0 | 0 1 | 0 1 | 1 0 | 1 0 1 - Semantic Scholar
ON A CLASS OF AUTOMATA GROUPS GENERALIZING LAMPLIGHTER GROUPS P. V. SILVA AND B. STEINBERG Dedicated to Professor Rostislav Grigorchuk on the occasion of his 50th birthday. Abstract. We study automata groups generated by reset automata. Every lamplighter group Z/nZ wr Z can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.
1. Introduction ˙ Grigorchuk and Zuk [6] showed that the lamplighter group Z/2Z wr Z can be constructed as the automata group of the following automaton:
1|1
0|0 0
1|0 1
0|1 Figure 1. Automaton generating the lamplighter group. They used this automaton to compute the spectrum and spectral measure associated to random walks on the lamplighter group [6], leading to a Date: September 21, 2003. 1991 Mathematics Subject Classification. 20F32,20E08,20F10,20M35. Key words and phrases. Automata groups, Cayley machines, reset automata, free semigroups, lamplighter groups. The authors were supported in part by the FCT and POCTI approved project POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by INTAS project 99–1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science. The first author was supported in part by FCT through the Centro de Matem´ atica da Universidade do Porto. The second author also gratefully acknowledges the support of NSERC. 1
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P. V. SILVA AND B. STEINBERG
counterexample to the strong form of the Atiyah conjecture [4]. Notice that each input letter resets the automaton to a particular state: 0 takes each state to the state labelled 0 and 1 takes each state to the state labelled 1. In this paper we study groups of automatic transformations generated by invertible Mealy automata all of whose inputs are resets, that is, so-called reset automata. Under mild conditions – say the resets are to distinct states and the automaton is reduced, for instance – we show that the semigroup of automatic transformations generated by a reset automaton is free on the set of states; this easily allows one to construct automata generating a free semigroup of any finite rank. Moreover, the group generated by such an automaton is either finite or locally-finite-by-infinite cyclic and hence amenable. Recall that a group is locally finite if its finitely generated subgroups are finite. It is well known that the class of amenable groups is closed under extensions and direct limits and contains all finite and Abelian groups; hence each locally-finite-by-cyclic group is amenable. Thus the groups generated by reset automata are similar in nature to lamplighter groups (and the analogy can be made more precise). In [10], Krohn and Rhodes associate to each finite monoid an automaton called its Cayley machine (see also [3]). These automata play a fundamental role in Krohn-Rhodes theory [8, 9, 10], which studies the relationship between decomposition of transformations computed by finite state automata and wreath product decompositions of finite semigroups. This paper is inspired by the idea of using Krohn-Rhodes theory to understand better groups generated by finite state automata. In particular, reset automata form the bottom of a natural hierarchy of counter-free (=aperiodic) automata [10, 3]. If G is a finite group, then the Cayley machine of G is an invertible automaton whose inverse is a reset automaton. We show that if G is an Abelian group, then the group generated by its Cayley machine is the restricted wreath product G wr Z. In particular, any lamplighter group Z/nZ wr Z can be generated by a reset automata: the inverse of the Cayley machine of Z/nZ. Our construction of Z/nZ wr Z via automata thus constructs a set of n generators that generate a free subsemigroup. The authors, together with M. Kambites [7], have used these automata to compute the spectra of such lamplighters along the lines of [6], where the Cayley Machine of Z/2Z is used (see Figure 6); a different approach to the spectra of wreath product groups can be found in [2].
2. Automata and Reset Automata A (finite state Mealy) automaton [3, 5] A is a 4-tuple A = (Q, A, δ, λ) where Q is a finite set of states, A is an alphabet, δ : Q × A → Q is the transition function and λ : Q × A → A is the output function. We use the notation: δ(q, a) = q · a, λ(q, a) = q ◦ a = λq (a).
