© Journal Algebra and Discrete Mathematics. We add to the classification of groups generated by 3-state automata over a 2-letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic

Davis, N

Elder, M

Reeves, L

Algebra and Discrete Mathematics

English

OPUS - University of Technology Sydney

Non-contracting groups generated by (3,2)-automata

We add to the classification of groups generated by 3-state automata over a 2-letter alphabet given by Bondarenko et al., by showing that a number of the groups in the classification are non-contracting. We show that the criterion we use to prove a self-similar action is non-contracting also implies that the associated self-similarity graph introduced by Nekrashevych is non-hyperbolic

Davis, N.

Elder, M.

Reeves, L.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 17 (2014). Number 1. pp. 20 – 32c© Journal “Algebra and Discrete Mathematics”Non-contracting groups generated by(3,2)-automataNick Davis, Murray Elder and Lawrence ReevesCommunicated by R. I. GrigorchukAbstract. We add to the classification of groups generatedby 3-state automata over a 2-letter alphabet given by Bondarenkoet al., by showing that a number of the groups in the classificationare non-contracting. We show that the criterion we use to prove aself-similar action is non-contracting also implies that the associatedself-similarity graph introduced by Nekrashevych is non-hyperbolic.IntroductionIn [1] a list of automaton groups generated by 3-state automata overa 2-letter alphabet is given and a great deal of information is listed foreach. Amongst the data given for each group was whether the group wascontracting or non-contracting. For ten automata the classification did notdetermine whether or not the group was contracting. In the numberingsystem of [1, page 17] the ten automata are:749, 861, 882, 887, 920, 969, 2361, 2365, 2402, 2427.Later Muntyan [3] showed that three of these are conjugate to othergroups in the classification, specifically 920 ∼= 2401, 2361 ∼= 939, and2365 ∼= 939.The first author was supported by an Australian Postgraduate Award. The secondand third authors are supported by Australian Research Council grants FT110100178,DP120100996 and DP1096912.2010 MSC: 20E08, 20F65, 68Q45.Key words and phrases: automaton group, self-similar group, contracting action,self-similarity graph.N. Davis, M. Elder, L. Reeves 21The purpose of this note is to show that all of the automaton groupslisted above are non-contracting. We first establish a criterion for a groupto be non-contracting, and then apply it in each case.We refer to [4] (Section 1.5) for the basic definitions of self-similaractions and automaton groups. A faithful action of a group G on the setof words X∗ over a finite alphabet X is called self-similar if for everyx ∈ X, g ∈ G there exist y ∈ X,h ∈ G so that g(xw) = yh(w) for allw ∈ X∗. The element h is called the restriction of g to x and denoted g |x.For g ∈ G and a finite word v ∈ X∗, the restriction of g to v, denoted g |v,is the element of G determined by the condition:g(vw) = g(v)g |v (w)for all w ∈ X∗. We will make use of the following basic properties:(gh) |v= g |h(v) h |v and g |uv= (g |u) |v .We denote by Xω the set of infinite words over X. The length of aword v ∈ X∗ is denoted |v|. The set X∗ is naturally the vertex set of abinary rooted tree and Xω corresponds to the set of ends of that tree.The action of G on X∗ determines an action of G on