Cayley automatic representations of wreath products
arXiv:1511.01630v1 [math.GR] 5 Nov 2015
Cayley automatic representations of wreath products D. Berdinsky
∗
B. Khoussainov
†
Abstract. We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed. Keywords: Finite automata, pushdown automata, nested stack automata, Cayley graphs, wreath products.
1
Introduction
In this paper we study representations of Cayley graphs of wreath products of groups using finite automata, pushdown automata and nested stack automata. The representations to be used are related to the notion of Cayley automatic groups that was introduced in [16]. The notion of Cayley automatic groups appeared to be a natural generalization of automatic groups in the sense of Thurston [9]. The set of Cayley automatic groups properly contains the set of automatic groups. In particular, the set of Cayley automatic groups includes finitely generated nilpotent groups of class at most two, lamplighter group [16] and Baumslag– Solitar groups [3]. The Cayley automatic groups retain some nice properties of automatic groups. They are closed under direct product, free product and finite extensions. The word problem in Cayley automatic groups is decidable in quadratic time. We assume that the reader is familiar with the notions of finite automaton and regular language. Let Σ be a finite alphabet. Put Σ = Σ ∪ {}, where ∈ / Σ. The convolution of two words w1 , w2 ∈ Σ∗ is the string w1 ⊗ w2 of length max{|w1 |, |w2 |} over the alphabet Σ × Σ defined as follows. The kth symbol of the string is (σ1 , σ2 ), where σi , i = 1, 2 is the kth symbol of wi if k 6 |wi | and otherwise. The convolution ⊗R of a binary relation R ⊂ Σ∗ × Σ∗ is defined as follows: ⊗R = {w1 ⊗ w2 |(w1 , w2 ) ∈ R}. We say that a binary ∗ The † The
University of Auckland, Department of Computer Science; [email protected] University of Auckland, Department of Computer Science; [email protected]
1
relation R ⊂ Σ∗ × Σ∗ is automatic if there exists a finite automaton in the alphabet Σ × Σ that accepts the convolution ⊗R. Such an automaton is called two–tape synchronous finite automaton. Let G be a group generated by a finite subset S ⊂ G. Consider the labeled and directed Cayley graph Γ(G, S). We say that G is Cayley automatic if there is a regular language L ⊂ Σ∗ for some finite alphabet Σ that uniquely represents elements of G, and for every s ∈ S the binary relation which is the set of directed edges of Γ(G, S) labeled by s is automatic. That is to say, the group G is Cayley automatic if the labeled digraph Γ(G, S) is an automatic structure [17]. In [7] the authors considered the extensions of the notion of Cayley automatic groups replacing the regular languages by more powerful languages. We denote by C a class of languages; for example, it can be the class of regular languages, context–free languages or context–sensitive languages. The notion of Cayley automatic groups can be extended as follows. Definition 1. Let G be a finitely generated group. We say that G is C Cayley automatic if there exists a finite set S ⊂ G generating G for which the following properties hold: • There exist a bijection ψ : L → G between a language L ⊂ Σ∗ which is in the class C and the group G; • For each h ∈ S the language Lh = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )h = ψ(w2 )} is in the class C. If C is the class of regular languages then C Cayley automatic groups are Cayley automatic groups. The notion of Cayley automatic groups is invariant under a choice of generators. Recall that the context–free and indexed languages are the ones that are recognizable by pushdown and nested stack automata respectively. Notice that for context–free and indexed Cayley automatic groups Definition 1 depends on the choice of generators. Notice that for a group G, a finite subset S ⊂ G and a finite alphabet Σ in Definition 1 the triple (G, S, Σ) is C–graph automatic according to the terminology introduced in [7]. Some results on C Cayley automatic representations of groups are as follows. In [16] the authors constructed the Cayley automatic representations for nilpotent groups of class at most two. In [3] we constructed the Cayley automatic representations for all Baumslag–Solitar groups. In [7] the authors constructed the blind deterministic 3–counter Cayley automatic representation for Baumslag–Solitar groups BS(m, n), n > m > 2. In [8] the authors constructed the deterministic non–blind 1–counter Cayley automatic representation for Thompson’s group F . We consider the cases when C is the class of regular languages, context– free languages and indexed languages. We assume that the reader is familiar with the notion of pushdown automata and context–free languages, and nested stack automata and indexed languages. For definitions of pushdown automata and context–free languages see, for example, [14]. For definitions of nested stack automata and indexed languages we refer the reader to [13]. For the introduction 2
to formal languages theory that covers regular languages, context–free languages and indexed languages we refer the reader to [12]. In this paper we construct: Cayley automatic representation for wreath products G o Z, context–free Cayley automatic representation for wreath products GoFn and indexed Cayley automatic representation for the wreath product Z2 o Z2 . In each case we specify the set of generators for wreath products with respect to which we consider their Cayley graphs. In addition, for the representations constructed we prove the inequalities of the form: λ|w| + µ 6 |g| 6 ξ|w| + δ, where |g| is the length of a group element g with respect to chosen set of generators and |w| is the length of the word w which is the representative of g, i.e., ψ(w) = g. In [2] the authors introduced the notion of parallel poly–pushdown groups. The definition of poly–pushdown groups uses two–tape asynchronous automata instead of synchronous ones used in Definition 1. It was shown that the set of poly–pushdown groups is closed under wreath product [2]. This implies that all wreath products to be considered in this paper are parallel poly–pushdown groups. For wreath products there is an abundance of results on quantitative characteristics such as growth rate [18], isoperimetric profiles [11] and drift of simple random walks [10, 6]. This makes studying representations of Cayley graphs of wreath products relevant to seeking connections between characteristics of groups and a computational power of automata that is sufficient to represent their Cayley graphs. In this paper we focus on the aforementioned hierarchy of languages, i.e., regular, context–free and indexed. The rest of this paper is organized as follows. In Section 2 we briefly recall the definitions and notations for wreath products of groups. In Section 3 we consider the Cayley automatic representation for wreath products G o Z. In Section 4 we consider the context–free Cayley automatic representation for wreath products G o Fn . In Section 5 we consider the indexed Cayley automatic representation for the wreath product Z2 o Z2 .
2
Wreath products of groups: definitions and notations
Recall the definition of the restricted wreath product of two groups A and B. For more details on wreath products see, for example, [15]. Given two groups A and B. We denote by A(B) the group of all functions B → A having finite supports and the usual multiplication rule. Recall that a function f : B → A has finite support if f (x) 6= e for only finite number of x ∈ B, where e is the identity of A. Let us define the homomorphism τ : B → Aut(A(B) ) as follows: for a given f ∈ A(B) , the automorphism τ (b) maps f to f b ∈ A(B) , where f b (x) = f (bx) for all x ∈ B. For given groups A and B, the restricted wreath 3
product A o B is defined to be the semidirect product A(B) oτ B. Thus, A o B is the set product B × A(B) with the group multiplication given by: 0
(b, f ) · (b0 , f 0 ) = (bb0 , f b f 0 ),
(1)
0
where f b (x) = f (b0 x). For our purposes we will use the converse order for representing elements of a wreath product. Namely, we will represent an element of A o B as a pair (f, b), where f ∈ A(B) and b ∈ B. For such a representation, the group multiplication is given by: −1 (2) (f, b) · (f 0 , b0 ) = (f f 0 b , bb0 ), −1
where f 0 b (x) = f 0 (b−1 x). There exist natural embeddings B → A o B and A(B) → A o B mapping b to (e, b) and f to (f, e) respectively, where e is the identity of A(B) and e is the identity of B. For the sake of simplicity, we will identify B and A(B) with the corresponding subgroups of A o B. Recall that by [1] we have that: the wreath product A o B of two finitely presented groups A and B is finitely presented iff either A is the trivial group or B is finite. Therefore, the wreath products to be considered: G o Z, G o F n and Z2 o Z2 are not finitely presented for a nontrivial group G. Thus, these groups are not automatic [9].
3
The wreath products of groups with the infinite cyclic group
Denote by t the generator of Z = hti, and by a the nontrivial element of Z2 . Let us consider the Cayley graph of the lamplighter group L2 = Z2 o Z with respect to the generators a and t. We have the following theorem. Theorem 1. The lamplighter group L2 is Cayley automatic. Proof. Recall that an element of L2 is a pair (f, z), where f is a function f : Z → Z2 that has finite support and z ∈ Z is the position of the lamplighter. The automatic representations for the Cayley graph of L2 with respect to the generators a and t were constructed in [16] and [3]. Here we simplify the representation used in [3]. In order to present f (i) ∈ Z2 we use the symbols 0 and 1: 0 means that a lamp in the position z = i is unlit, i.e., f (i) = e; 1 means that the lamp is lit, i.e., f (i) = a. By abuse of notation we will write f (i) = 0 instead of f (i) = e and f (i) = 1 instead of f (i) = a. Recall that only a finite number of lamps are lit for each element of the group L2 . To show the position of the origin z = 0 we use the symbols A0 and A1 if the lamp in the origin is unlit and lit respectively. To show the position of the lamplighter we use the symbols C0 and C1 if the lamp in the position of the lamplighter is unlit and lit respectively. In case the lamplighter is in the origin we use the symbols B0 and B1 . 4
Given an element (f, z) ∈ L2 . Let m be the smallest i ∈ Z such that f (i) = 1; if f (i) = 0 for all i ∈ Z, then put m = 0. Put ` = min{m, z, 0}. Let n be the largest j such that f (j) = 1; if f (j) = 0 for all j ∈ Z, then put n = 0. Put r = max{n, z, 0}. Let us represent (f, z) as follows: f (`)f (` + 1) . . . f (−1)Af (0) f (1) . . . f (z − 1)Cf (z) f (z + 1) . . . f (r − 1)f (r), (3) where Af (0) = A0 and Af (0) = A1 if f (0) = 0 and f (0) = 1, respectively; also, Cf (z) = C0 and Cf (z) = C1 if f (z) = 0 and f (z) = 1, respectively. In case z = 0 the word representing an element (f, z) is as follows: f (`)f (` + 1) . . . f (−1)Bf (0) f (1) . . . f (r − 1)f (r),
(4)
where Bf (0) = B0 and Bf (0) = B1 if f (0) = 0 and f (0) = 1, respectively. It can be seen that the language of the words representing all elements (f, z) of the group L2 is regular. Let an element g = (f, z) is represented by a word (3). Writing the words representing g and gt = (f, z + 1) one under another we have: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) f (z − 1)
Cf (z) f (z)
f (z + 1) Cf (z+1)
... ...
