Representations of a Free Group of Rank Two by Time-varying Mealy Automata

In the group theory various representations of free groups are used. A representation of a free group of rank two by the so-called time-varying Mealy automata over the changing alphabet is given. Two different constructions of such automata are presented.


Introduction
The theory of Mealy automata and groups generated by them is a part of geometric group theory which describes groups acting on a rooted tree.The study of the groups of automata was initiated in 70th years by several outstanding mathematicians (mainly by R. Grigorchuk).It has rapidly expanded in recent years and now plays an important role in algebra, theory of dynamical systems, spectral theory, ergodic theory and others (see [2]- [4], [7], [8]).
The Mealy automata considered so far had the same structure at every moment of a discrete time-scale (the so-called fixed Mealy automata).

A. Woryna
A concept of a time-varying Mealy automata over a changing alphabet was introduced for the first time in paper [9].In papers [10], [11], it is shown that time-varying Mealy automaton is a useful tool to describing groups acting on level homogeneous rooted trees which are not homogeneous.
One of key problems in the theory of groups of automata is the problem of embeddability of other known classes of groups into these groups.This problem was solved positively for free groups.The first example of such an automaton representation of a free group was suggested in [1].However, the complete proof is still unpublished.An example with the complete proof was first presented in [5].In paper [6] a representation of a free group of rank two by infinite unitriangular matrices is regarded as a group of transformations generated by fixed Mealy automata over a two-letter alphabet.In this paper we describe a representation of a free group of rank two by time-varying Mealy automata.Two different constructions will be presented.

Time-varying Mealy automata and groups generated by them
Let N 0 = {0, 1, 2, . ..} be a set of nonnegative integers.A changing alphabet is an infinite sequence where X t are nonempty, finite sets (sets of letters).A word over the changing alphabet X is a finite sequence x 0 x 1 . . .x l , where x i ∈ X i for i = 0, 1, . . ., l.
We denote by X * the set of all words (including the empty word ∅).By |w| we denote the length of the word w ∈ X * .The set of words of the length t we denote by X (t) .For any t ∈ N 0 we also consider the set X (t) of finite sequences in which the i-th letter (i = 1, 2, . ..) belongs to the set X t+i−1 .

Definition 2.1 . A time-varying Mealy automaton is a quintuple
where: Representations of a free group of rank two by ...
4. ϕ = (ϕ t ) t∈N 0 is a sequence of transitions functions of the form 5. ψ = (ψ t ) t∈N 0 is a sequence of output functions of the form We say that an automaton It is convenient to present a time-varying Mealy automaton as a labelled, directed, locally finite graph with vertices corresponding to the inside states of the automaton.For every t ∈ N 0 and every letter x ∈ X t an arrow labelled by x starts from every state q ∈ Q t and is going to the state ϕ t (q, x).Each vertex q ∈ Q t is labelled by the corresponding state function (1) σ t,q : X t → Y t , σ t,q (x) = ψ t (q, x).
To make the graph of the automaton clear, the sets of vertices V t and V t corresponding to the sets Q t and Q t respectively, are disjoint whenever t = t (in particular, different vertices may correspond to the same inside state).Moreover, we will substitute a large number of arrows connecting two fixed states and having the same direction for a one multi-arrow labelled by suitable letters and if the labelling of such a multi-arrow is obvious we will omit this labelling.
Example of a time-varying automaton.
In the Figure the state functions 1 and σ t constitute respectively the identity function and the cyclical permutation (0, 1, . . ., m t − 1) of X t .
A time-varying automaton may be interpreted as a machine, which being at a moment t ∈ N 0 in a state q ∈ Q t and reading on the input tape a letter x ∈ X t , goes to the state ϕ t (q, x), types on the output tape the letter ψ t (q, x), moves both tapes to the next position and then proceeds further to the next moment t + 1.
The automaton A with a fixed initial state q ∈ Q 0 is called the initial automaton and is denoted by A q .The above interpretation defines a natural action of A q on the words.Namely, the initial automaton A q defines a function f A q : X * → Y * as follows: , where the sequence q 0 , q 1 , . . ., q l of inside states is defined recursively: (2) Representations of a free group of rank two by ...