ON A CLASS OF AUTOMATA GROUPS
3
The functions λq : A → A, q ∈ Q, are called the state functions. One draws
(2.1)
a |b
q0
q
to mean q · a = q 0 , q ◦ a = b. That is, in state q with input a, the automaton outputs b and goes to state q 0 . The automaton is said to be invertible if each state function is a permutation. In this paper, all automata considered (except the Cayley machine of a non-group monoid) will be invertible. The inverse automaton is the automaton A−1 obtained by switching the input a and the output b on each arrow (2.1). The resulting transition in the inverse automaton is
(2.2)
b |a
q0
q
Given any state q ∈ Q, one obtains (sequential) functions [3, 5] (abusing notation) Aq : A∗ → A∗ and Aq : Aω → Aω defined by Aq (1) = 1,
Aq (a0 · · · an ) = λq (a0 )Aq·a0 (a1 · · · an ),
Aq (a0 a1 · · · ) = lim Aq (a0 · · · an ). n→∞
If A is invertible, then Aq is invertible [5]. Suppose λ0 ≥ λ1 ≥ λ2 . . . is a decreasing sequence of positive real numbers converging to 0. Then the topology of Aω can be defined by the following ultra-metric: for u, v ∈ Aω , d(u, v) = λn where n is the length of the largest common prefix of u and v. With this metric each Aq is a contraction [5, 11] and if A is invertible, then Aq is an isometry with inverse A−1 [5]. We q denote the group of isometries of Aω by Iso(Aω ); this group is isomorphic to the group of automorphisms of the standard rooted |A|-ary tree [1, 5]. The automaton A is called reduced if the functions Aq , q ∈ Q, are distinct. Since we view words from A∗ as acting on the right of Q and use notation such as q · a and q ◦ a, it seems natural to view Aq as acting on the left of A∗ (resp. Aω ); indeed, it is more convenient to write Aq (a0 a1 a2 · · · ) than (a0 a1 a2 · · · )Aq . Thus we view (Iso(Aω ), Aω ) as a left transformation group. There is a well-known [5, 1] isomorphism of left transformation groups: (2.3)
(Iso(Aω ), Aω ) ∼ = (S|A| , A) o (S|A| , A) o · · · ∼ = (S|A| , A) o (Iso(Aω ), Aω )
where o is the wreath product of left transformation groups; see also [11]. We remark that [5, 1] use right actions, so our notation is dual to theirs. Let |A| = n; then an element of the rightmost wreath product in (2.3) is of the form σ(f1 , . . . , fn ) where σ ∈ Sn and f1 , . . . , fn ∈ Iso(Aω ). Multiplication is given by σ(f1 , . . . , fn )τ (g1 , . . . , gn ) = στ (fτ (1) g1 , . . . , fτ (n) gn )
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P. V. SILVA AND B. STEINBERG
If A is an automaton, then in wreath product coordinates: Aq = λq (Aq·a1 , . . . , Aq·an )
(2.4)
where A = {a1 , . . . , an }. Moreover, if σ ∈ Sn , then
σ −1 Aq σ = σ −1 λq σ(Aq·σ(a1 ) , . . . , Aq·σ(an ) ).
The semigroup of automatic transformations generated by the automaton A, which we shall call the automata semigroup of A, is the semigroup S(A) generated by {Aq | q ∈ Q}. If A is invertible, then the group of automatic transformations generated by the automaton A, which we shall call the automata group of A, is the group G(A) generated by {Aq | q ∈ Q}. If A = (Q, A, δ, λ) is an automaton, then a ∈ A is called a reset if |Q · a| = 1; that is a resets the automaton to state Q · a. One says that A is a reset automaton if each letter is a reset. It is straightforward to see that two states q, q 0 of a reset automaton A are equivalent (i.e. Aq = Aq0 ) if and only if λq = λq0 , so A is reduced if and only if the state functions λq are distinct. The following notation will be convenient throughout the paper for dealing with reset automata. Define eq (a) = λ] ◦ a, a ∈ A, q ∈ Q (2.5) e a = Q · a, λ q (a) = qg
eq , q ∈ Q, the modified state functions. For example, in Figure 1 We call the λ eq = λq . Notice that if the state functions we have e 0 = 0, e 1 = 1 and so λ associated to states p and q are the same, then the modified state functions associated to the states are also the same – said differently, if two states have distinct modified state functions, then they have distinct state functions; in particular if the modified state functions are all distinct, then the reset automaton must be reduced. As another example, consider the automaton in Figure 2 below. Here e 0 = a, e 1=b=e 2 and so we calculate eb = 0 1 2 . ea = 0 1 2 , λ λ b b a a b b ea = 0 1 2 = λ eb . For the automaton in Figure 3 below, one checks λ b b a
1 | 1, 2 | 2
0|0 a
1 | 2, 2 | 0 b
0|1 Figure 2. A reset automaton with distinct modified state functions.
ON A CLASS OF AUTOMATA GROUPS
1 | 1, 2 | 0
0|2 a
5
1 | 2, 2 | 0 b
0|1 Figure 3. A reset automaton with non-distinct modified state functions generating a free semigroup. The following fact about computing with reset automata is a straightforward exercise that we leave to the reader. Lemma 2.1. Let A = (Q, A, δ, λ) be a reset automaton. Then, for q ∈ Q, Aq (a0 · · · an ) = (q ◦ a0 )(e a0 ◦ a1 )(e a1 ◦ a2 ) · · · (e an−1 ◦ an )
where the a0 , . . . , an are arbitrary letters from A.