f (r) . f (r)
(5)
Let an element g = (f, 0) is represented by a word (4). Writing the words representing g and gt = (f, 1) one under another we have: f (`) . . . f (`) . . .
f (−1) f (−1)
f (1) Cf (1)
Bf (0) Af (0)
f (2) f (2)
... ...
f (r) . f (r)
(6)
The other cases are considered similarly. From (5) and (6) it can be seen that the relation hg, gti is recognized by a synchronous two–tape finite automaton. Let an element g = (f, z) is represented by a word (3). Writing the words representing g and ga one under another we have: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) Cf (z) f (z − 1) Cf (z)
f (z + 1) f (z + 1)
... ...
f (r) , f (r)
(7)
where f (z) = 1 and f (z) = 0 if f (z) = 0 and f (z) = 1, respectively. Let an element g = (f, 0) is represented by a word (4). Writing the words representing g and ga one under another we have: f (`) . . . f (`) . . .
f (−1) f (−1)
Bf (0) Bf (0)
f (1) . . . f (1) . . .
f (r) , f (r)
(8)
where f (0) = 1 and f (0) = 0 if f (0) = 0 and f (0) = 1, respectively. From (7) and (8) it can be seen that the relation hg, gai is recognized by a synchronous two–tape finite automaton. Remark 1. It can be verified that with respect to the representation of L2 in Theorem 1 the left multiplication by the generators a and t are recognized by synchronous two–tape finite automata. This implies that L2 is Cayley biautomatic. For the definition of Cayley biatomatic groups see [16]. 5
For the automatic representation described in Theorem 1 the length of the word representing an element (f, z) is as follows: |w| = |r − `| + 1 = | max{n, z, 0} − min{m, z, 0}| + 1 = max{|n − m|, |n|, |m|, |n − z|, |m − z|, |z|} + 1.
(9)
For an element (f, z) denote by #supp f the cardinality of the set supp f = {j | f (j) = a}. The word length of an element (f, z) in L2 with respect to the generators a and t can be obtained as follows. Proposition 1. The word length of an element g = (f, z) with respect to the generators a and t is as follows: |g| = #supp f + min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}.
(10)
Proof. By [5], the left–first and the right–first normal forms of an element (f, z) ∈ L2 are as follows: ai1 . . . aip a−j1 . . . a−jq tz , a−j1 . . . a−jq ai1 . . . aip tz , where ip = n (if n > 0), jq = −m (if m 6 −1), ip > · · · > i1 > 0, jq > · · · > j1 > 0 and ak = tk at−k . It is proved [5] (see Proposition 3.6) that the word length of g = (f, z) ∈ L2 with respect to the generators a and t is as follows: |g| = p + q + min{2jq + ip + |z − ip |, 2ip + jq + |z + jq |}. Let us express |g| in terms of m 6 n for three different cases: • m 6 −1 and n > 0: |g| = p + q + min{−2m + n + |z − n|, 2n − m + |z − m|}, • m > 0: |g| = p + q + n + |z − n|, • n 6 −1: |g| = p + q − m + |z − m|. It can be seen that #supp f = p + q. Thus, we obtain (10). Corollary 1. For a given g ∈ L2 let w be the representative of g for the Cayley automatic representation of L2 constructed in Theorem 1. Then the following inequalities hold: |w| − 1 6 |g| 6 3|w| − 2. (11) Proof. First we prove the inequality: |g| 6 3|w| − 2. By (9), |w| > n − m + 1. Therefore, |w| > #supp f . Consider each of the three cases: m 6 −1 < 0 6 n, n 6 −1 and 0 6 m separately: • The case m 6 −1 < 0 6 n. If z > n, then we have: −2m + n + |z − n| = −2m + z 6 2(z − m). If z 6 m, then we have: 2n − m + |z − m| = 2n−m+m−z 6 2(n−z). If m < z < n, then we have: −2m+n+|z −n| = 2(n − m) − z and 2n − m + |z − m| = 2(n − m) + z. Therefore, by (9): min{−2m+n+|z −n|, 2n−m+|z −m|} 6 2(|w|−1). Thus, |g| 6 3|w|−2; 6
• The case m > 0. By (9) we have: n + |z − n| 6 2(|w| − 1). Thus, |g| 6 3|w| − 2; • The case n 6 −1. By (9), we have: −m + |z − m| 6 2(|w| − 1). Thus, |g| 6 3|w| − 2. Let us prove the inequality: |w| − 1 6 |g|. The identity e ∈ L2 is represented by the word: B0 . Therefore, the inequality holds for g = e. Suppose that the inequality holds for some g ∈ L2 . It follows from the proof of Theorem 1 that the length of the word representing ga equals |w|, and the lengths of the words representing gt and gt−1 are equal to either |w|, |w| + 1 or |w| − 1. This implies that the inequality holds for the elements ga, gt and gt−1 . Thus, the inequality holds for all g ∈ L2 . Remark 2. It is easy to construct an infinite sequence of elements Z2 o Z such that the upper bound in (11) is exact. Let us consider the infinite sequence of elements whose representatives are B1 , 1B1 1, 11B1 11, . . . . The lengths of such elements in the group Z2 o Z with respect to the generators a, t are 1, 7, 13, . . . ; so the equality |g| = 3|w| − 2 holds for all these elements. Remark 3. It is easy to construct an infinite sequence of elements Z2 o Z such that the lower bound in (11) is exact. Let us consider the infinite sequence of elements whose representatives are B0 , A0 C0 , A0 0C0 , A0 00C0 , . . . . The lengths of such elements in the group Z2 o Z with respect to the generators a, t are 0, 1, 2, 3, . . . ; so the equality |w| − 1 = |g| holds for all these elements. Let us consider wreath products GoZ for Cayley automatic groups G. Recall that the elements of a group GoZ can be obtained as the pairs of the form (f, z), where z ∈ Z and f : Z → G is a function which has finite support; i.e., f is an element of the direct product G(Z) [15]. Remark 4. The proof of Theorem 1 can be straightforwardly modified for a finite group G. For a group G that has n + 1 elements g0 = e, g1 , . . . , gn , we represent these elements by n+1 symbols 0, 1, . . . , n. Using the construction from Theorem 1 one can obtain an automatic representation for the Cayley graph of G o Z with respect to the set of generators g1 , . . . , gn and t, where Z = hti. For an infinite Cayley automatic group G we have the following theorem. Theorem 2. For an infinite Cayley automatic group G the group GoZ is Cayley automatic. Proof. Let {g1 , . . . , gn } ⊂ G be a finite set generating G. Since G is a Cayley automatic group there exists a bijection between a regular language LG and the group G such that the directed edges of the (right) Cayley graph labeled by g1 , . . . , gn are recognizable by synchronous two–tape finite automata Mg1 , . . . , Mgn respectively. For simplicity we may always assume that LG ⊂ {0, 1}∗ [4]. We will suppose that ∈ / LG . Let us construct an automatic representation for the Cayley graph 7
of G o Z similarly to that of for L2 (see Theorem 1). Since G is infinite we need to add some new features. We introduce two counterparts of the symbols 0 and 1: bold 0 and bold 1 which specify the beginning of a word of LG . In order to specify the position of the origin z = 0 we use the symbols A0 and A1 depending on whether the word that represents the element of G at z = 0 has 0 and 1 as the first letter. Similarly, we use the symbols C0 and C1 to specify the position of the lamplighter z. The symbols B0 and B1 are used in case when the lamplighter is at the origin z = 0. Let us show two simple examples. Take an element (f, 1) ∈ G o Z such that f (j) = e for j ∈ / [−1, 2] and f (−1) 6= e, f (0), f (1) and f (2) 6= e are represented by the words 011, 1001, 01 and 111 respectively. Then for our representation the element (f, 1) is represented as follows: 011A1 001C0 1111. Take an element (f, 0) ∈ G o Z such that f (j) = e for j ∈ / [−1, 1] and f (−1) 6= e, f (0) and f (1) 6= e are represented by the words 111, 000 and 01 respectively. Then for our representation the element (f, 0) is represented as follows: 111B0 0001. By abuse of notation use f (j) to denote the following modification of the word representing the element f (j) ∈ G: the first letter of the corresponding word in LG should be changed to the bold one. We denote by Af (0) and Bf (0) the following modifications of the word representing the element f (0) ∈ G: the first letter of the corresponding word in LG should be changed to A0 or A1 , and B0 or B1 respectively. We denote by Cf (z) the following modification of the word representing the element f (z) ∈ G: the first letter of the corresponding word in LG should be changed to C0 or C1 . Given an element (f, z) ∈ G o Z. Similarly to Theorem 1 put m to be the smallest i ∈ Z such that f (i) 6= e; if f (i) = e for all i ∈ Z, then put m = 0. Let ` = min{m, z, 0}. Let n be the largest j such that f (j) 6= e; if f (j) = e for all j ∈ Z, then put n = 0. Put r = max{n, z, 0}. Similarly to Theorem 1 represent an element (f, z) as follows: f (`) . . . Af (0) . . . Cf (z) . . . f (r).
(12)
In case z = 0 the word representing an element (f, z) is as follows: f (`) . . . Bf (0) . . . f (r).
