An Example of an Automatic Graph of Intermediate Growth
aa r X i v : . [ m a t h . G R ] F e b An Example of an Automatic Graph of IntermediateGrowth
Alexei Miasnikov ∗ Department of Mathematical Sciences,Stevens Institute of Technology,Castle Point, Hoboken, NJ, [email protected] Savchuk † Department of Mathematics and StatisticsUniversity of South Florida4202 E Fowler AveTampa, FL [email protected] 18, 2018
Abstract
We give an example of a 4-regular infinite automatic graph of intermediate growth.It is constructed as a Schreier graph of a certain group generated by 3-state automaton.The question was motivated by an open problem on the existence of Cayley automaticgroups of intermediate growth.
Introduction
Automatic groups were formally introduced by Thurston in 1986 motivated by earlier resultsof Cannon [6] on properties of Cayley graphs of hyperbolic groups. The latter results, inturn, were motivated by the pioneering work of Dehn on word problem in surface groups.All automatic groups have solvable in a quadratic time word problem and have at mostquadratic Dehn function. If, in addition, a group is bi-automatic, then it has solvable ∗ Partially Supported by the Marsden Fund of The Royal Society of New Zealand. † Partially Supported by the New Researcher Grant from University of South Florida. BS ( p, q ) (unless p = 0, q = 0, or p = ± q ), non-finitely presented groups, infinite torsiongroups, SL n ( Z ). This would be desirable to extend the class of automatic groups to somewider class while preserving the computational routines of automatic groups. Further, someof the very basic questions about the class of automatic groups have still not been solveddespite considerable efforts by the mathematical community. For example, it is not knownwhether each automatic group is bi-automatic.In view of the above arguments it was quite natural to search for possible generalizationsof the class of automatic groups. Several papers offered different approaches. Combablegroups share with automatic groups the fellow traveler property, but have weaker constraintson the language used in the definition. Bridson in [3] discusses the relation between thesetwo classes. The geometric generalization of the class of automatic groups, so-called, asyn-chronously indexed-combable groups, was defined and studied by Gillman and Bridson in [4].It uses indexed languages and covers the fundamental groups of all compact 3-manifold sat-isfying the geometrization conjecture. Unfortunately, this class looses certain importantalgorithmic features of automatic groups. Recently Brittenham and Hermiller [5] definedanother related class of stackable groups. They show, in particular, that every shortlex au-tomatic group, including every word hyperbolic group, is regularly stackable, and that eachstackable group is finitely presented. The exact relationship between these two classes is notyet fully understood.The notion of a Cayley automatic group was introduced and studied in [12] as a naturalgeneralization of the class of automatic groups. It has been observed that the Cayley graphsof automatic groups are automatic with respect to special encoding, in the sense of thetheory of automatic structures developed, in particular, by Khoussainov and Nerode [13].This theory can be traced back to works of Hodgson in the end of 1970’s – beginning of1980’s [10]. For a survey on the results in this theory we refer the reader to a paper byRubin [17]. A natural way to generalize the notion of automatic groups would be to removethe condition on the encoding on Cayley graphs. In other words, a group is called Cayleyautomatic, if its Cayley graph is automatic.The class of Cayley automatic groups retains many algorithmic properties of the classof automatic groups, but is much wider. In particular, it includes many examples of nilpo-tent and solvable groups, which are not automatic in the standard sense. Some of Cayleyautomatic groups are not finitely presented. For example, the restricted wreath product ofa nontrivial finite group G by Z is Cayley automatic. Further, it was recently shown byMiasnikov and ˇSuni´c in [14] that there exist Cayley automatic groups that are not Cayleybiautomatic, thus resolving an analogue of a longstanding question of the theory of auto-matic groups. At the same time, main algorithmic tools of automatic groups still work. Inparticular, the word problem in each Cayley automatic group can be decided in a quadratic2ime.Even further generalization of Cayley automatic groups, was recently introduced andstudied by Elder and Taback in [7]. For each class of languages C they define C -graphautomatic groups in exactly the same way as Cayley automatic groups with the differencethat the formal languages used in the definition must belong to class C . In particular, if C is the class of regular languages, one simply obtains the class of Cayley automatic groups.One of the motivations to consider other classes of languages is the fact proved in [7] thatpolynomial time word problem algorithm is still preserved if one replaces the class of regularlanguages by the class of counter languages.This paper was motivated by the following natural question regarding possible limitationsof the class of Cayley automatic groups. Question 1.