Xω. The group ofall automorphisms of the rooted tree is denoted Aut(Xω).A self-similar action of a group G is called finite-state if the set{g |v : v ∈ X∗}is finite for every g ∈ G. An automaton group is a finite-state self-similaraction of a group generated by the states of a finite state transducer withtransitions from state g to state h labeled x|y if g(x) = y and g |x= h forall x ∈ X.A (3, 2)-automaton group is an automaton group generated by anautomaton with three states and a 2-letter alphabet X = {0, 1}. We labelthe states a, b, and c. The automata are represented by a Moore diagram,which is given below for each automaton (see for example Figure 1).Recall the following definition from [4].Definition 1. A self-similar group G 6 Aut(Xω) is called contracting ifthere exists a finite subset N ⊆ G such that for all g ∈ G there existsk ∈ N such that g |v∈ N for all v ∈ X∗ with |v| > k. The minimal N iscalled the nucleus of the action.22Non-contracting groups generated by (3,2)-automataNote that if a self-similar group is not finite-state, then it is non-contracting.We make use of the following criterion, which was used in [1] to showthat 744 is non-contracting, and many times after that.Lemma 1. Let G 6 Aut(Xω) be a self-similar action. Suppose that thereexist g ∈ G and v ∈ X∗ such that:1) g |v= g,2) g(v) = v,3) g has infinite order.Then G is non-contracting.Proof. Assume for induction that g |vk= g and g(vk) = vk for k ≥ 1. Theng |vk+1= g |vkv= (g |vk) |v= g |v= g and g(vk+1) = g(v)g |v (vk) = vg(vk) =vvk.Next assume for induction that gn |vk= gn for n ≥ 1 and fixed k. Thengn+1 |vk= g |gn(vk) gn |vk= g |vk gn = ggn.It follows that a nucleus must contain gn for infinitely many n and so,since g has infinite order, the action is not contracting.Alternatively, though less directly, the lemma follows from Theorem 2below and Theorem 3.8.6 of [4].In the next section we apply this criterion to the ten automata listedabove. In Section 2 we prove that a self-similar group satisfying thiscriterion has a non-hyperbolic self-similarity graph.The authors would like to thank Dima Savchuk for helpful conversa-tions, and the anonymous referee for helpful corrections and suggestions.1. The automataTheorem 1. The automaton groups generated by the automata749, 861, 882, 887, 920, 969, 2361, 2365, 2402, 2427in [1] are non-contracting.Proof. For each automaton group we give an element g ∈ G and a wordv ∈ {0, 1}∗ with the (easily verifiable) property that g(v) = v and g |v= g.The Moore diagram of the automaton is given for reference. States thatact non-trivially on {0, 1} are shaded in the diagram. We then proveN. Davis, M. Elder, L. Reeves 23that g has infinite order, so that the criterion of Lemma 1 applies. Theapproach to showing that g has infinite order is to find another string v′that is not fixed by any power of g. We found the candidates for suitableelements and strings using some simple computer code and observingvarious patterns.It is convenient to introduce the equivalence relation on {0, 1}ω givenby left shift equivalence, that is, u ∼ v if there are finite prefixes u′ and v′of u and v respectively, and w ∈ {0, 1}ω such that u = u′w and v = v′w.For a finite word u ∈ {0, 1}∗, we denote by u∞ the element of {0, 1}ωformed by repeating u infinitely many times.Automaton 749We have a2bc(0100) = 0100 and (a2bc) |0100= a2bc.acb1|00|11|10|00|01|1Figure 1. Automaton 749To see that g = a2bc has infinite order