(13)
Let g be an element g = (f, z) ∈ G o Z. The words representing g and gt written one under another are either of the form (5) or (6). Therefore, the relation {hg, gti | g ∈ G o Z} is recognizable by a two–tape synchronous finite automaton. The words representing g and ggj for some j ∈ [1, n] written one under another are either of the form: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) Cf (z) f (z − 1) Cf (z)gj 8
f (z + 1) f (z + 1)
... ...
f (r) , f (r)
(14)
or of the form: f (`) . . . f (`) . . .
f (−1) Bf (0) f (−1) Bf (0)gj
f (1) f (1)
... ...
f (r) . f (r)
(15)
Since G is Cayley automatic, the differences ||Cf (z) | − |Cf (z)gj || and ||Bf (0) | − |Bf (0)gj || are bounded by a constant from above. By (14) and (15) this implies that the relation {hg, ggj i | g ∈ G o Z} is recognizable by a two–tape synchronous finite automaton. Remark 5. Suppose that G is Cayley biautomatic; for the definition of Cayley biautomatic groups we refer the reader to [16]. It can be verified that with respect to the representation of G o Z in the proof of Theorem 2 the left multiplication by the generators g1 , . . . , gn and t are recognizable by two–tape synchronous finite automata. Therefore, G o Z is Cayley biautomatic. For P a function f : Z → G with finite support we denote by |suppf | the sum j |f (j)|, where |f (j)| is the length of an element f (j) in the group G with respect to the generators g1 , . . . , gn . The generalization of Proposition 1 is straightforward. Proposition 2. The word length of an element g = (f, z) with respect to the generators g1 , . . . gn and t is as follows: |g| = |supp f | + min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}.
(16)
If G is a finite group the inequality (11) holds for GoZ. Suppose now that G is an infinite Cayley automatic group. For a given Cayley automatic representation of G we will suppose that non of the elements of G is represented by the empty word. Suppose that the following inequality holds for some constants C and D: |g| 6 C|u| + D,
(17)
where |u| is the length of the word representing g ∈ G with respect to a given Cayley automatic representation, and |g| is the word length of g with respect to generators g1 , . . . , gn . The analog of Corollary 1 for the case of an infinite Cayley automatic group G is as follows. Corollary 2. For a given g ∈ G o Z let w be the representative of g for the Cayley automatic representation of G o Z constructed in Theorem 2. Then the following inequalities hold: K0 1 |w| − 6 |g| 6 M |w| − 2, K K where M = C + D + 2 and K0 , K are integers.
9
(18)
Proof. First we prove the lower bound in (18). It is equivalent to the inequality: |w| 6 K|g| + K0 . Put K0 to be the length of the word that represents the identity e ∈ G. We denote by Mg1 , . . . , Mgn the synchronous two–tape automata that recognize the relations hg, gg1 i, . . . , hg, ggn i respectively. For an automaton Mgj put dj to be the maximum number of the padding symbols ♦ in the representations of the pairs hg, ggj i. Put K = max{K0 , dj | j ∈ [1, n]}. The inequality |w| 6 K|g| + K0 holds for g = e. Suppose that it holds for some g ∈ G o Z. It follows from the proof of Theorem 2 that the words representing ggj and ggj−1 have the lengths at most |w| + max{dj | j ∈ [1, n]}. The words gt and gt−1 are equal to either |w|, |w|+K0 or |w|−K0 . This implies that the inequality |w| 6 K|g| + K0 holds for the elements ggj , ggj−1 , gt and gt−1 as well. Let us prove the upper bound in (18). We have |suppf | 6 C|w| + D#suppf and #suppf 6 |w|. Therefore, |suppf | 6 (C + D)|w|. We have the following upper bound for the second summand of the right–hand side of (16): 2(|w| − 1) > min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}. Therefore, |g| 6 (C + D + 2)|w| − 2 which proves the upper bound in (18).
4
The wreath products of groups with a free group
Denote by a and b the generators of the free group F2 = ha, bi, and by h the nontrivial element of Z2 . Let us consider the Cayley graph of the wreath product Z2 oF2 with respect to the generators a, b and h. We have the following theorem. Theorem 3. The group Z2 o F2 is context–free Cayley automatic. Proof. Recall that an element of Z2 o F2 is a pair (f, z) where f is a function f : F2 → Z2 that has finite support and z ∈ F2 is the position of the lamplighter. The representation of Z2 o F2 to be introduced is a modification of that of Z2 o Z obtained in Theorem 1. Recall that in Theorem 1 we used the symbols 0, 1, A0 , A1 , B0 , B1 , C0 , C1 . In the representation of Z2 o F2 we will add the symbols D0 , D1 , E0 , E1 , the brackets (, ) and [, ], and the symbols D0A , D1A , D0B , D1B , D0C , D1C , E0C , E1C that are used to specify the positions of the origin e ∈ F2 and the lamplighter z ∈ F2 . Put Σ = {0, 1, A0 , A1 , B0 , B1 , C0 , C1 , D0 , D1 , E0 , E1 , (, ), [, ], D0A , D1A , D0B , D1B , C D0 , D1C , E0C , E1C }. Let L ⊂ Σ∗ be the language consisting of the words that satisfy the following properties: • The configuration of brackets (, ), [, ] is balanced; • Consider the pushdown automaton verifying that the configuration of brackets is balanced. It is not allowed to have ( on the top of the stack 10
while reading off (. It is not allowed to have [ on the top of the stack or the empty stack while reading off [; • For each pair of matched brackets ( and ) there exists only one of the symbols D0 , D1 , D0A , D1A , D0B , D1B , D0C , D1C between these brackets which is not inside any other pair of matched brackets between them. There exists one–to–one correspondence between matched brackets ( and ) and the set of symbols D0 , D1 , D0A , D1A , D0B , D1B , D0C , D1C in a word of L; • For each pair of matched brackets [ and ] there exists only one of the symbols E0 , E1 , E0C , E1C between these brackets which is not inside any other pair of matched brackets between them. There exists one–to–one correspondence between matched brackets [ and ] and the set of symbols E0 , E1 , E0C , E1C in a word of L; • Between each pair of matched brackets there exist at least two symbols; • The subwords of the form (0, 0), [0, 0] are not allowed. Also, a word cannot begin or end with 0; • For every word of L: either the letter B appears only one time or each of the letters A and C appears only one time. We construct representatives of Z2 o F2 in a recurrent manner. Let the elements of the subgroup Z2 o Z 6 Z2 o F2 , where Z = hai, have the same representatives as in Theorem 1. Reading off the letters between two matched brackets ( and ) corresponds to moving along the generator b from bottom to top. While reading off the letters between two matched brackets [ and ] corresponds to moving along the generator a from left to right. See Fig. 1 for an explanation of choosing directions for the generators a and b. For any word of L and for each pair of matched brackets ( and ) the structure of the subword between them is as follows ([(. . . ) . . . (. . . )Ei (. . . ) . . . (. . . )] . . . [. . . ]Dj [. . . ] . . . [. . . ]). Similarly, for each pair of matched brackets [ and ] the structure of the subword between them is as follows [([. . . ] . . . [. . . ]Di [. . . ] . . . [. . . ]) . . . (. . . )Ej (. . . ) . . . (. . . )]. The representatives for two nontrivial elements are shown in Fig. 1. It can be seen that L is recognizable by a deterministic pushdown automaton. Thus, L is a context–free language. The right multiplication by h either interchanges C0 and C1 , D0B and D1B , C D0 and D1C , or E0C and E1C . Therefore, the language Lh = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)h} is context–free. The right multiplication by a (or, b) moves the lamplighter by one step to the right (or, up). It is easily verified that the languages La = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)a} and Lb = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)b} are context–free. Remark 6. The pushdown automata which recognize the languages L, Lh , La and Lb in Theorem 3 are deterministic. In particular, this implies that Z2 o F2 is a parallel poly–pushdown group in the sense of [2]. That is because a language recognized by a two–tape synchronous automaton with a pushdown stack 11
b a
b a
a
b
a b
Figure 1: A white box means that the value of a function f : F2 → Z2 is e ∈ Z2 , a black box means that it is h ∈ Z2 , a black disk specifies the position of the lamplighter and tells us that the value of f is h. For the element to the left the word representing it is 11(1[1E0 1]D0A [E0 (C1 D1 )])([1E0 ]D0 [1E1 ])1, for the element to the right it is (D0 1)A1 ([1E0 ]D0 [1E1C ])1. is always recognized by a two–tape asynchronous automaton with a pushdown stack. However, the result that Z2 o F2 is a parallel poly–pushdown group can be obtained straightforwardly from the fact that the set of parallel poly–pushdown groups is closed under wreath products [2] (see Theorem 5.4). Corollary 3. For a given g ∈ Z2 o F2 let w be the representative of g for the context–free Cayley automatic representation of Z2 o F2 constructed in Theorem 3. Then the following inequalities hold: 1 1 |w| − 6 |g| 6 3|w| − 2. 3 3
(19)
Proof. First we prove the lower bound in (19). It is equivalent to the inequality: |w| 6 3|g| + 1. The representative for e ∈ Z2 o F2 is the word A0 of length 1, so the inequality holds for g = e. For every g ∈ Z2 o F2 the lengths of the representatives for g and gh are the same. It is easily verified that for every g ∈ Z2 o F2 the lengths of the representatives for g and ga (or gb) differ by at most 3. Therefore, the inequality |w| 6 3|g| + 1 holds for all g ∈ Z2 o F2 . Let us prove by induction the upper bound in (19): |g| 6 3|w| − 2. Consider the subgroup Z2 o Z 6 Z2 o F2 . For the elements of this subgroup the representatives are the same as in Theorem 1. By Corollary 1 we have that for these representatives the inequality |g| 6 3|w| − 2 is satisfied. In order to make an inductive step we observe that a word w that represents any element g ∈ Z2 o F2 has the form v0 (w1 )v1 (w2 )v2 . . . vn−1 (wn )vn , where the words v0 , v1 , . . . , vn do not contain brackets. By induction we obtain that 12
Pn |g| 6 i=1 (3|wi | − 2) + 3(|v0 . . . vn | + n) − 2 6 3|w| − 2. Thus, the inequality |g| 6 3|w| − 2 is proved. Remark 7. It is easy to construct an infinite sequence of elements of Z2 o F2 for which the lower bound in (19) is exact. Let us consider the infinite sequence of elements e, b, ba, bab, baba, . . . . For these elements the representatives are A0 , (D0A C0 ), (D0A [E0 C0 ]), (D0A [E0 (D0 C0 )]), (D0A [E0 (D0 [E0 C0 ])]), . . . , so for each element g of the sequence the length of the representative is exactly 3|g|+1. An infinite sequence for which the upper bound in (19) is exact was constructed in Remark 2. Theorem 3 and Corollary 3 can be generalized to the following theorem. Theorem 4. There exists a context–free Cayley automatic representation of the group Z2 o Fn for n > 2 such that for every g ∈ Z2 o Fn and its representative w with respect to this representation the following inequalities hold: 1 1 |w| − 6 |g| 6 3|w| − 2. 2n − 1 2n − 1
(20)
Remark 8. It is easily verified that Theorem 4 holds for the wreath product G o Fn for every finite group G and n > 2. Let ψ : L → G be a context–free Cayley automatic representation of a group G with respect to a finite set of generators {g1 , . . . , gm } ⊂ G such that for some constant N the following holds: for every u, v ∈ L such that ψ(u)gj = ψ(v), j ∈ {1, . . . , m} we have that ||u| − |v|| 6 N . Similarly to Theorem 2 and Corollary 2 one can obtain the following theorem. Theorem 5. Suppose that for the representation ψ : L → G the following inequality holds for all g ∈ G: |g| 6 C|u| + D, where C, D are constants and u is the representative for g, i.e., ψ(u) = g. There exists a context–free Cayley automatic representation of G o Fn for n > 2 such that for every g ∈ G o Fn and its representative w with respect to this representation the following inequalities hold: 1 K0 |w| − 6 |g| 6 M |w| − 2, (21) K K where M = C + D + 2, K0 is the length of the representative of e ∈ G with respect to ψ, and K = max{K0 + 2(n − 1), N }.