Notice, in particular, that only the first letter of the output actually depends on the state q. In fact, in wreath product coordinates, using (2.4) (2.6)
Aq = λq (Aq·a1 , . . . , Aq·an ) = λq (Aea1 , . . . , Aean )
where A = {a1 , . . . an }. The following theorem is one of the main result of this paper. Theorem 2.2. If A = (Q, A, δ, λ) is an invertible reset automaton with |Q| > 1 and with distinct modified state functions, then S(A) is a free semigroup on {Aq | q ∈ Q}. This occurs, in particular, if A is reduced and the resets are to distinct states. Notice that the condition in the theorem can be checked in linear time. For example, the automaton in Figure 2 satisfies the conditions of Theorem 2.2 and hence generates a free semigroup on two generators. The automaton from Figure 1 also satisfies the conditions of Theorem 2.2 and so generates a free semigroup of rank 2; this particular case was proved by ˙ Grigorchuk and Zuk [6, 5]. The automaton in Figure 3 does not meet the criteria of the theorem; nonetheless, we shall see later that it, too, generates a free semigroup. Theorem 2.2 gives an easy way to construct an automaton A so that S(A) is free on any prescribed number of generators n. Indeed, choose n distinct permutations from Sn : σ1 , . . . , σn . The states and alphabet of A are {1, . . . , n}; input i is, of course, a reset to state i. We set λi = σi . The resulting automaton clearly satisfies the hypotheses of Theorem 2.2; more natural constructions shall occur in Section 4. We mention the following corollary of our theorem.
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Corollary 2.3. Let A be an automaton satisfying the conditions of Theorem 2.2. Then G(A) has exponential growth. In particular, if A is a reduced reset automaton with at least two states and, moreover, the resets are to distinct states, then G(A) has exponential growth. Proof. It is well known that a group containing a free subsemigroup on two generators has exponential growth. The proof of Theorem 2.2 will require some preparation; we begin with some notation. Given pn , . . . , p1 ∈ Q and u ∈ A+ with |u| = m + 1, we define an (n + 1) × (m + 1) matrix with entries in A a00 a01 · · · a0m a10 a11 · · · a1m (2.7) M (pn , . . . , p1 ; u) = .. .. . . . . . an0 an1 · · · an,m recursively by the following Pascal Array type rules: a00 · · · a0m = u ai0 = pi ◦ ai−1,0 (2.8) aij = e ai−1,j−1 ◦ ai−1,j
where i = 1, . . . , n, j = 1, . . . m. For example, if we use the automaton from Figure 1, and u = 011, then, 0 1 1 1 1 0 M (1, 0, 1; 011) = 1 0 1 0 1 1
It will turn out, as the next lemma shows, that the ith row of our array (2.7) is the image of u under Api · · · Ap1 . We now proceed to establish several useful facts about these matrices. Lemma 2.4. For all i = 0, . . . , n, ai0 · · · aim = Api · · · Ap1 (u). Proof. The case i = 0 being trivial, assume that i > 0 and that the result holds for i − 1. Then, using Lemma 2.1 and (2.8), we obtain Api · · · Ap1 (u) = Api (Api−1 · · · Ap1 (u)) = Api (ai−1,0 · · · ai−1,m ) = (pi ◦ ai−1,0 )(e ai−1,0 ◦ ai−1,1 ) · · · (e ai−1,m−1 ◦ ai−1,m ) = ai0 ai1 · · · aim ,
Set u = u0 u1 · · · um with ui ∈ A. The following lemma is an observation about the structure of our Pascal-type Array. Lemma 2.5. Let i ≤ j and consider the function fij : A → A which on input a outputs the entry aij of M (pn , . . . , p1 ; u0 · · · uj−1 auj+1 · · · um ). Then fij is a permutation that depends only on uj−i · · · uj−1 (and not on any of the data: p1 , . . . , pn , u1 , . . . , uj−i−1 , uj+1 , . . . , um ).
ON A CLASS OF AUTOMATA GROUPS
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Proof. Formula (2.8) shows that each entry not belonging to the first row or column of the matrix depends only on the entry directly above it and the entry above it and one position to the left. From this it is immediate that fij (a) depends only on uj−i · · · uj−1 . That fij is a permutation follows since each row is obtained from the previous by applying an automorphism of the rooted Cayley tree of A∗ . The following corollary is immediate. Corollary 2.6. The entries aij with i ≤ j depend only on uj−i · · · uj . The next lemma is a simple fact from combinatorics on words. Lemma 2.7. Suppose X is an alphabet with at least two elements. Let ≡ be a congruence on X ∗ such that, for u, v ∈ X ∗ : u ≡ v, |u| = |v| ⇒ u = v.
