vector and c1 is even, then c1 v1 = 12 c1 v12 = 12 c1 (v2 )(1) . As vk = (vk−1 )(1) for 2 ≤ k ≤ n − 1, vn = (v1 )(1) , we (1) obtain w = 1 c v + v3 + · · · + v1 = (Xw)(1) where X is the monomial matrix  0 0 0 . . . 1 2 1 2 1

0 1 . . 0 0 0 0

 02 .  .

0 0 1 . 0 0

. . . . . .

. . . . . .

. . . . . 1

0 0 . . 0 0

  . 

Lemma 1. Let B, wj (j = 1, . . . , n) and X be as above. Then B fixes 0∞ if and only if f (k) = X k−1 w1 + X k−2 w2 + · · ·+ wk+1 are integral vectors for 1 ≤ k ≤ n− 1 and f (n − 1) = X n−1 w1 + X n−2 w2 + · · · + wn = 0. −1

Proof. The automorphism B fixes 0∞ if and only if B0 = (A(I − B)v1 ) · B A −1 fixes 0∞ ; in particular, B0 must be inactive. As B A is already inactive, the vector w1 = A(I − B)v1 also must be inactive. Since 1 w1 = A(I − B)v1 = −b21 v1 − b31 v2 + · · · − (b11 − 1)vn , 2 the condition that w1 is inactive is equivalent to the requirement that b21 is an even integer. Thus, w1 = (Xw1 )(1) . −1 The second level development of B = (w1 , 1)(B A )(1) is B = (w1 , 1)(w2 , 1)(1) (A2 BA−2 )(2) = (w1 + (w2 , 1), (w2 , 1))(A2 BA−2 )(2) = ((Xw1 + w2 , Xw1 ), (w2 , 1))(A2 BA−2 )(2) . Our next condition is that the vector 1 1 (1 − b11 ) − b32 v1 − b21 + b42 v2 Xw1 + w2 = 2 2 1 − · · · − bn1 − (1 − b22 ) vn 2


00255


1211

is inactive; that is, 12 (1 − b11 ) − b32 is an even integer. Therefore,


Xw1 + w2 = (X(Xw1 + w2 ))(1) = (X 2 w1 + Xw2 )(1) . This analysis leads to the conclusion that B fixes 0∞ if and only if f (k) = X k w1 + X k−1 w2 + · · · + wk+1 is integral for all k ≥ 1. We note that f (n) = Xf (n − 1) + w1, f (n + 1) = X 2 f (n − 1) + (Xw1 + w2 ), . . . , and f (2n − 1) = X n f (n − 1) + f (n − 1) = 32 f (n − 1), since X n = 12 In . More s+1 generally, f (2sn − 1) = 2 2s−1 f (n − 1) for all s ≥ 1. Since f (2sn − 1) is an integral vector for all s, we conclude that f (n − 1) = 0. Now, let ci−2 be the (i − 2)th entry of the vector f (n − 1) = X n−1 w1 + X n−2 w2 + · · · + wn . When this entry is written in terms of the canonical basis, we find that it is a linear combination of B(i+2k)(1+k) for k = 1 − i and of 12 (1 − B(2−i)(2−i) ) where all the coefficients are 12 -integers different from 0. The condition f (n − 1) = 0 translates into n linear equations ci = 0 for i = 1, . . . , n. We define an equivalence relation #n on {(i, j)|1 ≤ i, j ≤ n} by (i, j)#n (l, m) if and only if there exists r such that i + 2r ≡ l, j + r ≡ m modulo n. Proposition 3. Given q ≥ 2 there exists a prime p > q and K a subgroup of B(p, Z), isomorphic to B(q, Z) such that stabK (0∞ ) is trivial. Proof. Let ti = 3i−1 for i = 1, . . . , q. Then, it can be checked that (ti , tj ) = (tl , tm ) implies ti − 2tj = tl − 2tm . Choose a prime p > 3q and let r = p − tq . Define the subgroup K of B(p, Z) of sparse matrices B such that Bij = 0 when i = j and i or j = 3s with s < q, and Bii = 1 for i = 3s with s < q. Thus, B has the form   0 2b13 0 · · · 0 2b19 · · · 2b1tn 01×r b11  0 1 0 0···0 0 ··· 0 01×r    b 0 b33 0 · · · 0 2b39 · · · 2b3tn 01×r   31    . . ··· . ··· . .   .  .  b91 0 b93 0 · · · 0 b99 · · · 2b9tn 01×r     . . . 0···0 . ··· . .     btn 1 0 btn 3 ··· btn 9 · · · btn tn 01×r  0r×1 0r×1 0r×1 ··· 0r×1 · · · 0r×1 Ir×r Then, (ti , tj )#p (tl , tm ) implies (i, j) = (l, m). Also, each Bti ,tj occurs in a unique coefficient ck of f (n − 1). Hence, for this choice of the sequence ts and prime p, the condition f (n − 1) = 0 leads to B = Ip . Hence, K0∞ = 1. 3. Concluding Remarks The tree-wreathing construction was used in conjunction with the representation in F of B(n, Z) to obtain branch subgroups of F , containing free subgroups of rank 2 (see, [7]).


1212

00255

S. Sidki


Given a countable residually finite 2-group K and a chain of normal subgroups of K, having successive quotients of order 2, the representation of K on the coset tree associated to this chain produces a copy of K in A, which acts fixed point freely on the boundary. Therefore, in particular, the condition stabK (0∞ ) = 1 is satisfied. However, it is not known for what groups the representation on one of its coset trees produces a finite-state group. Let C be the infinite cyclic group and G = Cwr(CwrC). The question of whether G admits a faithful finite-state representation is open (see [8], Problem 15.19). Acknowledgment The author extends his thanks to Professor Pierre de la Harpe for his warm hospitality at the University of Geneva during May 2003 and acknowledges support from the Swiss National Science Foundation and the Brazilian Conselho Nacional de Pesquisa. References [1] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957) 29–62. [2] V. V. Nekrachevych and S. Sidki, Automorphisms of the binary tree: State-closed subgroups and dynamics of 1/2 endomorphisms, in Groups — Topological, Combinatorial and Arithmetic Aspects, ed. T. M¨ uller (Cambridge University Press, 2004). [3] A. M. Brunner and S. Sidki, Wreath operations in the group of automorphisms of the binary tree, J. Algebra 257 (2002) 51–64. [4] R. I. Grigorchuk, V. I. Nekrachevych and V. I. Suschanskii, Automata, dynamical systems and groups, Proc. Steklov Inst. 231 (2000) 128–203. [5] S. Sidki, Automorphisms of one-rooted trees: growth, circuit structure and acyclicity, J. Math. Sci. 100 (2000) 1925–1943. [6] S. Sidki, Finite automata of polynomial growth do not generate a free group, Geom. Dedicata. 108 (2004) 193–204. [7] S. Sidki and J. S. Wilson, Free subgroups of branch groups, Arch. Math. 80 (2003) 458–463. [8] Unsolved Problems in Group Theory: The Kourovka Notebook, 15th Edn. (Novosibirsk, 2002).