5
The wreath product Z2 o Z2
Denote by x and y the standard generators of Z2 , and by h the nontrivial element of Z2 . Let us consider the Cayley graph of the wreath product Z2 o Z2 with respect to the generators x, y and h. We have the following theorem. Theorem 6. There exists an indexed Cayley automatic representation ψ : L → Z2 o Z2 for the wreath product Z2 o Z2 such that L is a regular language. 13
(0,3) (0,2) (0,1) (-3,0) (-2,0) (-1,0) (0,0)
(1,0)
(2,0)
(3,0)
(0,-1) (0,-2) (0,-3)
Figure 2: A white box means that the value of a function f : Z2 → Z2 is e ∈ Z2 , a black box means that it is h ∈ Z2 , a black disk at the point (0, −2) specifies the position of the lamplighter and tells us that the value of f is h. For the element shown in this figure the word representing it is 0100011000000100001000C1 000101111000011000101100001. Proof. Recall that an element of Z2 o Z2 is a pair (f, z), where f is a function f : Z2 → Z2 that has finite support and z ∈ Z2 is the position of the lamplighter. Put Σ = {0, 1, C0 , C1 }. Let us consider the map t : N → Z2 such that: t(1) = (0, 0), t(2) = (1, 0), t(3) = (1, 1), t(4) = (0, 1), t(5) = (−1, 1), t(6) = (−1, 0), t(7) = (−1, −1), t(8) = (0, −1), and etc.; the map t : N → Z2 is depicted in Fig. 2. For a given element (f, z) ∈ Z2 o Z2 we represent it as the word for which the kth symbol is 0 if f (t(k)) = e, 1 if f (t(k)) = h, C0 if f (t(k)) = e and z = t(k), and C1 if f (t(k)) = h and z = t(k). The last letter of a word is not allowed to be 0, i.e., it can be either 1, C0 or C1 . It can be seen that the language L ⊂ Σ∗ of all representatives and the language Lh = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )h = ψ(w2 )} are regular. Let us consider the languages Lx = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )x = ψ(w2 )} and Ly = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )y = ψ(w2 )}. We will show that Lx and Ly are indexed languages. Put the memory alphabet Ξ = {I, B, T }. The symbols B and T are used to denote the bottom and the top of the stack respectively. The symbol I is used for all other intermediate positions. Let w be a word in the language Lx . Let us consider the stack automaton Mx that works as follows until it meets first time the letter that contains C0 or C1 : • Initially the stack is empty; • Mx reads off the first letter of w and pushes the symbol B onto the stack;
14
T I
I
B
T B
T B
T B
B
I B
T I B
T I B
T I B
T I B
I B
Figure 3: The content of the stack is shown for the first several iterations of Mx . In general situation, the content of the stack is shown to the right. • Mx reads off the second letter of w and pushes the symbol T onto the stack; • Mx reads off the third letter of w and makes one step down; • Mx reads off the fourth letter of w and makes one step up; • Mx makes two silent moves popping the symbol T and pushing I; • Mx reads off the fifth letter of w and pushes T onto the stack; • Mx reads off the sixth letter of w and makes one step down; • Mx reads off the seventh letter of w and makes one step down; • Mx reads off the eights letter of w and makes one step up; • The process continues by moving up and down along the stack between the bottom symbol B and the top symbol T . Each time the top is reached, it is raised up by one. This process is shown in Fig. 3. It is easy to verify the following: if a letter being read at a position m contains C0 (or, C1 ) first time then either the next letter at the position m + 1 contains C0 (or, C1 ) or the letter at the position m + (4n + 1) contains C0 (or, C1 ), where n is the current height of the stack. In order to verify the latter case, the automaton Mx , after it meets the letter containing C0 (or, C1 ) first time, works as follows: • Mx makes silent moves moving up until it reaches the top of the stack; • Then the automaton Mx pops a symbol out of stack each time it reads off four consecutive letters; 15
• After the stack is emptied the automaton keeps working without using it. It can be seen that the automaton Mx recognizes the language Lx . In a similar way one can obtain the nested stack automaton My that recognizes the language Ly . Remark 9. It can be seen that the nested stack automata Mx and My are deterministic and have limited erasing. Remark 10. For a given g ∈ Z2 o Z2 let w be the representative of g for the indexed Cayley automatic representations of Z2 o Z2 constructed in Theorem 6. Then the following inequality holds: |g| 6 2|w| − 1.
(22)
However, it is easily verified that the inequality of the form: λ|w| + µ 6 |g|
(23)
is not satisfied for all g ∈ Z2 o Z2 .
References [1] Gilbert Baumslag. Wreath products and finitely presented groups. Mathematische Zeitschrift, 75(1):22–28, 1961. [2] Gilbert Baumslag, Michael Shapiro, and Hamish Short. Parallel poly– pushdown groups. Journal of Pure and Applied Algebra, 140(3):209–227, 1999. [3] Dmitry Berdinsky and Bakhadyr Khoussainov. On automatic transitive graphs. Lecture Notes in Computer Science, A.M. Shur and M.V. Volkov (Eds.): Developments in Language Theory 2014, 8633:1–12, 2014. [4] A Blumensath. Automatic Structures. Diploma Thesis, RWTH, 1999. [5] Sean Cleary and Jennifer Taback. Dead end words in lamplighter groups and other wreath products. The Quarterly Journal of Mathematics, 56(2):165–178, 2005. [6] A Dyubina. An example of the rate of growth for a random walk on a group. Russian Mathematical Surveys, 54(5):1023–1024, 1999. [7] Murray Elder and Jennifer Taback. C–graph automatic groups. Journal of Algebra, 413:289–319, 2014. [8] Murray Elder and Jennifer Taback. Thompson’s group F is 1–counter graph automatic. arXiv:1501.04313 [math.GR], 2015.
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[9] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word Processing in Groups. Jones and Barlett Publishers. Boston, MA, 1992. [10] A Erschler. On the Asymptotics of Drift. Journal of Mathematical Sciences, 121(3):2437–2440, 2004. [11] Anna Erschler. On Isoperimetric Profiles of Finitely Generated Groups. Geometriae Dedicata, 100(1):157–171, 2003. [12] Robert Gilman. Formal languages and their application to combinatorial group theory. In Alexandre V. Borovik, editor, Groups, Languages, Algorithms, volume 378 of Contemporary Mathematics, pages 1–36. American Mathematical Society, 2005. [13] Robert Gilman and Michael Shapiro. On groups whose word problem is solved by a nested stack automaton. arXiv:math/9812028 [math.GR], 1998. [14] J.E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison–Wesley, 2001. [15] M. I. Kargapolov and Ju. I. Merzljakov. Fundamentals of the theory of groups. Springer–Verlag New York Inc., 1979. [16] Olga Kharlampovich, Bakhadyr Khoussainov, and Alexei Miasnikov. From automatic structures to automatic groups. Groups, Geometry, and Dynamics, 8(1):157–198, 2014. [17] Bakhadyr Khoussainov and Anil Nerode. Automatic presentations of structures. In Daniel Leivant, editor, Logic and Computational Complexity, volume 960 of Lecture Notes in Computer Science, pages 367–392. Springer Berlin Heidelberg, 1995. [18] Walter Parry. Growth series of some wreath products. Transactions of the American Mathematical Society, 331(2):751–759, 1992.
17
Cayley automatic representations of wreath products D. Berdinsky
∗
B. Khoussainov
†
Abstract. We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed. Keywords: Finite automata, pushdown automata, nested stack automata, Cayley graphs, wreath products.