Then ≡ is the trivial congruence.
Proof. Suppose u ≡ v; if |u| = |v|, we are done, so assume without loss of generality that |u| < |v|. Let a 6= b ∈ X. Then uav ≡ vau and ubv ≡ vbu. Since all these words have equal length, it follows from our assumptions that uav = vau. But since |u| < |v|, this says that ua is a prefix of v. Similarly, ubv = vbu and so ub is a prefix of v. But this contradicts a 6= b, completing the proof. We may now proceed to the proof of Theorem 2.2. Proof of Theorem 2.2. Suppose S(A) is not free on {Aq | q ∈ Q}. By Lemma 2.7 (using |Q| > 1 and A reduced), there must be a non-trivial relation of the form (2.9)
A p n · · · A p 1 = A qn · · · A q1
holding in S(A). Since S(A) is cancellative (being embeddable in G(A)), we may assume without loss of generality that p1 6= q1 . Let u = u0 · · · un ∈ A∗ be a word and set A = M (Apn , . . . , Ap1 ; u),
B = M (Aqn , . . . , Aq1 ; u)
as per (2.7). By Corollary 2.6, aij = bij for i ≤ j and they depend only on u0 · · · uj . We view the matrices associated to any prefix of u as submatrices of A and B. We prove by induction on i that for all i ≥ 0, one can choose letters u0 , . . . , ui such that e aj+1,j 6= ebj+1,j for all j = 0, . . . , i. ep 6= λ eq , there is a letter u0 ∈ A such that λ ep (u0 ) 6= λ eq (u0 ). Since λ 1 1 1 1 Setting a00 = u0 = b00 , we have e e e ^ e a10 = p^ 1 ◦ a00 = λp1 (u0 ) 6= λq1 (u0 ) = q1 ◦ b00 = b10 .
Suppose, inductively, we have chosen u0 , . . . , ui so that e aj+1,j 6= ebj+1,j for 0 ≤ j ≤ i. By assumption, the modified state functions are distinct, so there exists c ∈ A such that
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P. V. SILVA AND B. STEINBERG
eea ee λ (c) 6= λ (c) i+1,i bi+1,i
(2.10)
Recalling (2.8) and using Corollary 2.6, we have ai+2,i+1 = e ai+1,i ◦ ai+1,i+1
bi+2,i+1 = ebi+1,i ◦ bi+1,i+1 ai+1,i+1 = bi+1,i+1 .
(2.11)
Lemma 2.5 says that, for fixed u0 · · · ui , the function fi+1,i+1 obtained from assigning to a choice of ui+1 ∈ A the entry ai+1,i+1 = bi+1,i+1 is a permutation. Thus we can choose ui+1 so that ai+1,i+1 = c = bi+1,i+1 . But, by (2.10), (2.11) and the definition of the modified state functions (2.5), eea ee e ai+2,i+1 = λ (c) 6= λ (c) = ebi+2,i+1 , i+1,i bi+1,i
as required. To complete the proof, recall from Lemma 2.4 that an0 · · · ann = Apn · · · Ap1 (u)
(2.12)
bn0 · · · bnn = Aqn · · · Aq1 (u)
But e an,n−1 6= ebn,n−1 implies an,n−1 6= bn,n−1 . We conclude from (2.12) that (2.9) does not hold – a contradiction – proving S(A) is free on the Aq , q ∈ Q. 2.1. Examples of Reset Automata. We now provide examples to show what can happen when the hypotheses of Theorem 2.2 fail. Of course an invertible reset automaton with a single state generates a finite group. More generally, if all resets go to the same state, the group will be finite; this will follow from Proposition 2.8 below, which gives an easy condition forcing the group to be finite. We begin with an illustrative example:
1 | 2, 2 | 1, 3 | 3
0|0 a
1 | 2, 2 | 3, 3 | 1
b
0|0 Figure 4. A reset automaton generating a finite group.
ON A CLASS OF AUTOMATA GROUPS
9
Let A be the automaton in Figure 4; abusing notation, set a = Aa , ea = 0 1 2 3 = λ eb and so Theorem 2.2 does b = Ab . One computes λ a b b b not apply. In wreath product coordinates, a = (12)(a, b, b, b),
b = (123)(a, b, b, b).