Is there a Cayley Automatic group of intermediate growth?
Recall that the growth function of a finitely generated group G with respect to a gener-ating set S is a function γ G,S : N → N such that γ G,S ( n ) is equal to the number of elementsof G that can be expressed as a product of at most n elements of S ∪ S − . More generally,the growth function γ Γ ,x ( n ) of a locally finite graph Γ with respect to the selected base point x is a function such that γ Γ ,x ( n ) is the number of elements in the ball of radius n in Γ cen-tered at the base point. The growth function γ G,S can then be defined as γ Cay(
G,S ) ,e , whereCay( G, S ) is the Cayley graph of G with respect to a generating set S and e is the identityelement in G . The growth function of any finitely generated group cannot grow faster thanthe exponential function and, according to Gromov’s celebrated theorem, γ G,S ( n ) grows asa polynomial function if and only if G is virtually nilpotent. It was a longstanding questionposed by Milnor if there is a group whose growth function is intermediate, i.e. it growsfaster than any polynomial function, but slower than the exponential function [15]. Thefirst example of such group was constructed by Grigorchuk in [9], but even up to now allknown constructions of such groups are based to certain extent on ideas from the originalconstruction in [9]. It is known that automatic groups cannot have intermediate growth,which is a limitation that might be overcame by passing to a bigger class. Also, there are noknown examples of finitely presented groups of intermediate growth, which makes the classof Cayley automatic groups particularly appealing. Unlike the class of automatic groups itcontains many groups that are not finitely presented.By the definition, a group is Cayley automatic if and only if its Cayley graph is automatic(i.e. admits automatic structure). The main purpose of this paper is the following theorem. Theorem 0.1.
There is an automatic graph of intermediate growth.
The graph that we use to prove the main theorem belongs to the family of graphs ofintermediate growth constructed in [2] as the family of Schreier graphs of the action of agroup generated by the two nontrivial states of the 3-state automaton depicted in Figure 1on the boundary of a binary rooted tree. One of the graphs in this family was constructedearlier by Benjamini and Hoffman in [1]. The last paper also gives credit to Bartholdi whopointed out that the graph under consideration was, in fact, a Schreier graph of a groupgenerated by automaton. 3he paper is organized as follows. In Section 1 we recall the main definitions related togroups generated by automata. The main example of a graph of intermediate growth Γ (01) ∞ and the group G acting on this graph is given in Section 2. Section 3 introduces the notionsof an automatic graph and of a Cayley automatic group. Finally, Section 4 contains theproof that the graph Γ (01) ∞ is automatic. Acknowledgement.
The authors are grateful to Thomas Colcombet for useful discussionsand to the anonymous referee whose valuable suggestions have enhanced the exposition ofthe paper and optimized some proofs.
Let X be a finite set of cardinality d and let X ∗ denote the set of all finite words over X (that can be though as the free monoid generated by X ). This set can be naturally endowedwith a structure of a rooted d -ary tree by declaring that v is adjacent to vx for any v ∈ X ∗ and x ∈ X . The empty word corresponds to the root of the tree and X n corresponds tothe n -th level of the tree. We will be interested in the groups of graph automorphisms andsemigroups of graph homomorphisms of X ∗ . Any such homomorphism can be defined viathe notion of initial automaton. Definition 1. A Mealy automaton (or simply automaton ) is a tuple ( Q, X, π, λ ) , where Q is a set (a set of states), X is a finite alphabet, π : Q × X → Q is a transition function and λ : Q × X → X is an output function. If the set of states Q is finite the automaton is called finite . If for every state q ∈ Q the output function λ ( q, x ) induces a permutation of X , theautomaton A is called invertible. Selecting a state q ∈ Q produces an initial automaton A q . Automata are often represented by the
Moore diagrams . The Moore diagram of anautomaton A = ( Q, X, π, λ ) is a directed graph in which the vertices are the states from Q and the edges have form q x | λ ( q,x ) −→ π ( q, x ) for q ∈ Q and x ∈ X . If the automaton is invertible,then it is common to label vertices of the Moore diagram by the permutation λ ( q, · ) andleave just first components from the labels of the edges. An example of Moore diagram isshown in Figure 1.Any initial automaton induces a homomorphism of X ∗ . Given a word v = x x x . . . x n ∈ X ∗ it scans its first letter x and outputs λ ( x ). The rest of the word is handled in a similarfashion by the initial automaton A π ( x ) . Formally speaking, the functions π and λ can beextended to π : Q × X ∗ → Q and λ : Q × X ∗ → X ∗ via π ( q, x x . . . x n ) = π ( π ( q, x ) , x x . . . x n ) ,λ ( q, x x . . . x n ) = λ ( q, x ) λ ( π ( q, x ) , x x . . . x n ) . By construction any initial automaton acts on X ∗ as a homomorphism and every invert-ible initial automaton acts on X ∗ as an automorphism. Definition 2.