1
Introduction
In this paper we study representations of Cayley graphs of wreath products of groups using finite automata, pushdown automata and nested stack automata. The representations to be used are related to the notion of Cayley automatic groups that was introduced in [16]. The notion of Cayley automatic groups appeared to be a natural generalization of automatic groups in the sense of Thurston [9]. The set of Cayley automatic groups properly contains the set of automatic groups. In particular, the set of Cayley automatic groups includes finitely generated nilpotent groups of class at most two, lamplighter group [16] and Baumslag– Solitar groups [3]. The Cayley automatic groups retain some nice properties of automatic groups. They are closed under direct product, free product and finite extensions. The word problem in Cayley automatic groups is decidable in quadratic time. We assume that the reader is familiar with the notions of finite automaton and regular language. Let Σ be a finite alphabet. Put Σ = Σ ∪ {}, where ∈ / Σ. The convolution of two words w1 , w2 ∈ Σ∗ is the string w1 ⊗ w2 of length max{|w1 |, |w2 |} over the alphabet Σ × Σ defined as follows. The kth symbol of the string is (σ1 , σ2 ), where σi , i = 1, 2 is the kth symbol of wi if k 6 |wi | and otherwise. The convolution ⊗R of a binary relation R ⊂ Σ∗ × Σ∗ is defined as follows: ⊗R = {w1 ⊗ w2 |(w1 , w2 ) ∈ R}. We say that a binary ∗ The † The
University of Auckland, Department of Computer Science; [email protected] University of Auckland, Department of Computer Science; [email protected]
1
relation R ⊂ Σ∗ × Σ∗ is automatic if there exists a finite automaton in the alphabet Σ × Σ that accepts the convolution ⊗R. Such an automaton is called two–tape synchronous finite automaton. Let G be a group generated by a finite subset S ⊂ G. Consider the labeled and directed Cayley graph Γ(G, S). We say that G is Cayley automatic if there is a regular language L ⊂ Σ∗ for some finite alphabet Σ that uniquely represents elements of G, and for every s ∈ S the binary relation which is the set of directed edges of Γ(G, S) labeled by s is automatic. That is to say, the group G is Cayley automatic if the labeled digraph Γ(G, S) is an automatic structure [17]. In [7] the authors considered the extensions of the notion of Cayley automatic groups replacing the regular languages by more powerful languages. We denote by C a class of languages; for example, it can be the class of regular languages, context–free languages or context–sensitive languages. The notion of Cayley automatic groups can be extended as follows. Definition 1. Let G be a finitely generated group. We say that G is C Cayley automatic if there exists a finite set S ⊂ G generating G for which the following properties hold: • There exist a bijection ψ : L → G between a language L ⊂ Σ∗ which is in the class C and the group G; • For each h ∈ S the language Lh = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )h = ψ(w2 )} is in the class C. If C is the class of regular languages then C Cayley automatic groups are Cayley automatic groups. The notion of Cayley automatic groups is invariant under a choice of generators. Recall that the context–free and indexed languages are the ones that are recognizable by pushdown and nested stack automata respectively. Notice that for context–free and indexed Cayley automatic groups Definition 1 depends on the choice of generators. Notice that for a group G, a finite subset S ⊂ G and a finite alphabet Σ in Definition 1 the triple (G, S, Σ) is C–graph automatic according to the terminology introduced in [7]. Some results on C Cayley automatic representations of groups are as follows. In [16] the authors constructed the Cayley automatic representations for nilpotent groups of class at most two. In [3] we constructed the Cayley automatic representations for all Baumslag–Solitar groups. In [7] the authors constructed the blind deterministic 3–counter Cayley automatic representation for Baumslag–Solitar groups BS(m, n), n > m > 2. In [8] the authors constructed the deterministic non–blind 1–counter Cayley automatic representation for Thompson’s group F . We consider the cases when C is the class of regular languages, context– free languages and indexed languages. We assume that the reader is familiar with the notion of pushdown automata and context–free languages, and nested stack automata and indexed languages. For definitions of pushdown automata and context–free languages see, for example, [14]. For definitions of nested stack automata and indexed languages we refer the reader to [13]. For the introduction 2
to formal languages theory that covers regular languages, context–free languages and indexed languages we refer the reader to [12]. In this paper we construct: Cayley automatic representation for wreath products G o Z, context–free Cayley automatic representation for wreath products GoFn and indexed Cayley automatic representation for the wreath product Z2 o Z2 . In each case we specify the set of generators for wreath products with respect to which we consider their Cayley graphs. In addition, for the representations constructed we prove the inequalities of the form: λ|w| + µ 6 |g| 6 ξ|w| + δ, where |g| is the length of a group element g with respect to chosen set of generators and |w| is the length of the word w which is the representative of g, i.e., ψ(w) = g. In [2] the authors introduced the notion of parallel poly–pushdown groups. The definition of poly–pushdown groups uses two–tape asynchronous automata instead of synchronous ones used in Definition 1. It was shown that the set of poly–pushdown groups is closed under wreath product [2]. This implies that all wreath products to be considered in this paper are parallel poly–pushdown groups. For wreath products there is an abundance of results on quantitative characteristics such as growth rate [18], isoperimetric profiles [11] and drift of simple random walks [10, 6]. This makes studying representations of Cayley graphs of wreath products relevant to seeking connections between characteristics of groups and a computational power of automata that is sufficient to represent their Cayley graphs. In this paper we focus on the aforementioned hierarchy of languages, i.e., regular, context–free and indexed. The rest of this paper is organized as follows. In Section 2 we briefly recall the definitions and notations for wreath products of groups. In Section 3 we consider the Cayley automatic representation for wreath products G o Z. In Section 4 we consider the context–free Cayley automatic representation for wreath products G o Fn . In Section 5 we consider the indexed Cayley automatic representation for the wreath product Z2 o Z2 .
2
Wreath products of groups: definitions and notations
Recall the definition of the restricted wreath product of two groups A and B. For more details on wreath products see, for example, [15]. Given two groups A and B. We denote by A(B) the group of all functions B → A having finite supports and the usual multiplication rule. Recall that a function f : B → A has finite support if f (x) 6= e for only finite number of x ∈ B, where e is the identity of A. Let us define the homomorphism τ : B → Aut(A(B) ) as follows: for a given f ∈ A(B) , the automorphism τ (b) maps f to f b ∈ A(B) , where f b (x) = f (bx) for all x ∈ B. For given groups A and B, the restricted wreath 3
product A o B is defined to be the semidirect product A(B) oτ B. Thus, A o B is the set product B × A(B) with the group multiplication given by: 0
(b, f ) · (b0 , f 0 ) = (bb0 , f b f 0 ),
(1)
0
where f b (x) = f (b0 x). For our purposes we will use the converse order for representing elements of a wreath product. Namely, we will represent an element of A o B as a pair (f, b), where f ∈ A(B) and b ∈ B. For such a representation, the group multiplication is given by: −1 (2) (f, b) · (f 0 , b0 ) = (f f 0 b , bb0 ), −1
where f 0 b (x) = f 0 (b−1 x). There exist natural embeddings B → A o B and A(B) → A o B mapping b to (e, b) and f to (f, e) respectively, where e is the identity of A(B) and e is the identity of B. For the sake of simplicity, we will identify B and A(B) with the corresponding subgroups of A o B. Recall that by [1] we have that: the wreath product A o B of two finitely presented groups A and B is finitely presented iff either A is the trivial group or B is finite. Therefore, the wreath products to be considered: G o Z, G o F n and Z2 o Z2 are not finitely presented for a nontrivial group G. Thus, these groups are not automatic [9].
3
The wreath products of groups with the infinite cyclic group
Denote by t the generator of Z = hti, and by a the nontrivial element of Z2 . Let us consider the Cayley graph of the lamplighter group L2 = Z2 o Z with respect to the generators a and t. We have the following theorem. Theorem 1. The lamplighter group L2 is Cayley automatic. Proof. Recall that an element of L2 is a pair (f, z), where f is a function f : Z → Z2 that has finite support and z ∈ Z is the position of the lamplighter. The automatic representations for the Cayley graph of L2 with respect to the generators a and t were constructed in [16] and [3]. Here we simplify the representation used in [3]. In order to present f (i) ∈ Z2 we use the symbols 0 and 1: 0 means that a lamp in the position z = i is unlit, i.e., f (i) = e; 1 means that the lamp is lit, i.e., f (i) = a. By abuse of notation we will write f (i) = 0 instead of f (i) = e and f (i) = 1 instead of f (i) = a. Recall that only a finite number of lamps are lit for each element of the group L2 . To show the position of the origin z = 0 we use the symbols A0 and A1 if the lamp in the origin is unlit and lit respectively. To show the position of the lamplighter we use the symbols C0 and C1 if the lamp in the position of the lamplighter is unlit and lit respectively. In case the lamplighter is in the origin we use the symbols B0 and B1 . 4
Given an element (f, z) ∈ L2 . Let m be the smallest i ∈ Z such that f (i) = 1; if f (i) = 0 for all i ∈ Z, then put m = 0. Put ` = min{m, z, 0}. Let n be the largest j such that f (j) = 1; if f (j) = 0 for all j ∈ Z, then put n = 0. Put r = max{n, z, 0}. Let us represent (f, z) as follows: f (`)f (` + 1) . . . f (−1)Af (0) f (1) . . . f (z − 1)Cf (z) f (z + 1) . . . f (r − 1)f (r), (3) where Af (0) = A0 and Af (0) = A1 if f (0) = 0 and f (0) = 1, respectively; also, Cf (z) = C0 and Cf (z) = C1 if f (z) = 0 and f (z) = 1, respectively. In case z = 0 the word representing an element (f, z) is as follows: f (`)f (` + 1) . . . f (−1)Bf (0) f (1) . . . f (r − 1)f (r),
(4)
where Bf (0) = B0 and Bf (0) = B1 if f (0) = 0 and f (0) = 1, respectively. It can be seen that the language of the words representing all elements (f, z) of the group L2 is regular. Let an element g = (f, z) is represented by a word (3). Writing the words representing g and gt = (f, z + 1) one under another we have: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) f (z − 1)
Cf (z) f (z)
f (z + 1) Cf (z+1)
... ...