Notice that (12) and (123) centralize c = (a, b, b, b) ∈ Iso(Aω ). So the map a 7→ ((12), c), b 7→ ((123), c) gives an injective homomorphism from G to S3 × hci (where we view S3 ≤ S4 as the permutations fixing 0). Therefore, it will follow that G(A) is finite if we can show that c has finite order. But, a6 = (12)6 c6 = c6 b6 = (123)6 c6 = c6 c6 = (a6 , b6 , b6 , b6 ) = (c6 , c6 , c6 , c6 ) so c6 = 1. It is easy to see that ord(c) = 6, so G(A)
1. Introduction ˙ Grigorchuk and Zuk [6] showed that the lamplighter group Z/2Z wr Z can be constructed as the automata group of the following automaton:
1|1
0|0 0
1|0 1
0|1 Figure 1. Automaton generating the lamplighter group. They used this automaton to compute the spectrum and spectral measure associated to random walks on the lamplighter group [6], leading to a Date: September 21, 2003. 1991 Mathematics Subject Classification. 20F32,20E08,20F10,20M35. Key words and phrases. Automata groups, Cayley machines, reset automata, free semigroups, lamplighter groups. The authors were supported in part by the FCT and POCTI approved project POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by INTAS project 99–1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science. The first author was supported in part by FCT through the Centro de Matem´ atica da Universidade do Porto. The second author also gratefully acknowledges the support of NSERC. 1
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P. V. SILVA AND B. STEINBERG
counterexample to the strong form of the Atiyah conjecture [4]. Notice that each input letter resets the automaton to a particular state: 0 takes each state to the state labelled 0 and 1 takes each state to the state labelled 1. In this paper we study groups of automatic transformations generated by invertible Mealy automata all of whose inputs are resets, that is, so-called reset automata. Under mild conditions – say the resets are to distinct states and the automaton is reduced, for instance – we show that the semigroup of automatic transformations generated by a reset automaton is free on the set of states; this easily allows one to construct automata generating a free semigroup of any finite rank. Moreover, the group generated by such an automaton is either finite or locally-finite-by-infinite cyclic and hence amenable. Recall that a group is locally finite if its finitely generated subgroups are finite. It is well known that the class of amenable groups is closed under extensions and direct limits and contains all finite and Abelian groups; hence each locally-finite-by-cyclic group is amenable. Thus the groups generated by reset automata are similar in nature to lamplighter groups (and the analogy can be made more precise). In [10], Krohn and Rhodes associate to each finite monoid an automaton called its Cayley machine (see also [3]). These automata play a fundamental role in Krohn-Rhodes theory [8, 9, 10], which studies the relationship between decomposition of transformations computed by finite state automata and wreath product decompositions of finite semigroups. This paper is inspired by the idea of using Krohn-Rhodes theory to understand better groups generated by finite state automata. In particular, reset automata form the bottom of a natural hierarchy of counter-free (=aperiodic) automata [10, 3]. If G is a finite group, then the Cayley machine of G is an invertible automaton whose inverse is a reset automaton. We show that if G is an Abelian group, then the group generated by its Cayley machine is the restricted wreath product G wr Z. In particular, any lamplighter group Z/nZ wr Z can be generated by a reset automata: the inverse of the Cayley machine of Z/nZ. Our construction of Z/nZ wr Z via automata thus constructs a set of n generators that generate a free subsemigroup. The authors, together with M. Kambites [7], have used these automata to compute the spectra of such lamplighters along the lines of [6], where the Cayley Machine of Z/2Z is used (see Figure 6); a different approach to the spectra of wreath product groups can be found in [2].
2. Automata and Reset Automata A (finite state Mealy) automaton [3, 5] A is a 4-tuple A = (Q, A, δ, λ) where Q is a finite set of states, A is an alphabet, δ : Q × A → Q is the transition function and λ : Q × A → A is the output function. We use the notation: δ(q, a) = q · a, λ(q, a) = q ◦ a = λq (a).