The semigroup (group) generated by all states of an automaton A is calledan automaton semigroup ( automaton group ) and denoted by S ( A ) (respectively G ( A ) ). X ∗ can be encoded by the action of an initial automa-ton. In order to show this we need a notion of a section of a homomorphism at a vertex ofthe tree. Let g be a homomorphism of the tree X ∗ and x ∈ X . Then for any v ∈ X ∗ wehave g ( xv ) = g ( x ) v ′ for some v ′ ∈ X ∗ . Then the map g | x : X ∗ → X ∗ given by g | x ( v ) = v ′ defines a homomorphism of X ∗ and is called the section of g at vertex x . Furthermore, forany x x . . . x n ∈ X ∗ we define g | x x ...x n = g | x | x . . . | x n . Given a homomorphism g of X ∗ we construct an initial automaton A ( g ) whose actionon X ∗ coincides with that of g as follows. The set of states of A ( g ) is the set { g | v : v ∈ X ∗ } of different sections of g at the vertices of the tree. The transition and output functions aredefined by π ( g | v , x ) = g | vx ,λ ( g | v , x ) = g | v ( x ) . Throughout the paper we will use the following convention. If g and h are the elementsof some (semi)group acting on set A and a ∈ A , then gh ( a ) = h ( g ( a )) . (1)Taking into account convention (1) one can compute sections of any element of an au-tomaton semigroup as follows. If g = g g · · · g n and v ∈ X ∗ , then g | v = g | v · g | g ( v ) · · · g n | g g ··· g n − ( v ) . (2)For any automaton group G there is a natural embedding G ֒ → G ≀ Sym( X )defined by G ∋ g ( g , g , . . . , g d ) λ ( g ) ∈ G ≀ Sym( X ) , where g , g , . . . , g d are the sections of g at the vertices of the first level, and λ ( g ) is apermutation of X induced by the action of g on the first level of the tree.The above embedding is convenient in computations involving the sections of automor-phisms, as well as for defining automaton groups. Sometimes it is called the wreath recursion defining the group.Finally, we note that any homomorphism of X ∗ induces an action on the set X ∞ of allinfinite words over X that can be viewed as a boundary of the tree X ∗ .5Sfrag replacements 1 / / / / / , / / / ab e Figure 1: Automaton generating group G The main graph studied in this paper is a Schreier graph of a certain group G defined below.We start this section from recalling the definition of a Schreier graph. Definition 3.
Let G be a group generated by a finite generating set S acting on a set M . The (orbital) Schreier graph Γ( G, S, M ) of the action of G on M with respect to the generatingset S is an oriented labeled graph defined as follows. The set of vertices of Γ( G, S, M ) is M and there is an arrow from x ∈ M to y ∈ M labeled by s ∈ S if and only if x s = y , where x s denotes the image of x under the action of s . We will call a Schreier graph with a selectedbasepoint a pointed Schreier graph.