f (r) . f (r)
(5)
Let an element g = (f, 0) is represented by a word (4). Writing the words representing g and gt = (f, 1) one under another we have: f (`) . . . f (`) . . .
f (−1) f (−1)
f (1) Cf (1)
Bf (0) Af (0)
f (2) f (2)
... ...
f (r) . f (r)
(6)
The other cases are considered similarly. From (5) and (6) it can be seen that the relation hg, gti is recognized by a synchronous two–tape finite automaton. Let an element g = (f, z) is represented by a word (3). Writing the words representing g and ga one under another we have: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) Cf (z) f (z − 1) Cf (z)
f (z + 1) f (z + 1)
... ...
f (r) , f (r)
(7)
where f (z) = 1 and f (z) = 0 if f (z) = 0 and f (z) = 1, respectively. Let an element g = (f, 0) is represented by a word (4). Writing the words representing g and ga one under another we have: f (`) . . . f (`) . . .
f (−1) f (−1)
Bf (0) Bf (0)
f (1) . . . f (1) . . .
f (r) , f (r)
(8)
where f (0) = 1 and f (0) = 0 if f (0) = 0 and f (0) = 1, respectively. From (7) and (8) it can be seen that the relation hg, gai is recognized by a synchronous two–tape finite automaton. Remark 1. It can be verified that with respect to the representation of L2 in Theorem 1 the left multiplication by the generators a and t are recognized by synchronous two–tape finite automata. This implies that L2 is Cayley biautomatic. For the definition of Cayley biatomatic groups see [16]. 5
For the automatic representation described in Theorem 1 the length of the word representing an element (f, z) is as follows: |w| = |r − `| + 1 = | max{n, z, 0} − min{m, z, 0}| + 1 = max{|n − m|, |n|, |m|, |n − z|, |m − z|, |z|} + 1.
(9)
For an element (f, z) denote by #supp f the cardinality of the set supp f = {j | f (j) = a}. The word length of an element (f, z) in L2 with respect to the generators a and t can be obtained as follows. Proposition 1. The word length of an element g = (f, z) with respect to the generators a and t is as follows: |g| = #supp f + min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}.
(10)
Proof. By [5], the left–first and the right–first normal forms of an element (f, z) ∈ L2 are as follows: ai1 . . . aip a−j1 . . . a−jq tz , a−j1 . . . a−jq ai1 . . . aip tz , where ip = n (if n > 0), jq = −m (if m 6 −1), ip > · · · > i1 > 0, jq > · · · > j1 > 0 and ak = tk at−k . It is proved [5] (see Proposition 3.6) that the word length of g = (f, z) ∈ L2 with respect to the generators a and t is as follows: |g| = p + q + min{2jq + ip + |z − ip |, 2ip + jq + |z + jq |}. Let us express |g| in terms of m 6 n for three different cases: • m 6 −1 and n > 0: |g| = p + q + min{−2m + n + |z − n|, 2n − m + |z − m|}, • m > 0: |g| = p + q + n + |z − n|, • n 6 −1: |g| = p + q − m + |z − m|. It can be seen that #supp f = p + q. Thus, we obtain (10). Corollary 1. For a given g ∈ L2 let w be the representative of g for the Cayley automatic representation of L2 constructed in Theorem 1. Then the following inequalities hold: |w| − 1 6 |g| 6 3|w| − 2. (11) Proof. First we prove the inequality: |g| 6 3|w| − 2. By (9), |w| > n − m + 1. Therefore, |w| > #supp f . Consider each of the three cases: m 6 −1 < 0 6 n, n 6 −1 and 0 6 m separately: • The case m 6 −1 < 0 6 n. If z > n, then we have: −2m + n + |z − n| = −2m + z 6 2(z − m). If z 6 m, then we have: 2n − m + |z − m| = 2n−m+m−z 6 2(n−z). If m < z < n, then we have: −2m+n+|z −n| = 2(n − m) − z and 2n − m + |z − m| = 2(n − m) + z. Therefore, by (9): min{−2m+n+|z −n|, 2n−m+|z −m|} 6 2(|w|−1). Thus, |g| 6 3|w|−2; 6
• The case m > 0. By (9) we have: n + |z − n| 6 2(|w| − 1). Thus, |g| 6 3|w| − 2; • The case n 6 −1. By (9), we have: −m + |z − m| 6 2(|w| − 1). Thus, |g| 6 3|w| − 2. Let us prove the inequality: |w| − 1 6 |g|. The identity e ∈ L2 is represented by the word: B0 . Therefore, the inequality holds for g = e. Suppose that the inequality holds for some g ∈ L2 . It follows from the proof of Theorem 1 that the length of the word representing ga equals |w|, and the lengths of the words representing gt and gt−1 are equal to either |w|, |w| + 1 or |w| − 1. This implies that the inequality holds for the elements ga, gt and gt−1 . Thus, the inequality holds for all g ∈ L2 . Remark 2. It is easy to construct an infinite sequence of elements Z2 o Z such that the upper bound in (11) is exact. Let us consider the infinite sequence of elements whose representatives are B1 , 1B1 1, 11B1 11, . . . . The lengths of such elements in the group Z2 o Z with respect to the generators a, t are 1, 7, 13, . . . ; so the equality |g| = 3|w| − 2 holds for all these elements. Remark 3. It is easy to construct an infinite sequence of elements Z2 o Z such that the lower bound in (11) is exact. Let us consider the infinite sequence of elements whose representatives are B0 , A0 C0 , A0 0C0 , A0 00C0 , . . . . The lengths of such elements in the group Z2 o Z with respect to the generators a, t are 0, 1, 2, 3, . . . ; so the equality |w| − 1 = |g| holds for all these elements. Let us consider wreath products GoZ for Cayley automatic groups G. Recall that the elements of a group GoZ can be obtained as the pairs of the form (f, z), where z ∈ Z and f : Z → G is a function which has finite support; i.e., f is an element of the direct product G(Z) [15]. Remark 4. The proof of Theorem 1 can be straightforwardly modified for a finite group G. For a group G that has n + 1 elements g0 = e, g1 , . . . , gn , we represent these elements by n+1 symbols 0, 1, . . . , n. Using the construction from Theorem 1 one can obtain an automatic representation for the Cayley graph of G o Z with respect to the set of generators g1 , . . . , gn and t, where Z = hti. For an infinite Cayley automatic group G we have the following theorem. Theorem 2. For an infinite Cayley automatic group G the group GoZ is Cayley automatic. Proof. Let {g1 , . . . , gn } ⊂ G be a finite set generating G. Since G is a Cayley automatic group there exists a bijection between a regular language LG and the group G such that the directed edges of the (right) Cayley graph labeled by g1 , . . . , gn are recognizable by synchronous two–tape finite automata Mg1 , . . . , Mgn respectively. For simplicity we may always assume that LG ⊂ {0, 1}∗ [4]. We will suppose that ∈ / LG . Let us construct an automatic representation for the Cayley graph 7
of G o Z similarly to that of for L2 (see Theorem 1). Since G is infinite we need to add some new features. We introduce two counterparts of the symbols 0 and 1: bold 0 and bold 1 which specify the beginning of a word of LG . In order to specify the position of the origin z = 0 we use the symbols A0 and A1 depending on whether the word that represents the element of G at z = 0 has 0 and 1 as the first letter. Similarly, we use the symbols C0 and C1 to specify the position of the lamplighter z. The symbols B0 and B1 are used in case when the lamplighter is at the origin z = 0. Let us show two simple examples. Take an element (f, 1) ∈ G o Z such that f (j) = e for j ∈ / [−1, 2] and f (−1) 6= e, f (0), f (1) and f (2) 6= e are represented by the words 011, 1001, 01 and 111 respectively. Then for our representation the element (f, 1) is represented as follows: 011A1 001C0 1111. Take an element (f, 0) ∈ G o Z such that f (j) = e for j ∈ / [−1, 1] and f (−1) 6= e, f (0) and f (1) 6= e are represented by the words 111, 000 and 01 respectively. Then for our representation the element (f, 0) is represented as follows: 111B0 0001. By abuse of notation use f (j) to denote the following modification of the word representing the element f (j) ∈ G: the first letter of the corresponding word in LG should be changed to the bold one. We denote by Af (0) and Bf (0) the following modifications of the word representing the element f (0) ∈ G: the first letter of the corresponding word in LG should be changed to A0 or A1 , and B0 or B1 respectively. We denote by Cf (z) the following modification of the word representing the element f (z) ∈ G: the first letter of the corresponding word in LG should be changed to C0 or C1 . Given an element (f, z) ∈ G o Z. Similarly to Theorem 1 put m to be the smallest i ∈ Z such that f (i) 6= e; if f (i) = e for all i ∈ Z, then put m = 0. Let ` = min{m, z, 0}. Let n be the largest j such that f (j) 6= e; if f (j) = e for all j ∈ Z, then put n = 0. Put r = max{n, z, 0}. Similarly to Theorem 1 represent an element (f, z) as follows: f (`) . . . Af (0) . . . Cf (z) . . . f (r).
(12)
In case z = 0 the word representing an element (f, z) is as follows: f (`) . . . Bf (0) . . . f (r).
(13)
Let g be an element g = (f, z) ∈ G o Z. The words representing g and gt written one under another are either of the form (5) or (6). Therefore, the relation {hg, gti | g ∈ G o Z} is recognizable by a two–tape synchronous finite automaton. The words representing g and ggj for some j ∈ [1, n] written one under another are either of the form: f (`) . . . f (`) . . .