ON A CLASS OF AUTOMATA GROUPS
3
The functions λq : A → A, q ∈ Q, are called the state functions. One draws
(2.1)
a |b
q0
q
to mean q · a = q 0 , q ◦ a = b. That is, in state q with input a, the automaton outputs b and goes to state q 0 . The automaton is said to be invertible if each state function is a permutation. In this paper, all automata considered (except the Cayley machine of a non-group monoid) will be invertible. The inverse automaton is the automaton A−1 obtained by switching the input a and the output b on each arrow (2.1). The resulting transition in the inverse automaton is
(2.2)
b |a
q0
q
Given any state q ∈ Q, one obtains (sequential) functions [3, 5] (abusing notation) Aq : A∗ → A∗ and Aq : Aω → Aω defined by Aq (1) = 1,
Aq (a0 · · · an ) = λq (a0 )Aq·a0 (a1 · · · an ),
Aq (a0 a1 · · · ) = lim Aq (a0 · · · an ). n→∞
If A is invertible, then Aq is invertible [5]. Suppose λ0 ≥ λ1 ≥ λ2 . . . is a decreasing sequence of positive real numbers converging to 0. Then the topology of Aω can be defined by the following ultra-metric: for u, v ∈ Aω , d(u, v) = λn where n is the length of the largest common prefix of u and v. With this metric each Aq is a contraction [5, 11] and if A is invertible, then Aq is an isometry with inverse A−1 [5]. We q denote the group of isometries of Aω by Iso(Aω ); this group is isomorphic to the group of automorphisms of the standard rooted |A|-ary tree [1, 5]. The automaton A is called reduced if the functions Aq , q ∈ Q, are distinct. Since we view words from A∗ as acting on the right of Q and use notation such as q · a and q ◦ a, it seems natural to view Aq as acting on the left of A∗ (resp. Aω ); indeed, it is more convenient to write Aq (a0 a1 a2 · · · ) than (a0 a1 a2 · · · )Aq . Thus we view (Iso(Aω ), Aω ) as a left transformation group. There is a well-known [5, 1] isomorphism of left transformation groups: (2.3)
(Iso(Aω ), Aω ) ∼ = (S|A| , A) o (S|A| , A) o · · · ∼ = (S|A| , A) o (Iso(Aω ), Aω )
where o is the wreath product of left transformation groups; see also [11]. We remark that [5, 1] use right actions, so our notation is dual to theirs. Let |A| = n; then an element of the rightmost wreath product in (2.3) is of the form σ(f1 , . . . , fn ) where σ ∈ Sn and f1 , . . . , fn ∈ Iso(Aω ). Multiplication is given by σ(f1 , . . . , fn )τ (g1 , . . . , gn ) = στ (fτ (1) g1 , . . . , fτ (n) gn )
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P. V. SILVA AND B. STEINBERG
If A is an automaton, then in wreath product coordinates: Aq = λq (Aq·a1 , . . . , Aq·an )
(2.4)
where A = {a1 , . . . , an }. Moreover, if σ ∈ Sn , then
σ −1 Aq σ = σ −1 λq σ(Aq·σ(a1 ) , . . . , Aq·σ(an ) ).
The semigroup of automatic transformations generated by the automaton A, which we shall call the automata semigroup of A, is the semigroup S(A) generated by {Aq | q ∈ Q}. If A is invertible, then the group of automatic transformations generated by the automaton A, which we shall call the automata group of A, is the group G(A) generated by {Aq | q ∈ Q}. If A = (Q, A, δ, λ) is an automaton, then a ∈ A is called a reset if |Q · a| = 1; that is a resets the automaton to state Q · a. One says that A is a reset automaton if each letter is a reset. It is straightforward to see that two states q, q 0 of a reset automaton A are equivalent (i.e. Aq = Aq0 ) if and only if λq = λq0 , so A is reduced if and only if the state functions λq are distinct. The following notation will be convenient throughout the paper for dealing with reset automata. Define eq (a) = λ] ◦ a, a ∈ A, q ∈ Q (2.5) e a = Q · a, λ q (a) = qg
eq , q ∈ Q, the modified state functions. For example, in Figure 1 We call the λ eq = λq . Notice that if the state functions we have e 0 = 0, e 1 = 1 and so λ associated to states p and q are the same, then the modified state functions associated to the states are also the same – said differently, if two states have distinct modified state functions, then they have distinct state functions; in particular if the modified state functions are all distinct, then the reset automaton must be reduced. As another example, consider the automaton in Figure 2 below. Here e 0 = a, e 1=b=e 2 and so we calculate eb = 0 1 2 . ea = 0 1 2 , λ λ b b a a b b ea = 0 1 2 = λ eb . For the automaton in Figure 3 below, one checks λ b b a
1 | 1, 2 | 2
0|0 a
1 | 2, 2 | 0 b
0|1 Figure 2. A reset automaton with distinct modified state functions.
ON A CLASS OF AUTOMATA GROUPS
1 | 1, 2 | 0
0|2 a
5
1 | 2, 2 | 0 b
0|1 Figure 3. A reset automaton with non-distinct modified state functions generating a free semigroup. The following fact about computing with reset automata is a straightforward exercise that we leave to the reader. Lemma 2.1. Let A = (Q, A, δ, λ) be a reset automaton. Then, for q ∈ Q, Aq (a0 · · · an ) = (q ◦ a0 )(e a0 ◦ a1 )(e a1 ◦ a2 ) · · · (e an−1 ◦ an )
where the a0 , . . . , an are arbitrary letters from A.