An equivalent view on Schreier graphs goes back to Schreier, who called these graphs coset graphs . For any subgroup H of G , the group G acts on the right H -cosets G/H byright multiplication. This action gives rise to the Schreier graph Γ(
G, S, G/H ). Conversely, if G acts on M transitively, then Γ( G, S, M ) is canonically isomorphic to Γ(
G, S, G/
Stab G ( x ))for any x ∈ M , where the vertex y ∈ M in Γ( G, S, M ) corresponds to the coset from G/ Stab G ( x ) consisting of all elements of G that move x to y . Also, to simplify notation, wewill refer to Γ( G, S, G/
Stab G ( x )) as the Schreier graph of x and denote it by Γ x when thegroup, the set, and the action are clear from the context.Consider a group G generated by two nontrivial states of a 3-state automaton A over2-letter alphabet X = { , } defined by the following wreath recursion a = ( e, a ) σ,b = ( b, a ) , where e denotes the identity of G and σ is a nontrivial permutation of { , } . The Moorediagram of this automaton is shown in Figure 1.This group acts on the boundary { , } ∞ of a tree { , } ∗ and this action induces anuncountable family of pointed orbital Schreier graphs Γ ω for each ω ∈ { , } ∞ . Namely, Γ ω is an orbital Schreier graph of the action of G on the orbit of ω with respect to the generatingset S = { a, b } with the basepoint ω . This family of graphs was completely described in [2]and we borrow our notation from this paper.The structure of Γ ω is as follows. The vertices of Γ ω are identified with integers and theset of edges E ω consists of countably many families E nω , n ≥ Figure 2: Graph Γ (01) ∞ at one of the vertices of Γ ω . The family E ω is defined as E ω = { ( n, n + 1) : n ∈ Z } . Each successive E nω , n > ω = x x x . . . and x n = 0, then let c ωn bethe largest nonpositive integer that is not the endpoint of any of the edges in E ω , . . . , E n − ω .If x n = 1, then let c ωn be the smallest positive integer that is not the endpoint of any of theedges in E ω , . . . , E n − ω . The family E nω is now defined as E nω = { (2 n z + c ωn , n ( z + 1) + c ωn ) : z ∈ Z } . By construction, if there are both infinitely many 0’s and infinitely many 1’s in ω , then eachvertex in Γ ω will be adjacent to exactly 4 edges in ∪ n ≥ E nω . In this case we simply have E ω = ∪ n ≥ E nω . If there is only a finite number of 0’s or 1’s in ω , then all vertices in Γ ω except exactly onevertex t will have four adjacent edges in ∪ n ≥ E nω , while t will be an endpoint of only twoedges from E ω . In this case E ω = { loop at t } ∪ ( ∪ n ≥ E nω ) . In particular, graph Γ (01) ∞ is shown in Figure 2.The following theorem has been proved in [2]:7 heorem 2.1 ([2]) . All orbital Schreier graphs Γ ω for ω ∈ { , } ∞ of the group G haveintermediate growth. More specifically, the growth function satisfies n log n (cid:22) | B ( ω, n ) | (cid:22) n log n The above theorem is a generalization of an earlier result of Benjamini and Hoffman [1]who, in particular, proved that Γ ∞ has intermediate growth. Let X by a finite alphabet. For a special symbol ⋄ / ∈ X we define an extended alphabet X ⋄ = X ∪ {⋄} . For a pair ( w , w ) of finite words over X we define a convolution or a paddedpair (see, for example, [11]) ⊗ ( w , w ) to be the word over ( X ⋄ ) of length max {| w | , | w |} ,whose j -th symbol is ( σ , σ ), where σ i = (cid:26) the j -th symbol of w i , if j ≤ | w i |⋄ , otherwiseFor example, if X = { , } , then ⊗ (011 , (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) ⋄ (cid:19) (cid:18) ⋄ (cid:19) , where letters of ( X ⋄ ) are written for convenience as columns. We note that w and w canbe empty.Let R be a binary relation on X ∗ . The convolution of R is the language over ( X ⋄ ) defined by ⊗ R = {⊗ ( w , w ) | ( w , w ) ∈ R } . A binary relation R on X ∗ is called regular if its convolution ⊗ R is a regular languageover ( X ⋄ ) , i.e. ⊗ R is recognizable by a finite automaton acceptor over ( X ⋄ ) . To avoid pos-sible confusion we emphasize that the automata acceptors here are different from automatatransducers defined in Section 1.Now we proceed to the definition of automatic graphs and Cayley automatic groups.Let Γ = ( V, E, σ : E → S ) be graph whose edges are labeled by elements of finite set S = { s , s , . . . , s n } according to the map σ . This graph can be interpreted as a system of | S | binary relations E s on V , for s ∈ S , where E s = { ( v, v ′ ) | ( v, v ′ ) ∈ E and the label of ( v, v ′ ) is s } . Each map : V → X ∗ induces | S | binary relations E s on X ∗ given by E s = { ( v, v ′ ) | ( v, v ′ ) ∈ E s } . The definition of automatic graph below is a particular instance of an automatic struc-ture [13, 12]. 8 efinition 4.