we consider the string 0∞.Observe that since g |000= babc, babc(000) = 101, and (babc) |000= babc,we have g(0∞) = 001(101)∞. Then note that a |101= b |101= c |101= aand a4(101) = 101. It follows that for any n > 1, gn(0∞) = un(101)∞where u1 = 001 and un = g(un−1101). In other words gn(0∞) is left-shiftequivalent to (101)∞. We now note that g−1(0∞) is not of this form,which establishes that g has infinite order. Observe that:g−1 |0000= a−1b−1a−2, g−1(0000) = 0011,a−1b−1a−2 |0000= a−1b−1a−2, a−1b−1a−2(0000) = 1011.Therefore g−1(0∞) = 0011(1011)∞.24Non-contracting groups generated by (3,2)-automataAutomaton 861We have c(010) = 010 and c |010= c.acb0|11|01|10|00|01|1Figure 2. Automaton 861Since x |11= b for any x ∈ {a, b, c} and b(1∞) = 1∞, it follows thatcn(1∞) ∼ 1∞ for any n > 0. But c−1(1∞) = (10)∞, so c has infinite order.Automaton 882We have acacbc(11) = 11 and (acacbc) |11= acacbc.acb1|0, 0|10|01|10|01|1Figure 3. Automaton 882To show that g = acacbc has infinite order we claim that:1) g2n(02n+1) = 02n+1,2) g2n|02n+1= cacb,3) cacb(0∞) = 110∞.from which it follows that g2n(0∞) = 02n+1110∞ for all n > 1.We prove the first and second claims by induction on n. Note firstthat b2 is the identity in the group, as can be seen from the automatonfor b2 in Figure 4.N. Davis, M. Elder, L. Reeves 25a2c2b21|1, 0|00|01|10|01|1Figure 4. Action of b2 in the group 882We have g(0) = 0 and g |0= cacbbb = cacb. Then inductively,g2n+1(02n+3) = g2ng2n(02n+100)= g2n(02n+1cacb(00)) = g2n(02n+111)= 02n+1cacb(11) = 02n+100,g2n+1|02n+3 = (g2ng2n) |02n+3= g2n |g2n (02n+3) g2n |02n+3= g2n|02n+111 g2n |02n+3= (g2n |02n+1) |11 (g2n |02n+1) |00= (cacb) |11 (cacb) |00= bbcacbbb = cacb.For the third claim observe that cacb(00) = 11, cacb |00= cb3, andcb3(0∞) = 0∞. Thencacb(00∞) = cacb(00)(cacb) |00 (0∞) = 11cb3(0∞) = 110∞.Automaton 887We have bc(00) = 00 and (bc) |00= bc.acb0|11|00|0,1|10|01|1Figure 5. Automaton 887To establish that bc has infinite order we show that for all n > 1,(bc)n(1∞) 6= 1∞ as follows.Since bc(1) = 1 and (bc) |1= ca, we have (bc)n(1∞) = 1(ca)n(1∞) andit suffices to show that (ca)n(1∞) 6= 1∞. We show that for n > 2:1) (ca)4n(111) = 111 and (ca)4n(110) = 110,26Non-contracting groups generated by (3,2)-automata2) (ca)2n|111= (ca)2n−1 ,3) (ca)2n(1∞) = (111)n−1(1010)1∞ .It’s clear that the third claim implies that (ca)n(1∞) 6= 1∞ for alln > 1.The first claim follows from (ca)4(111) = 111 and (ca)4(110) = 110.For the second claim, note first that a, b and c all have order 2, ascan be seen from the automaton for 〈a2, b2, c2〉 in Figure 6a2c2b21|10|00|01|10|01|1Figure 6. Action of the elements a2, b2, c2 in the group 887Then (ca) |111= aa = 1 and (ca) |110= bb = 1. Also,(ca)2 |111= (ca) |ca(111) (ca) |111= (ca) |011 (ca) |111= ca,and(ca)4 |111= (ca)2 |(ca)2(111) (ca)2 |111= (ca)2 |101 ca = caaaca = caca.Inductively, for n > 3,(ca)2n|111 = ((ca)2n−1(ca)2n−1) |111= (ca)2n−1 |(ca)2n−1(111)(ca)2n−1|111= (ca)2n−1|111 (ca)2n−2 = (ca)2n−2(ca)2n−2= (ca)2n−1.For the third claim, note that ca(1∞) = 0(bb)(1∞) = 01∞ and(ca)4(1∞) = (ca)4(111)(ca)4 |111 (1∞) = 111(ca)2(1∞)= 111101(ca)2 |111 (1∞) = 111101(ca)(1∞) = 11110101∞.Then for n > 3(ca)2n(1∞) = (ca)2n(111)(ca)2n|111 (1∞) = 111(ca)2n−1(1∞)= 111(111)n−1(1010)1∞ = (111)n(1010)1∞.N. Davis, M. Elder, L. Reeves 27Automaton 920We have b(1) = 1 and b |1= b.acb0|11|00|01|11|10|0Figure 7. Automaton 920Since a−1 |1= b−1 |1= b−1 and b−1(1) = 1, it follows that b−n(01∞) ∼1∞. But