123
The function f A q is called the automaton function defined by A q .The image of a word w = x 0 x 1 . . .x l under a map f A q can be easily found using the graph of the automaton.One must find a directed path starting in a vertex q ∈ Q 0 and with consecutive labels x 0 , x 1 , . . ., x l .Such a path will be unique.If σ 0 , σ 1 , . . ., σ l are the labels of consecutive vertices in this path, then the word f A q (w) is equal to σ 0 (x 0 )σ 1 (x 1 ) . . .σ l (x l ).In the set of words over a changing alphabet, we consider for any k ∈ N 0 the equivalence relation ∼ k as follows: is called a remainder of f on the word w or simply a w-remainder of f .Definition 2.3 .Let A = (Q, X, Y, ϕ, ψ) be a time-varying Mealy automaton.For any t 0 ∈ N 0 the automaton A| t 0 = (Q , X , Y , ϕ , ψ ) defined as follows If f is generated by the initial automaton A q and the word w = x 0 x 1 . . .x l , then the w-remainder f w is an automaton function generated by the automaton B q l , where B = A| l and the initial state q l is obtained from (2).

A. Woryna
Definition 2.4 .An automaton A in which input and output alphabets coincide and every its state function σ t,q : X t → X t is a permutation of X t is called a permutational automaton.
If A is a permutational automaton, then for every q ∈ Q 0 the transformation f A q defines a permutation of X * .The set SA(X) of automaton functions defined by all initial automata over a common input and output alphabet X forms a monoid with the identity function as the neutral element.The subset GA(X) of functions generated by permutational automata is a group of invertible elements in SA(X).The group GA(X) is an example of residually finite groups (see [10]).Definition 2.5 .Let A = (Q, X, X, ϕ, ψ) be a time-varying permutational automaton.The group of the form is called the group generated by automaton A.
For any permutational automaton A, the group G(A) is residually finite, as a subgroup of GA(X).It turns out that groups of this form include the class of finitely generated residually finite groups.

Theorem 2.2 ([10]
).For any n-generated residually finite group G, there is an n-state time-varying automaton A such that G ∼ = G(A).

The embedding into the permutational wreath product
Let X = (X t ) t∈N 0 be a changing alphabet and let G be any subgroup of GA(X).For any i ∈ N 0 we consider the group which is a group generated by w-remainders of functions from G on all words w ∈ X (i) .In particular If we additionally assume that A is accessible, that is every state of A may be obtained from the recurrence (2) for some initial state q ∈ Q 0 and some word w = x 0 x 1 . . .x l , then the equality G i = G(A| i ) holds for every i ∈ N 0 .
Proposition 3.1 .For any f, g ∈ SA(X) and any word w ∈ X * , we have On the other hand what gives (3) from the previous equality.The formula (4) follows by substitution of f for g −1 in (3).
Let us arrange the letters of X i in the sequence: x 0 , x 1 , . . ., x m−1 .
Proposition 3.2 .The group G i embeds into the permutational wreath product G i+1 X i S(X i ) by the mapping where σ g ∈ S(X i ) is defined by the equality σ g (x) = g(x).
P roof.The mapping Ψ is one-to-one, what follows from the equalities g(xu) = σ g (x)g x (u) for x ∈ X i and u ∈ X (i+1) .By Proposition 3.1, we have: Hence Ψ is a homomorphism.

Representations of a free group of rank two by time-varying Mealy automata
In this chapter we describe a representation of a free group of rank two by time-varying Mealy automata.Two different constructions of such automata will be presented.
The first construction gives a representation by a 2-state automaton.It uses the following reverse order relation ≺ among freely reduced group words in symbols a, b: and w 1 , w 2 first differ (counting from the right side) in their k-th terms, then the order of these words depends on their k-th terms.
The crucial point of this construction constitute two permutations a, b of the set N with the following property: if w is a group word in a and b on the l-th position (l = 1, 2, . ..) in the above ordering, then the permutation of N defined by w maps the number 1 into l.The permutations a, b may be defined by the following formulas: A. Woryna and the mappings āt : are any bijections.It is not hard to see that a t , b t ∈ S(X t ).In particular, the automaton A is permutational.The graph of this automaton is presented in Figure 2. Theorem 4.1 .The group G(A) generated by the functions f A 0 and f A 1 is a free group of rank two which is freely generated by these functions.