The labeled graph
Γ = (
V, E, σ : E → S ) is called automatic , if there is afinite alphabet X and an injective map : V → X ∗ such that • V is a regular language over X and • E s is a regular binary relation on X ∗ for each s ∈ S .In such a case, the tuple ( V , E s , E s , . . . , E s k ) is called an automatic structure on graph Γ with respect to S = { s , s , . . . , s k } . A rich source of examples of labeled graphs comes from group actions. Let G be a finitelygenerated group with a finite generating set S . The (right) Schreier graph Γ( G, S, Y ) of theaction of G on Y is a graph, whose vertex set is Y , and for each y ∈ Y and s ∈ S there isan edge labelled by s from y to y s . The (right) Cayley graph of G can be thought of as aSchreier graph of the regular action of G on itself by multiplication on right.The definition of Cayley automatic groups from [12] is as follows: Definition 5.
A finitely generated group G with finite generating set S is Cayley automatic if its Cayley graph
Cay(
G, S ) with respect to S is automatic. We note that even though the property of being Cayley automatic depends only on agroup, and not on a finite generating set (see [12]), the same group can have both automaticand non-automatic Schreier graphs. For example, a group which is not Cayley automaticcertainly acts trivially on the one element set producing an automatic Schreier graph.
It is an open question whether there is a Cayley automatic group of intermediate growth.We do not answer this question here, however, we construct an automatic Schreier graph ofintermediate growth. The main purpose of this note is the following theorem.
Theorem 4.1.
The Schreier graph Γ (01) ∞ of intermediate growth is automatic. The proof of this theorem will be elaborated through the lemmas below. Throughoutthe proof we will denote Γ (01) ∞ simply by Γ. First, we produce an injection : V (Γ) → X ∗ .This amounts to labelling the vertices of Γ by different words over X . Since Γ is a Schreiergraph of the action of G on the orbit of (01) ∞ , the vertices of Γ are already labelled byinfinite words over X .Recall that two infinite words ω = x x x . . . and ω ′ = y y y . . . in X ∞ are called cofinal if there exist N > x n = y n for all n ≥ N . It is proved in [2] that the orbit of ω ∈ X ∞ coincides with the cofinality class of ω except the case if ω is cofinal to 0 ∞ or 1 ∞ ,when the orbit coincides with the union of cofinality classes of 0 ∞ and 1 ∞ . Therefore, in thecase of ω = (01) ∞ , each vertex of Γ is initially labelled by an infinite word over X that iscofinal with (01) ∞ . We define an injection : V (Γ) → X ∗ by sending each vertex v to theprefix of its label of length l with the property that l is the largest nonnegative integer such9Sfrag replacements i A V accepting V (Γ (01) ∞ )that the l -th digit of the label of v differs from the l -th digit in (01) ∞ . So, for example, wehave 0110(01) ∞ = 0110 , (01) ∞ = ∅ , where ∅ denotes the empty word over X . Lemma 4.2.
The set V (Γ) ⊂ X ∗ is a regular language over X .Proof. First of all, we observe that w = x x . . . x l ∈ V (Γ) if and only if either w = ∅ or thelast letter x l of w is different from the l -th letter of (01) ∞ .It is straightforward to verify now that V (Γ) is accepted by the automaton A V depictedin Figure 3, where the initial state is labelled by i and the terminal states are marked bydouble circles. Indeed, when automaton reads word w = x x . . . x l over X starting fromthe initial (top left) state, we can keep track of whether the last letter of w is different fromthe l -th lettter of (01) ∞ by looking at the state in which we end up after reading w . If weend up in one of the top two states, then w ends with the letter opposite to the l -th letterin (01) ∞ and is accepted by A V . On the contrary, if we end up in one of the bottom twostates, then w ends with the l -th letter in (01) ∞ and is not accepted by A V .Note that more generally, we can similarly define : V (Γ ω ) → X ∗ for any ω ∈ X ∞ . Inthe case of preperiodic ω the analog of Lemma 4.2 can be proved by constructing a similarautomaton. But we will not need this more general result here.Before proving that E a and E b are regular relations on X ∗ we prove the following auxiliarylemma. Lemma 4.3.