Af (0) Af (0)
... ...
f (z − 1) Cf (z) f (z − 1) Cf (z)gj 8
f (z + 1) f (z + 1)
... ...
f (r) , f (r)
(14)
or of the form: f (`) . . . f (`) . . .
f (−1) Bf (0) f (−1) Bf (0)gj
f (1) f (1)
... ...
f (r) . f (r)
(15)
Since G is Cayley automatic, the differences ||Cf (z) | − |Cf (z)gj || and ||Bf (0) | − |Bf (0)gj || are bounded by a constant from above. By (14) and (15) this implies that the relation {hg, ggj i | g ∈ G o Z} is recognizable by a two–tape synchronous finite automaton. Remark 5. Suppose that G is Cayley biautomatic; for the definition of Cayley biautomatic groups we refer the reader to [16]. It can be verified that with respect to the representation of G o Z in the proof of Theorem 2 the left multiplication by the generators g1 , . . . , gn and t are recognizable by two–tape synchronous finite automata. Therefore, G o Z is Cayley biautomatic. For P a function f : Z → G with finite support we denote by |suppf | the sum j |f (j)|, where |f (j)| is the length of an element f (j) in the group G with respect to the generators g1 , . . . , gn . The generalization of Proposition 1 is straightforward. Proposition 2. The word length of an element g = (f, z) with respect to the generators g1 , . . . gn and t is as follows: |g| = |supp f | + min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}.
(16)
If G is a finite group the inequality (11) holds for GoZ. Suppose now that G is an infinite Cayley automatic group. For a given Cayley automatic representation of G we will suppose that non of the elements of G is represented by the empty word. Suppose that the following inequality holds for some constants C and D: |g| 6 C|u| + D,
(17)
where |u| is the length of the word representing g ∈ G with respect to a given Cayley automatic representation, and |g| is the word length of g with respect to generators g1 , . . . , gn . The analog of Corollary 1 for the case of an infinite Cayley automatic group G is as follows. Corollary 2. For a given g ∈ G o Z let w be the representative of g for the Cayley automatic representation of G o Z constructed in Theorem 2. Then the following inequalities hold: K0 1 |w| − 6 |g| 6 M |w| − 2, K K where M = C + D + 2 and K0 , K are integers.
9
(18)
Proof. First we prove the lower bound in (18). It is equivalent to the inequality: |w| 6 K|g| + K0 . Put K0 to be the length of the word that represents the identity e ∈ G. We denote by Mg1 , . . . , Mgn the synchronous two–tape automata that recognize the relations hg, gg1 i, . . . , hg, ggn i respectively. For an automaton Mgj put dj to be the maximum number of the padding symbols ♦ in the representations of the pairs hg, ggj i. Put K = max{K0 , dj | j ∈ [1, n]}. The inequality |w| 6 K|g| + K0 holds for g = e. Suppose that it holds for some g ∈ G o Z. It follows from the proof of Theorem 2 that the words representing ggj and ggj−1 have the lengths at most |w| + max{dj | j ∈ [1, n]}. The words gt and gt−1 are equal to either |w|, |w|+K0 or |w|−K0 . This implies that the inequality |w| 6 K|g| + K0 holds for the elements ggj , ggj−1 , gt and gt−1 as well. Let us prove the upper bound in (18). We have |suppf | 6 C|w| + D#suppf and #suppf 6 |w|. Therefore, |suppf | 6 (C + D)|w|. We have the following upper bound for the second summand of the right–hand side of (16): 2(|w| − 1) > min{2 max{−m, 0} + max{n, 0} + |z − max{n, 0}|, 2 max{n, 0} + max{−m, 0} + |z + max{−m, 0}|}. Therefore, |g| 6 (C + D + 2)|w| − 2 which proves the upper bound in (18).
4
The wreath products of groups with a free group
Denote by a and b the generators of the free group F2 = ha, bi, and by h the nontrivial element of Z2 . Let us consider the Cayley graph of the wreath product Z2 oF2 with respect to the generators a, b and h. We have the following theorem. Theorem 3. The group Z2 o F2 is context–free Cayley automatic. Proof. Recall that an element of Z2 o F2 is a pair (f, z) where f is a function f : F2 → Z2 that has finite support and z ∈ F2 is the position of the lamplighter. The representation of Z2 o F2 to be introduced is a modification of that of Z2 o Z obtained in Theorem 1. Recall that in Theorem 1 we used the symbols 0, 1, A0 , A1 , B0 , B1 , C0 , C1 . In the representation of Z2 o F2 we will add the symbols D0 , D1 , E0 , E1 , the brackets (, ) and [, ], and the symbols D0A , D1A , D0B , D1B , D0C , D1C , E0C , E1C that are used to specify the positions of the origin e ∈ F2 and the lamplighter z ∈ F2 . Put Σ = {0, 1, A0 , A1 , B0 , B1 , C0 , C1 , D0 , D1 , E0 , E1 , (, ), [, ], D0A , D1A , D0B , D1B , C D0 , D1C , E0C , E1C }. Let L ⊂ Σ∗ be the language consisting of the words that satisfy the following properties: • The configuration of brackets (, ), [, ] is balanced; • Consider the pushdown automaton verifying that the configuration of brackets is balanced. It is not allowed to have ( on the top of the stack 10
while reading off (. It is not allowed to have [ on the top of the stack or the empty stack while reading off [; • For each pair of matched brackets ( and ) there exists only one of the symbols D0 , D1 , D0A , D1A , D0B , D1B , D0C , D1C between these brackets which is not inside any other pair of matched brackets between them. There exists one–to–one correspondence between matched brackets ( and ) and the set of symbols D0 , D1 , D0A , D1A , D0B , D1B , D0C , D1C in a word of L; • For each pair of matched brackets [ and ] there exists only one of the symbols E0 , E1 , E0C , E1C between these brackets which is not inside any other pair of matched brackets between them. There exists one–to–one correspondence between matched brackets [ and ] and the set of symbols E0 , E1 , E0C , E1C in a word of L; • Between each pair of matched brackets there exist at least two symbols; • The subwords of the form (0, 0), [0, 0] are not allowed. Also, a word cannot begin or end with 0; • For every word of L: either the letter B appears only one time or each of the letters A and C appears only one time. We construct representatives of Z2 o F2 in a recurrent manner. Let the elements of the subgroup Z2 o Z 6 Z2 o F2 , where Z = hai, have the same representatives as in Theorem 1. Reading off the letters between two matched brackets ( and ) corresponds to moving along the generator b from bottom to top. While reading off the letters between two matched brackets [ and ] corresponds to moving along the generator a from left to right. See Fig. 1 for an explanation of choosing directions for the generators a and b. For any word of L and for each pair of matched brackets ( and ) the structure of the subword between them is as follows ([(. . . ) . . . (. . . )Ei (. . . ) . . . (. . . )] . . . [. . . ]Dj [. . . ] . . . [. . . ]). Similarly, for each pair of matched brackets [ and ] the structure of the subword between them is as follows [([. . . ] . . . [. . . ]Di [. . . ] . . . [. . . ]) . . . (. . . )Ej (. . . ) . . . (. . . )]. The representatives for two nontrivial elements are shown in Fig. 1. It can be seen that L is recognizable by a deterministic pushdown automaton. Thus, L is a context–free language. The right multiplication by h either interchanges C0 and C1 , D0B and D1B , C D0 and D1C , or E0C and E1C . Therefore, the language Lh = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)h} is context–free. The right multiplication by a (or, b) moves the lamplighter by one step to the right (or, up). It is easily verified that the languages La = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)a} and Lb = {u ⊗ v|u, v ∈ L, ψ(v) = ψ(u)b} are context–free. Remark 6. The pushdown automata which recognize the languages L, Lh , La and Lb in Theorem 3 are deterministic. In particular, this implies that Z2 o F2 is a parallel poly–pushdown group in the sense of [2]. That is because a language recognized by a two–tape synchronous automaton with a pushdown stack 11
b a
b a
a
b
a b
Figure 1: A white box means that the value of a function f : F2 → Z2 is e ∈ Z2 , a black box means that it is h ∈ Z2 , a black disk specifies the position of the lamplighter and tells us that the value of f is h. For the element to the left the word representing it is 11(1[1E0 1]D0A [E0 (C1 D1 )])([1E0 ]D0 [1E1 ])1, for the element to the right it is (D0 1)A1 ([1E0 ]D0 [1E1C ])1. is always recognized by a two–tape asynchronous automaton with a pushdown stack. However, the result that Z2 o F2 is a parallel poly–pushdown group can be obtained straightforwardly from the fact that the set of parallel poly–pushdown groups is closed under wreath products [2] (see Theorem 5.4). Corollary 3. For a given g ∈ Z2 o F2 let w be the representative of g for the context–free Cayley automatic representation of Z2 o F2 constructed in Theorem 3. Then the following inequalities hold: 1 1 |w| − 6 |g| 6 3|w| − 2. 3 3
(19)
Proof. First we prove the lower bound in (19). It is equivalent to the inequality: |w| 6 3|g| + 1. The representative for e ∈ Z2 o F2 is the word A0 of length 1, so the inequality holds for g = e. For every g ∈ Z2 o F2 the lengths of the representatives for g and gh are the same. It is easily verified that for every g ∈ Z2 o F2 the lengths of the representatives for g and ga (or gb) differ by at most 3. Therefore, the inequality |w| 6 3|g| + 1 holds for all g ∈ Z2 o F2 . Let us prove by induction the upper bound in (19): |g| 6 3|w| − 2. Consider the subgroup Z2 o Z 6 Z2 o F2 . For the elements of this subgroup the representatives are the same as in Theorem 1. By Corollary 1 we have that for these representatives the inequality |g| 6 3|w| − 2 is satisfied. In order to make an inductive step we observe that a word w that represents any element g ∈ Z2 o F2 has the form v0 (w1 )v1 (w2 )v2 . . . vn−1 (wn )vn , where the words v0 , v1 , . . . , vn do not contain brackets. By induction we obtain that 12
Pn |g| 6 i=1 (3|wi | − 2) + 3(|v0 . . . vn | + n) − 2 6 3|w| − 2. Thus, the inequality |g| 6 3|w| − 2 is proved. Remark 7. It is easy to construct an infinite sequence of elements of Z2 o F2 for which the lower bound in (19) is exact. Let us consider the infinite sequence of elements e, b, ba, bab, baba, . . . . For these elements the representatives are A0 , (D0A C0 ), (D0A [E0 C0 ]), (D0A [E0 (D0 C0 )]), (D0A [E0 (D0 [E0 C0 ])]), . . . , so for each element g of the sequence the length of the representative is exactly 3|g|+1. An infinite sequence for which the upper bound in (19) is exact was constructed in Remark 2. Theorem 3 and Corollary 3 can be generalized to the following theorem. Theorem 4. There exists a context–free Cayley automatic representation of the group Z2 o Fn for n > 2 such that for every g ∈ Z2 o Fn and its representative w with respect to this representation the following inequalities hold: 1 1 |w| − 6 |g| 6 3|w| − 2. 2n − 1 2n − 1
(20)
Remark 8. It is easily verified that Theorem 4 holds for the wreath product G o Fn for every finite group G and n > 2. Let ψ : L → G be a context–free Cayley automatic representation of a group G with respect to a finite set of generators {g1 , . . . , gm } ⊂ G such that for some constant N the following holds: for every u, v ∈ L such that ψ(u)gj = ψ(v), j ∈ {1, . . . , m} we have that ||u| − |v|| 6 N . Similarly to Theorem 2 and Corollary 2 one can obtain the following theorem. Theorem 5. Suppose that for the representation ψ : L → G the following inequality holds for all g ∈ G: |g| 6 C|u| + D, where C, D are constants and u is the representative for g, i.e., ψ(u) = g. There exists a context–free Cayley automatic representation of G o Fn for n > 2 such that for every g ∈ G o Fn and its representative w with respect to this representation the following inequalities hold: 1 K0 |w| − 6 |g| 6 M |w| − 2, (21) K K where M = C + D + 2, K0 is the length of the representative of e ∈ G with respect to ψ, and K = max{K0 + 2(n − 1), N }.