b(01∞) = 0∞, so b has infinite order.Automaton 969We have c(0) = 0 and c |0= c.acb0|11|00|01|11|10|0Figure 8. Automaton 969To show that c has infinite order we claim that for n > 1,cn((101)∞) ∼{(100)∞ n even,(011)∞ n odd.To see this, note that c((101)∞) = 11c((110)∞) = 11(100)∞. If u ∼(100)∞, then c(u) ∼ (011)∞. If u ∼ (011)∞, then c(u) ∼ (100)∞. Both28Non-contracting groups generated by (3,2)-automatastatements follow from the observation that for any generator x ∈ {a, b, c},x |10= c.Finally, observe that c−1((101)∞) = 1∞, which proves that c hasinfinite order.Automaton 2361We have c(0) = 0 and c |0= c.acb0|11|0 0|11|01|10|0Figure 9. Automaton 2361Observe that a(0∞) = 10∞ and c(0∞) = 0∞. Therefore, for all n > 0,cn(10∞) ∼ 0∞. Also, c−1(10∞) = 1∞. It follows that c has infinite order.Automaton 2365We have c(0) = 0 and c |0= c.acb1|00|1 0|11|01|10|0Figure 10. Automaton 2365N. Davis, M. Elder, L. Reeves 29To see that c has infinite order, observe that a−1(0∞) = 10∞ andc−1(0∞) = 0∞. Therefore, for all n > 0, c−n(10∞) ∼ 0∞. As c(10∞) = 1∞,it follows that c has infinite order.Automaton 2402We have c(0) = 0 and c |0= c.acb1|00|11|00|11|10|0Figure 11. Automaton 2402Note that cn(10∞) ∼ 0∞ since x |00= c for any x ∈ {a, b, c}. Howeverc−2(10∞) = 101∞. Therefore c has infinite order.Automaton 2427We have c(0) = 0 and c |0= c.acb0|11|01|00|11|10|0Figure 12. Automaton 242730Non-contracting groups generated by (3,2)-automataTo see that c has infinite order note that a((101)∞) = 01(101)∞,b((101)∞) = 00(101)∞ and c((101)∞) = 11(101)∞. Therefore, for alln > 1, cn((101)∞ ∼ (101)∞. However, c−2((101)∞) = (100)∞.2. Non-hyperbolic self-similarity graphsNekrashevych introduced the notion of a self-similarity graph of aself-similar action. He proved that if a self-similar group is contracting,the corresponding self-similarity graph (endowed with the natural metric)is hyperbolic [4, Theorem 3.8.6]. The converse to this result is open.Here we provide a partial converse to this fact, which applies to self-similar actions that satisfy the criterion of Lemma 1. We know of noautomaton group that is non-contracting and doesn’t satisfy the condition.An example (suggested by the referee) of a self-similar group that is non-contracting but does not satisfy the criterion is the infinite cyclic groupgenerated by the element a ∈ Aut(Xω) determined by a(0) = 1, a(1) = 0,a |0= a and a |1= a2. This group is not finite-state. It can be shown thatthe self-similarity graph of this example is not hyperbolic.Definition 2 ([4] Defn. 3.7.1). The self-similarity graph Σ(G,S,X) ofa self-similar group G with generating set S acting on X∗ is the graphwith vertex set X∗ and an edge {u, v} whenever:• u = s(v) for some s ∈ S — these are the horizontal edges,• u = xv for some x ∈ X — these are the vertical edges.Observe that horizontal edges connect strings in X∗ of the same length,and vertical edges connect strings that differ in length by 1.We use the characterization of hyperbolic geodesic metric spacesinvolving the divergence of geodesics, see [2, p.412].Definition 3. Let Y be a geodesic metric space. A function e : N → Ris called a divergence function if for all y ∈ Y , for all R, r ∈ N and forall geodesics α : [0, a] → Y and β : [0, b] → Y with α(0) = β(0) = y,a > R + r and b > R + r the following holds: if dY (α(R), β(R)) > e(0)then any path from α(R+ r) to β(R+ r) that stays outside the open ballof radius R+ r about y has length