P roof. The generators f A
0 and f A 1 map any word For every n ∈ N we have: for t large enough.Indeed, if a group word which derives from w by substitution of all f A 0 for a and of all f A 1 for b is on the l-th position (in the above ordering), then l = 1 and the last letter of g (11 . . .
is equal to l for t large enough.
The construction of the automaton B is quite different from the automaton A. In particular, the automaton B is not finite.On the other hand, the labelling of its inside states is quite straight.Namely, every state function σ t,q is a power of a cyclical permutation α t .
We consider for any i ∈ N 0 the remainders a i and b i of the functions f B 0 and f B 1 respectively, on the word 00 . . .0 of the length i.Let G i = G(B| i ) be a group generated by an i-remainder of the automaton B. From the graph of B, we see that The embedding of G i into the permutational wreath product G i+1 X i S(X i ) is induced by the following equations: (5) Let x * = x 0 x 1 . . .x l−1 be any word over Y and let g x 0 , g x 0 x 1 , . . ., g x 0 x 1 ...x l−1 be remainders of g on the consecutive beginnings of x * .The remainder g x 0 ...x j ∈ G i+j+1 (j = 0, 1, . . .l − 1) is represented by some freely reduced group word w x 0 ...x j in the symbols a i+j+1 , b i+j+1 : Using the equations 5, we may derive w x 0 ...x j from w x 0 ...x j−1 (from w if j = 0) in the following way: (i) if x j = 0, then every syllable of the form a s i+j is substituted for a s i+j+1 and every syllable of the form b r i+j is substituted for b r i+j+1 , (ii) if x j = 0, then every syllable of the form a s i+j is substituted for a s i+j+1 and every syllable of the form b r i+j is substituted for b r i+j+1 or -in case of α s i+j (x j ) = i + j + 1 -for a r(i+j) i+j+1 , where s is the sum of all exponents on a i+j -syllables on the left of b r i+j .
Thus w x 0 ...x j is a freely reduction of the word derived from w x 0 ...x j−1 by the rules (i) and (ii).The word w x * is called an x * -remainder of w.The rules (i) and (ii) define the action of G i on the set Y * as follows: where S j is the sum of all exponents on a i+j -syllables in w x 0 ...x j−1 .
Theorem 4.2 .The group G(B) generated by the functions f B 0 and f B 1 is a free group of rank two which is freely generated by these functions.P roof.We show that for every i ∈ N 0 the group G i is freely generated by a i , b i .Let be any nonempty, freely reduced group word in a i , b i and let N (w) be the number of b i -syllables in w.We prove by induction on N (w) that w does not define the neutral element in G i .To this, we assume that for every j ∈ N 0 any nonempty, freely reduced group word v in a j , b j with N (v) < N (w) does not define the neutral element in G j .
If N (w) ≤ 1, then w = a s 1 i (s 1 = 0) or w = a s 1 i b r 1 i a s 2 i (r 1 = 0).In this case we easily check that none of the above words defines the neutral element in G i .Let, now assume N (w) > 1.Then k > 1.Let us denote:

A. Woryna
We consider the remainders w y * and w x * of w on sequences: We may derive w x * from w y * by substitution of every its a i+l -syllable for an appropriate a i+l+1 -syllable and every b i+l -syllable for b i+l+1 -syllable or else a i+l+1 -syllable -according to the rules (i) and (ii).The substitution of any b i+l -syllable for a i+l+1 -syllable we call simply as a i+l+1 -substitution.
There are no two consecutive syllables b i+l in w y * for which the a i+l+1 -substitutions hold.Otherwise α s j+1 i+l (i + l + 1) = i + l + 1 and since, as s j+1 = 0, we have consequently i+l is any subword in w y * such that the a i+l+1 -substitution holds for b r j i+l , then this subword will be substituted for a s i+l+1 in w x * , where s = s j + r j (i + l) + s j+1 .Since As a result of the above observation, we obtain that w x * is nonempty.Moreover, for the syllable b r k−1 i+l in w y * the a i+l+1 -substitution holds.As a result we have N (w x * ) < N (w).
By inductive assumption, w x * does not define the neutral element in G i+l+1 .As a consequence, w does not define the neutral element in G i .
and v have a common prefix of the length k.Let X and Y be changing alphabets and let f be a function of the form f : X * → Y * .If f preserves the relation ∼ k for any k, then we say that f preserves beginnings of the words.If |f (w)| = |w| for any w ∈ X * , then we say that f preserves lengths of the words.

Theorem 2 . 1 (
[9]).The function f : X * → Y * is an automaton function iff it preserves beginnings and lengths of the words.Definition 2.2 .Let w ∈ X * be a word of the length n = |w|.The function f w : X (n) → Y (n) defined by the equality

Figure 2 .
Figure 2. The automaton A which generates a free group of rank two.

Figure 3 .
Figure 3.The automaton B which generates a free group of rank two.