For every regular language L over X the language L pairs = {⊗ ( u, v ) | u, v ∈ L } is regular over ( X ⋄ ) .Proof. There is a natural one-to-one correspondence between finite words over ( X ⋄ ) andpairs of words over X ⋄ of the same length. Thus, for words u = x x . . . x n and v = y y . . . y n over X ⋄ of the same length we will sometimes denote by ( u, v ) a corresponding word over( X ⋄ ) whose j -th letter ( x j , y j ) for 1 ≤ j ≤ n .10Sfrag replacements (1 , ,
0) (0 ,
1) (0 , a ea a e e e e (1 , ⋄ )( ⋄ , ⋄ ,
1) (0 , ⋄ ) (0 , ⋄ )( ⋄ , A a accepting L a By definition of the convolution we have L pairs = L ∩ L ∩ L , where L = { ( u, v ) ∈ (( X ⋄ ) ) ∗ | u ∈ L ⋄ ∗ , v ∈ X ∗⋄ } ,L = { ( u, v ) ∈ (( X ⋄ ) ) ∗ | u ∈ X ∗⋄ , v ∈ L ⋄ ∗ } ,L = { ( u, v ) ∈ (( X ⋄ ) ) ∗ | ( u, v ) has no letter ( ⋄ , ⋄ )and has no subwords ( x, ⋄ )( y, z ) and ( ⋄ , x )( z, y ) , z ∈ X, x, y ∈ X ⋄ } . The languages L and L are regular. We can build an automaton A L over X ⋄ rec-ognizing L from the automaton A L over X with the state set Q recognizing L as follows.The set of states of A L is Q ∪ { t } , where t is a terminal state of A L not in Q . For eachtransition q x −→ q in A L for q , q ∈ Q and x ∈ X , we introduce | X | + 1 transitions of theform q x,y ) −→ q , y ∈ X ⋄ . Additionally, for each terminal state t ′ ∈ Q we introduce | X | + 1transitions of the form t ′ ( ⋄ ,y ) −→ t , y ∈ X ⋄ . The automaton recognizing L is constructedsimilarly.Since the language L is clearly regular over ( X ⋄ ) , we get that L pairs is regular as theintersection of three regular languages. Lemma 4.4.
For each s ∈ { a, b } the binary relation E s is regular over X .Proof. We have to prove that ⊗ E a and ⊗ E b are regular languages over ( X ⋄ ) . We willshow that both of these languages can be obtained as an intersection of a regular language L pairs with regular languages accepted by automata built by modifying the automaton A generating the group G .We start from ⊗ E a . Let L a be a regular language over ( X ⋄ ) recognized by an automaton A a shown in Figure 4, where the initial state is a and terminal states are marked by doublecircles. Of course, states e through e are equivalent, but we intentionally separate themto make the connection between automata A and A a more clear and to emphasize differentcases in the proof. We will show that ⊗ E a = L a ∩ L pairs , (3)11Sfrag replacements (1 , ,
1) (1 ,
0) (0 ,
1) (0 , , ab e Figure 5: Modified automaton ˜ A thus proving that ⊗ E a is regular language over ( X ⋄ ) .Recall that the vertices of Γ are labelled by infinite words over X cofinal with (01) ∞ . Wewill identify the vertices with their labels. By definition of : V (Γ) → X ∗ the preimage of v ∈ V (Γ) under is the vertex ξ v = x x x . . . , where x i = (cid:26) i -th letter of v, if i ≤ | v | , i -th letter of (01) ∞ , if i > | v | .Suppose ( u, v ) ∈ ⊗ E a for some u, v ∈ X ∗⋄ of the same length. Then there are words u ′ , v ′ ∈ V (Γ) such that ( u, v ) = ⊗ ( u ′ , v ′ ) and such that there is an edge in Γ labelled by a from ξ u ′ to ξ v ′ . Since ( u, v ) = ⊗ ( u ′ , v ′ ) by the definition of L pairs we immediately get that( u, v ) ∈ L pairs .We will now show that ( u, v ) is in L a , i.e. accepted by A a . By definition of the adjacencyin Γ, we can read an infinite sequence of pairs of letters in X corresponding to the pair ( ξ u ′ , ξ v ′ )of infinite words over X by following the transitions in automaton ˜ A over X depicted inFigure 5 starting from state a .Suppose first, that both u = x x . . . x k and v = y y . . . y k do not contain ⋄ , and, hence, u ′ = u and v ′ = v . In this case we can disregard all transitions in A a with labels containing ⋄ .After removing all such transitions and corresponding states from A a we get an automatonequivalent to automaton ˜ A with initial state a . In particular, we read the same words over X along paths in these automata. Therefore, we can read ( u, v ) along the path in A a . Thequestion is only if we end up in the accepting state of A a .Since the pair consisting of two empty words is not accepted by A a and is not in ⊗ E a , wecan assume that k >