at least e(r).Proposition 1 ([2, p.412]). Let Y be a geodesic metric space. Then Yis hyperbolic if and only if it admits an exponential divergence function.N. Davis, M. Elder, L. Reeves 31Theorem 2. Let G be a self-similar group with finite generating set Sacting on X∗, and suppose that there exist g ∈ G and v ∈ X∗ such that:1) g |v= g,2) g(v) = v,3) g has infinite order.Then the self-similarity graph Σ(G,S,X) is non-hyperbolic.Proof. The vertex in Σ(G,S,X) corresponding to the empty string islabelled ∅. A vertex in the open ball based at ∅ of radius N corresponds toa string in X∗ of length less than N . Note that an element of X∗ uniquelydefines a vertical geodesic emanating from ∅ whose length is equal tothat of the word. Considering such geodesics, we show that Σ(G,S,X)does not admit an exponential divergence function, and is therefore nothyperbolic.Suppose for a contradiction that e : N → R is a divergence functionfor Σ(G,S,X) and that it is increasing and unbounded. If the maximumsize of an orbit of any w ∈ X∗ under g was N , then gN !(w) = w for allw ∈ X∗. Since g has infinite order, it follows that there are arbitrarilylarge orbits under its action on X∗. Vertices in Σ(G,S,X) have uniformlybounded degree. It follows that there is a bound on the number of verticesin any metric ball of fixed radius, so we can choose n ∈ N and w ∈ X∗such that dΣ(w, gn(w)) > e(0). More explicitly, choose w so that its orbitunder the action of g has size greater than the number of vertices in anyball in Σ(G,S,X) of radius e(0).For an element g ∈ G, denote by ‖g‖S the length of g with respect tothe word metric on G determined by the generating set S. For all k ∈ Nthe vertices vkw and gn(vkw) = vkgn(w) are connected by a horizontalpath of length exactly n‖g‖S , and this path lies outside the open ballof radius |vkw| centered at ∅. Choose k ∈ N such that e(|vk|) > n‖g‖S .Since e is a divergence function, any horizontal path connecting vkw andvkgn(w) must have length at least e(|vk|). This contradiction establishesthe result.References[1] Ievgen Bondarenko, Rostislav Grigorchuk, Rostyslav Kravchenko, Yevgen Muntyan,Volodymyr Nekrashevych, Dmytro Savchuk, and Zoran Šunić. On classiﬁcation ofgroups generated by 3-state automata over a 2-letter alphabet. Algebra DiscreteMath., (1):1–163, 2008.32Non-contracting groups generated by (3,2)-automata[2] Martin R. Bridson and André Haeﬂiger. Metric spaces of non-positive curvature,volume 319 of Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.[3] Yevgen Muntyan. Automata Groups. PhD thesis, Texas A&M University, 2009.[4] Volodymyr Nekrashevych. Self-similar groups, volume 117 of Mathematical Surveysand Monographs. American Mathematical Society, Providence, RI, 2005.Contact informationN. Davis Department of Mathematics and StatisticsUniversity of MelbourneParkville VIC 3010, AustraliaE-Mail: nickd@unimelb.edu.auM. Elder School of Mathematical and Physical SciencesThe University of NewcastleCallaghan NSW 2308, AustraliaE-Mail: murrayelder@gmail.comURL: sites.google.com/site/melderauL. Reeves Department of Mathematics and StatisticsUniversity of MelbourneParkville VIC 3010, AustraliaE-Mail: lreeves@unimelb.edu.auURL: www.ms.unimelb.edu.au/∼reevesReceived by the editors: 29.11.2013and in final form 22.12.2013.

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Non-contracting groups generated by (3,2)-automata

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