5
The wreath product Z2 o Z2
Denote by x and y the standard generators of Z2 , and by h the nontrivial element of Z2 . Let us consider the Cayley graph of the wreath product Z2 o Z2 with respect to the generators x, y and h. We have the following theorem. Theorem 6. There exists an indexed Cayley automatic representation ψ : L → Z2 o Z2 for the wreath product Z2 o Z2 such that L is a regular language. 13
(0,3) (0,2) (0,1) (-3,0) (-2,0) (-1,0) (0,0)
(1,0)
(2,0)
(3,0)
(0,-1) (0,-2) (0,-3)
Figure 2: A white box means that the value of a function f : Z2 → Z2 is e ∈ Z2 , a black box means that it is h ∈ Z2 , a black disk at the point (0, −2) specifies the position of the lamplighter and tells us that the value of f is h. For the element shown in this figure the word representing it is 0100011000000100001000C1 000101111000011000101100001. Proof. Recall that an element of Z2 o Z2 is a pair (f, z), where f is a function f : Z2 → Z2 that has finite support and z ∈ Z2 is the position of the lamplighter. Put Σ = {0, 1, C0 , C1 }. Let us consider the map t : N → Z2 such that: t(1) = (0, 0), t(2) = (1, 0), t(3) = (1, 1), t(4) = (0, 1), t(5) = (−1, 1), t(6) = (−1, 0), t(7) = (−1, −1), t(8) = (0, −1), and etc.; the map t : N → Z2 is depicted in Fig. 2. For a given element (f, z) ∈ Z2 o Z2 we represent it as the word for which the kth symbol is 0 if f (t(k)) = e, 1 if f (t(k)) = h, C0 if f (t(k)) = e and z = t(k), and C1 if f (t(k)) = h and z = t(k). The last letter of a word is not allowed to be 0, i.e., it can be either 1, C0 or C1 . It can be seen that the language L ⊂ Σ∗ of all representatives and the language Lh = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )h = ψ(w2 )} are regular. Let us consider the languages Lx = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )x = ψ(w2 )} and Ly = {w1 ⊗ w2 |w1 , w2 ∈ L, ψ(w1 )y = ψ(w2 )}. We will show that Lx and Ly are indexed languages. Put the memory alphabet Ξ = {I, B, T }. The symbols B and T are used to denote the bottom and the top of the stack respectively. The symbol I is used for all other intermediate positions. Let w be a word in the language Lx . Let us consider the stack automaton Mx that works as follows until it meets first time the letter that contains C0 or C1 : • Initially the stack is empty; • Mx reads off the first letter of w and pushes the symbol B onto the stack;
14
T I
I
B
T B
T B
T B
B
I B
T I B
T I B
T I B
T I B
I B
Figure 3: The content of the stack is shown for the first several iterations of Mx . In general situation, the content of the stack is shown to the right. • Mx reads off the second letter of w and pushes the symbol T onto the stack; • Mx reads off the third letter of w and makes one step down; • Mx reads off the fourth letter of w and makes one step up; • Mx makes two silent moves popping the symbol T and pushing I; • Mx reads off the fifth letter of w and pushes T onto the stack; • Mx reads off the sixth letter of w and makes one step down; • Mx reads off the seventh letter of w and makes one step down; • Mx reads off the eights letter of w and makes one step up; • The process continues by moving up and down along the stack between the bottom symbol B and the top symbol T . Each time the top is reached, it is raised up by one. This process is shown in Fig. 3. It is easy to verify the following: if a letter being read at a position m contains C0 (or, C1 ) first time then either the next letter at the position m + 1 contains C0 (or, C1 ) or the letter at the position m + (4n + 1) contains C0 (or, C1 ), where n is the current height of the stack. In order to verify the latter case, the automaton Mx , after it meets the letter containing C0 (or, C1 ) first time, works as follows: • Mx makes silent moves moving up until it reaches the top of the stack; • Then the automaton Mx pops a symbol out of stack each time it reads off four consecutive letters; 15
• After the stack is emptied the automaton keeps working without using it. It can be seen that the automaton Mx recognizes the language Lx . In a similar way one can obtain the nested stack automaton My that recognizes the language Ly . Remark 9. It can be seen that the nested stack automata Mx and My are deterministic and have limited erasing. Remark 10. For a given g ∈ Z2 o Z2 let w be the representative of g for the indexed Cayley automatic representations of Z2 o Z2 constructed in Theorem 6. Then the following inequality holds: |g| 6 2|w| − 1.
(22)
However, it is easily verified that the inequality of the form: λ|w| + µ 6 |g|
(23)
is not satisfied for all g ∈ Z2 o Z2 .
References [1] Gilbert Baumslag. Wreath products and finitely presented groups. Mathematische Zeitschrift, 75(1):22–28, 1961. [2] Gilbert Baumslag, Michael Shapiro, and Hamish Short. Parallel poly– pushdown groups. Journal of Pure and Applied Algebra, 140(3):209–227, 1999. [3] Dmitry Berdinsky and Bakhadyr Khoussainov. On automatic transitive graphs. Lecture Notes in Computer Science, A.M. Shur and M.V. Volkov (Eds.): Developments in Language Theory 2014, 8633:1–12, 2014. [4] A Blumensath. Automatic Structures. Diploma Thesis, RWTH, 1999. [5] Sean Cleary and Jennifer Taback. Dead end words in lamplighter groups and other wreath products. The Quarterly Journal of Mathematics, 56(2):165–178, 2005. [6] A Dyubina. An example of the rate of growth for a random walk on a group. Russian Mathematical Surveys, 54(5):1023–1024, 1999. [7] Murray Elder and Jennifer Taback. C–graph automatic groups. Journal of Algebra, 413:289–319, 2014. [8] Murray Elder and Jennifer Taback. Thompson’s group F is 1–counter graph automatic. arXiv:1501.04313 [math.GR], 2015.
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[9] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word Processing in Groups. Jones and Barlett Publishers. Boston, MA, 1992. [10] A Erschler. On the Asymptotics of Drift. Journal of Mathematical Sciences, 121(3):2437–2440, 2004. [11] Anna Erschler. On Isoperimetric Profiles of Finitely Generated Groups. Geometriae Dedicata, 100(1):157–171, 2003. [12] Robert Gilman. Formal languages and their application to combinatorial group theory. In Alexandre V. Borovik, editor, Groups, Languages, Algorithms, volume 378 of Contemporary Mathematics, pages 1–36. American Mathematical Society, 2005. [13] Robert Gilman and Michael Shapiro. On groups whose word problem is solved by a nested stack automaton. arXiv:math/9812028 [math.GR], 1998. [14] J.E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison–Wesley, 2001. [15] M. I. Kargapolov and Ju. I. Merzljakov. Fundamentals of the theory of groups. Springer–Verlag New York Inc., 1979. [16] Olga Kharlampovich, Bakhadyr Khoussainov, and Alexei Miasnikov. From automatic structures to automatic groups. Groups, Geometry, and Dynamics, 8(1):157–198, 2014. [17] Bakhadyr Khoussainov and Anil Nerode. Automatic presentations of structures. In Daniel Leivant, editor, Logic and Computational Complexity, volume 960 of Lecture Notes in Computer Science, pages 367–392. Springer Berlin Heidelberg, 1995. [18] Walter Parry. Growth series of some wreath products. Transactions of the American Mathematical Society, 331(2):751–759, 1992.
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