0. Observe that both x k and y k must be different from the k -th letterin (01) ∞ since otherwise u or v would not be in V (Γ). In particular, we get that x k = y k .But this means that we have to be in the accepting state e of A a after reading ( u, v ) startingfrom state a . Therefore, ( u, v ) is accepted by A a .Now assume that | u | = | u ′ | > | v ′ | . Then v has a form y y . . . y l ⋄ k − l for some 0 ≤ l < k ,and the pair of infinite words ( ξ u , ξ v ′ ) can be read along the path in ˜ A .Note that by definition of : V (Γ) → X ∗ the letter x k of u is different from the k -thsymbol of (01) ∞ , which coincides with the k -th symbol of ξ v ′ . Therefore, while reading apair containing x k by ˜ A with initial state a we must be at state a . But since as soon as weleave state a we never come back, it follows that we must remain in state a after reading k first pairs of letters in ( ξ u , ξ v ′ ). Consequently, we will be in the state a after reading( x x . . . x l , y y . . . y l ) along the path in A a . 12Sfrag replacements (1 , ,
0) (0 , ,
0) (0 , , ab ea a e e e e e e e e a a a a b b (1 , ⋄ ) (1 , ⋄ )(1 , ⋄ ) (1 , ⋄ )(1 , ⋄ )( ⋄ ,
1) ( ⋄ , ⋄ ,
1) ( ⋄ , ⋄ , ⋄ ,
1) (0 , ⋄ )(0 , ⋄ ) (0 , ⋄ )( ⋄ , ⋄ , A b accepting L b l pairs of letters of ( u, v ) by A a , the next pair we read is ( x l +1 , ⋄ ).Consider two cases:1. If x l +1 = 0, then since we are in the state a in ˜ A after reading first l pairs, we shift tothe state e in ˜ A after reading the pair containing x l +1 . After this, the automaton ˜ A will accept only pairs of identical letters. In particular, we get that ξ u coincides ξ v , andthus with (01) ∞ by construction of ξ v , at positions starting from l + 2. Therefore, x l +1 must be different from the ( l + 1)-st letter in (01) ∞ . Hence, u = x x . . . x l
0, and afterreading ( u, v ) = ( x x . . . x l , y y . . . y l ⋄ ) starting from the state a the automaton A a will shift to the accepting state e and will accept ( u, v ).2. If x l +1 = 1, then while reading the ( l + 1)-st letter of ( u, v ) by automaton ˜ A we muststay at the state a , because we have to follow the arrow, the first coordinate of whoselabel is 1. But as there is just one arrow whose label has the first coordinate 1 goingout of state a , this determines uniquely the second coordinate of this label, which mustbe 0. Therefore, the ( l + 1)-st letter of ξ v ′ , and thus of (01) ∞ is 0. But this implies thatthe ( l + 2)-nd letter in (01) ∞ is 1. Note that this is precisely the only place where weneed that ω = (01) ∞ , because we need the next letter of ω to be completely determinedby the previous one, so we have to choose ω from 0 ∞ , 1 ∞ , (01) ∞ and (10) ∞ .Now if ( l + 2)-nd letter of (01) ∞ , and thus of ξ v ′ , is 1, in the automaton ˜ A we have tofollow the arrow going out of a state a , whose label’s second coordinate is 1. Thereis again exactly one such arrow, that ends up in the state e and whose label is (0 , x l +2 = 0, and ξ u and (01) ∞ coincide at positions l + 3 and higher. Therefore, u = x x . . . x l
10, and after reading ( u, v ) = ( x x . . . x l , y y . . . y l ⋄ ⋄ ) starting fromthe state a the automaton A a will shift to the accepting state e and will accept ( u, v ).The case when | v | = | v ′ | > | u ′ | is analogous. In this case after reading ( u, v ) the automa-ton A a will end up either in the state e or in the state e . Therefore, each word in ⊗ E a isaccepted by A a .Conversely, if a word ( u, v ) over ( X ⋄ ) is accepted by A a , then after reading this wordthe automaton has to shift to one of the five terminal states. Consider all cases separately:1. If we end up in the state e , then ( u, v ) = (1 n ⋄ ⋄ , n