Notice, in particular, that only the first letter of the output actually depends on the state q. In fact, in wreath product coordinates, using (2.4) (2.6)
Aq = λq (Aq·a1 , . . . , Aq·an ) = λq (Aea1 , . . . , Aean )
where A = {a1 , . . . an }. The following theorem is one of the main result of this paper. Theorem 2.2. If A = (Q, A, δ, λ) is an invertible reset automaton with |Q| > 1 and with distinct modified state functions, then S(A) is a free semigroup on {Aq | q ∈ Q}. This occurs, in particular, if A is reduced and the resets are to distinct states. Notice that the condition in the theorem can be checked in linear time. For example, the automaton in Figure 2 satisfies the conditions of Theorem 2.2 and hence generates a free semigroup on two generators. The automaton from Figure 1 also satisfies the conditions of Theorem 2.2 and so generates a free semigroup of rank 2; this particular case was proved by ˙ Grigorchuk and Zuk [6, 5]. The automaton in Figure 3 does not meet the criteria of the theorem; nonetheless, we shall see later that it, too, generates a free semigroup. Theorem 2.2 gives an easy way to construct an automaton A so that S(A) is free on any prescribed number of generators n. Indeed, choose n distinct permutations from Sn : σ1 , . . . , σn . The states and alphabet of A are {1, . . . , n}; input i is, of course, a reset to state i. We set λi = σi . The resulting automaton clearly satisfies the hypotheses of Theorem 2.2; more natural constructions shall occur in Section 4. We mention the following corollary of our theorem.
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Corollary 2.3. Let A be an automaton satisfying the conditions of Theorem 2.2. Then G(A) has exponential growth. In particular, if A is a reduced reset automaton with at least two states and, moreover, the resets are to distinct states, then G(A) has exponential growth. Proof. It is well known that a group containing a free subsemigroup on two generators has exponential growth. The proof of Theorem 2.2 will require some preparation; we begin with some notation. Given pn , . . . , p1 ∈ Q and u ∈ A+ with |u| = m + 1, we define an (n + 1) × (m + 1) matrix with entries in A a00 a01 · · · a0m a10 a11 · · · a1m (2.7) M (pn , . . . , p1 ; u) = .. .. . . . . . an0 an1 · · · an,m recursively by the following Pascal Array type rules: a00 · · · a0m = u ai0 = pi ◦ ai−1,0 (2.8) aij = e ai−1,j−1 ◦ ai−1,j
where i = 1, . . . , n, j = 1, . . . m. For example, if we use the automaton from Figure 1, and u = 011, then, 0 1 1 1 1 0 M (1, 0, 1; 011) = 1 0 1 0 1 1
It will turn out, as the next lemma shows, that the ith row of our array (2.7) is the image of u under Api · · · Ap1 . We now proceed to establish several useful facts about these matrices. Lemma 2.4. For all i = 0, . . . , n, ai0 · · · aim = Api · · · Ap1 (u). Proof. The case i = 0 being trivial, assume that i > 0 and that the result holds for i − 1. Then, using Lemma 2.1 and (2.8), we obtain Api · · · Ap1 (u) = Api (Api−1 · · · Ap1 (u)) = Api (ai−1,0 · · · ai−1,m ) = (pi ◦ ai−1,0 )(e ai−1,0 ◦ ai−1,1 ) · · · (e ai−1,m−1 ◦ ai−1,m ) = ai0 ai1 · · · aim ,
Set u = u0 u1 · · · um with ui ∈ A. The following lemma is an observation about the structure of our Pascal-type Array. Lemma 2.5. Let i ≤ j and consider the function fij : A → A which on input a outputs the entry aij of M (pn , . . . , p1 ; u0 · · · uj−1 auj+1 · · · um ). Then fij is a permutation that depends only on uj−i · · · uj−1 (and not on any of the data: p1 , . . . , pn , u1 , . . . , uj−i−1 , uj+1 , . . . , um ).
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Proof. Formula (2.8) shows that each entry not belonging to the first row or column of the matrix depends only on the entry directly above it and the entry above it and one position to the left. From this it is immediate that fij (a) depends only on uj−i · · · uj−1 . That fij is a permutation follows since each row is obtained from the previous by applying an automorphism of the rooted Cayley tree of A∗ . The following corollary is immediate. Corollary 2.6. The entries aij with i ≤ j depend only on uj−i · · · uj . The next lemma is a simple fact from combinatorics on words. Lemma 2.7. Suppose X is an alphabet with at least two elements. Let ≡ be a congruence on X ∗ such that, for u, v ∈ X ∗ : u ≡ v, |u| = |v| ⇒ u = v.
Then ≡ is the trivial congruence.