01) for some n ≥
0. In the case n is even, the word 0 n
01 is not in V (Γ), and thus ( u, v ) / ∈ L pairs . If n is odd, then( u, v ) = ⊗ (1 n , n n and 0 n
01 are elements of V (Γ) corresponding tovertices ξ u = 1 n ∞ and ξ v = 0 n ∞ that are connected by the edge in Γ labelled by a , because one can read ( ξ u , ξ v ) alongthe path in the automaton ˜ A .2. The cases when we end up in states e , e and e are treated in the same way.14. If we end up in the state e , then both u and v do not contain ⋄ . We will be able toread ( ξ u , ξ v ) along the path in the automaton ˜ A . So there is an edge from ξ u to ξ v inΓ. Therefore, in this situation, ( u, v ) ∈ L pairs , if and only if ( u, v ) ∈ ⊗ E a .Thus, each word that is accepted by A a and is in L pairs must be in ⊗ E a . This finishesthe proof of the equality (3).Similarly to L a we define L b to be a regular language recognized by automaton A b de-picted in Figure 6. Similarly to the definition of an automaton A a , states e through e areequivalent, but we intentionally separate them to make the diagram of an automaton moreclear. The proof that ⊗ E b = L b ∩ L pairs , is analogous to the proof of equality (3). Proof of Theorem 4.1.
The main Theorem 4.1 now follows by the definition of an automaticgraph from Lemma 4.2 and Lemma 4.4.
References
References [1] Itai Benjamini and Christopher Hoffman. ω -periodic graphs. Electron. J. Combin. ,12:Research Paper 46, 12 pp. (electronic), 2005.[2] Ievgen Bondarenko, Tullio Ceccherini-Silberstein, Alfredo Donno, and VolodymyrNekrashevych. On a family of Schreier graphs of intermediate growth associated witha self-similar group.
European J. Combin. , 33(7):1408–1421, 2012.[3] Martin R. Bridson. Combings of groups and the grammar of reparameterization.
Com-ment. Math. Helv. , 78(4):752–771, 2003.[4] Martin R. Bridson and Robert H. Gilman. Formal language theory and the geometryof 3-manifolds.
Comment. Math. Helv. , 71(4):525–555, 1996.[5] Mark Brittenham and Susan Hermiller. Stackable groups, tame filling invariants, andalgorithmic properties of groups. Preprint: arXiv:1212.1230, 2011.[6] James W. Cannon. The combinatorial structure of cocompact discrete hyperbolicgroups.
Geom. Dedicata , 16(2):123–148, 1984.[7] Murray Elder and Jennifer Taback. C -graph automatic groups. J. Algebra , 413:289–319,2014. 158] 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 BartlettPublishers, Boston, MA, 1992.[9] R. I. Grigorchuk. Degrees of growth of finitely generated groups and the theory ofinvariant means.
Izv. Akad. Nauk SSSR Ser. Mat. , 48(5):939–985, 1984.[10] Bernard R. Hodgson. D´ecidabilit´e par automate fini.
Ann. Sci. Math. Qu´ebec , 7(1):39–57, 1983.[11] Derek F. Holt. Automatic groups, subgroups and cosets. In
The Epstein birthdayschrift , volume 1 of
Geom. Topol. Monogr. , pages 249–260 (electronic). Geom. Topol.Publ., Coventry, 1998.[12] Olga Kharlampovich, Bakhadyr Khoussainov, and Alexei Miasnikov. From automaticstructures to automatic groups.
Groups Geom. Dyn. , 8(1):157–198, 2014.[13] Bakhadyr Khoussainov and Anil Nerode. Automatic presentations of structures. In
Logic and computational complexity (Indianapolis, IN, 1994) , volume 960 of
LectureNotes in Comput. Sci. , pages 367–392. Springer, Berlin, 1995.[14] Alexei Miasnikov and Zoran ˇSuni´c. Cayley graph automatic groups are not necessarilyCayley graph biautomatic. In
Language and automata theory and applications , volume7183 of
Lecture Notes in Comput. Sci. , pages 401–407. Springer, Heidelberg, 2012.[15] J. Milnor. Problem 5603.
Amer. Math. Monthly , 75:685–686, 1968.[16] Volodymyr Nekrashevych.
Self-similar groups , volume 117 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, 2005.[17] Sasha Rubin. Automata presenting structures: A survey of the finite string case.