Proof. Suppose u ≡ v; if |u| = |v|, we are done, so assume without loss of generality that |u| < |v|. Let a 6= b ∈ X. Then uav ≡ vau and ubv ≡ vbu. Since all these words have equal length, it follows from our assumptions that uav = vau. But since |u| < |v|, this says that ua is a prefix of v. Similarly, ubv = vbu and so ub is a prefix of v. But this contradicts a 6= b, completing the proof. We may now proceed to the proof of Theorem 2.2. Proof of Theorem 2.2. Suppose S(A) is not free on {Aq | q ∈ Q}. By Lemma 2.7 (using |Q| > 1 and A reduced), there must be a non-trivial relation of the form (2.9)
A p n · · · A p 1 = A qn · · · A q1
holding in S(A). Since S(A) is cancellative (being embeddable in G(A)), we may assume without loss of generality that p1 6= q1 . Let u = u0 · · · un ∈ A∗ be a word and set A = M (Apn , . . . , Ap1 ; u),
B = M (Aqn , . . . , Aq1 ; u)
as per (2.7). By Corollary 2.6, aij = bij for i ≤ j and they depend only on u0 · · · uj . We view the matrices associated to any prefix of u as submatrices of A and B. We prove by induction on i that for all i ≥ 0, one can choose letters u0 , . . . , ui such that e aj+1,j 6= ebj+1,j for all j = 0, . . . , i. ep 6= λ eq , there is a letter u0 ∈ A such that λ ep (u0 ) 6= λ eq (u0 ). Since λ 1 1 1 1 Setting a00 = u0 = b00 , we have e e e ^ e a10 = p^ 1 ◦ a00 = λp1 (u0 ) 6= λq1 (u0 ) = q1 ◦ b00 = b10 .
Suppose, inductively, we have chosen u0 , . . . , ui so that e aj+1,j 6= ebj+1,j for 0 ≤ j ≤ i. By assumption, the modified state functions are distinct, so there exists c ∈ A such that
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eea ee λ (c) 6= λ (c) i+1,i bi+1,i
(2.10)
Recalling (2.8) and using Corollary 2.6, we have ai+2,i+1 = e ai+1,i ◦ ai+1,i+1
bi+2,i+1 = ebi+1,i ◦ bi+1,i+1 ai+1,i+1 = bi+1,i+1 .
(2.11)
Lemma 2.5 says that, for fixed u0 · · · ui , the function fi+1,i+1 obtained from assigning to a choice of ui+1 ∈ A the entry ai+1,i+1 = bi+1,i+1 is a permutation. Thus we can choose ui+1 so that ai+1,i+1 = c = bi+1,i+1 . But, by (2.10), (2.11) and the definition of the modified state functions (2.5), eea ee e ai+2,i+1 = λ (c) 6= λ (c) = ebi+2,i+1 , i+1,i bi+1,i
as required. To complete the proof, recall from Lemma 2.4 that an0 · · · ann = Apn · · · Ap1 (u)
(2.12)
bn0 · · · bnn = Aqn · · · Aq1 (u)
But e an,n−1 6= ebn,n−1 implies an,n−1 6= bn,n−1 . We conclude from (2.12) that (2.9) does not hold – a contradiction – proving S(A) is free on the Aq , q ∈ Q. 2.1. Examples of Reset Automata. We now provide examples to show what can happen when the hypotheses of Theorem 2.2 fail. Of course an invertible reset automaton with a single state generates a finite group. More generally, if all resets go to the same state, the group will be finite; this will follow from Proposition 2.8 below, which gives an easy condition forcing the group to be finite. We begin with an illustrative example:
1 | 2, 2 | 1, 3 | 3
0|0 a
1 | 2, 2 | 3, 3 | 1
b
0|0 Figure 4. A reset automaton generating a finite group.
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Let A be the automaton in Figure 4; abusing notation, set a = Aa , ea = 0 1 2 3 = λ eb and so Theorem 2.2 does b = Ab . One computes λ a b b b not apply. In wreath product coordinates, a = (12)(a, b, b, b),
b = (123)(a, b, b, b).
Notice that (12) and (123) centralize c = (a, b, b, b) ∈ Iso(Aω ). So the map a 7→ ((12), c), b 7→ ((123), c) gives an injective homomorphism from G to S3 × hci (where we view S3 ≤ S4 as the permutations fixing 0). Therefore, it will follow that G(A) is finite if we can show that c has finite order. But, a6 = (12)6 c6 = c6 b6 = (123)6 c6 = c6 c6 = (a6 , b6 , b6 , b6 ) = (c6 , c6 , c6 , c6 ) so c6 = 1. It is easy to see that ord(c) = 6, so G(A)
