Automaton groups and complete square complexes

Transcription

1 rxiv: v1 [mth.gr] 1 Jul 2017 Automton groups nd complete squre complexes Ievgen Bondrenko nd Bohdn Kivv July 4, 2017 Astrct The first exmple of non-residully finite group in the clsses of finitely presented smll-cnceltion groups, utomtic groups, nd CAT(0) groups ws constructed y Wise s the fundmentl group of complete squre complex (CSC for short) with twelve squres. At the sme time, Jnzen nd Wise proved tht CSCs with t most three squres, five or seven squres hve residully finite fundmentl group. The smllest open cses were CSCs with four squres nd directed complete VH complexes with six squres. We prove tht the CSC with four squres studied y Jnzen nd Wise hs non-residully finite fundmentl group. In prticulr, this gives nonresidully finitecat(0) group isometric to F 2 F 2. For the clss of complete directed VH complexes, we prove tht there re exctly two complexes with six squres hving non-residully finite fundmentl group. In prticulr, this positively nswers to question of Wise on whether the min exmple from his PhD thesis is non-residully finite. As y-product, we get finitely presented torsion-free simple groups which decompose into n mlgmted free product of free groups F 7 F49 F 7. Our pproch relies on the connection etween squre complexes nd utomt discovered y Glsner nd Mozes, where complete VH complexes with one vertex correspond to ireversile utomt. We prove tht the squre complex ssocited to ireversile utomton with two sttes or over the inry lphet generting n infinite utomton group hs non-residully finite fundmentl group. We descrie utomton groups ssocited to CSCs with four squres nd get two simple utomton representtions of the free group F 2 nd the first utomton representtion of the free product C 3 C Mthemtics Suject Clssifiction: 20F65, 20M35, 20E08 Keywords: squre complex, ireversile utomton, residul finiteness, utomton group 1 Introduction One of the outstnding open prolems in geometric group theory is whether wordhyperolic groups re residully finite, which is importnt in understnding of the topology 1

2 c 0 c c c c , 1 1 Figure 1: The Wise complex W nd the Aleshin utomton of hyperolic spces. Before the introduction of word-hyperolic groups, Schupp nd lter Pride, Gersten, nd others sked out the residul properties of groups in neighorhood of word-hyperolic groups: for finitely presented smll-cnceltion groups, utomtic groups, CAT(0) groups. The first exmple of non-residully finite group elonging to ech of these clsses ws constructed y Wise in his disserttion [28]. Shortly fter tht Burger nd Mozes [4, 5] constructed finitely presented torsion-free simple groups. All these exmples re the fundmentl groups of complete squre complexes (non-positively curved squre complexes covered y the direct product of two trees). The construction of Wise is sed on specil squre complex W, which is clled the min exmple in [28, 30]. This complex is otined y gluing the six unit squres shown in Figure 1. Wise studied tiling properties of these squres nd proved tht they dmit specil non-periodic tiling of the plne with periodiclly leled xes (n nti-torus). This oservtion ws the key ingredient in severl interesting exmples (see [11, 12, 29]). In prticulr, Wise proved tht the sugroup of π 1 ( W ) generted y the loops,,c is not seprle, nd therefore the mlgmted free product π 1 ( W ),,c π 1 ( W ) is non-residully finite CAT(0) group. The question ws rised [30, Prolem 10.19] whether π 1 ( W ) itself is non-residully finite. Below we give positive nswer to this question s corollry of more generl sttement. Our pproch nd initil interest in squre complexes lie through the theory of utomton groups, which is nother fscinting topic of modern group theory. The theory of utomton groups dels with specil clss of utomt-trnsducers (see exmple in Figure 1). The reltion etween such utomt nd squre complexes ws discovered y Glsner nd Mozes in [8]. Let A e n utomton with the set of sttes S nd n inputoutput lphet X. The squre complex A ssocited to A is otined y gluing the squres given y the rrows in A s shown in Figure 2. The complex A elongs to the clss of directed VH squre complexes with one vertex. The fundmentl group of A hs 2

3 x n rrow s x y t in A produces unit squre s t. Figure 2: A leled unit squre ssocited to n rrow in n utomton y presenttion π 1 ( A ) = S,X sx = yt for ech rrow s x y t in A, which immeditely suggest tht there should e connection etween geometric properties of A nd the comintoril structure of A. Indeed, Glsner nd Mozes noticed tht A is CSC exctly when A elongs to the clss of ireversile utomt introduced in [18] in reltion to commensurtors of free groups. The ireversile utomton corresponding to the Wise complex W is shown in Figure 1. Another lgeric oject relted to n utomton is the ssocited utomton group. An utomton group G A is defined y the ction of n utomton A on input words. Roughlyspeking, groupisnutomtongroupifonecnputnutomtonstructureon the group consistent with its group structure. The study of utomton groups ws initilly motivted y severl exmples, minly the Grigorchuk group, tht enjoy mny fscinting properties: torsion, intermedite growth, menle ut not elementry menle, nonuniformly exponentil growth, finite width, just-infiniteness, etc. Further investigtions showed tht utomton groups nturlly rise in diverse res of mthemtics (see [9, 10, 20]). In this pper we show tht utomton groups re useful in the study of squre complexes s well. Theorem 9. Let A = (X,S,λ) e ireversile utomton. 1. If G A is finite, then π 1 ( A ) is virtully direct product of two free groups nd therefore residully finite. 2. If G A is infinite, then π 1 ( A ) is not S -seprle nd not X -seprle. Theorem 11. Let A = (X,S,λ) e ireversile utomton with two sttes or over the inry lphet. If G A is infinite, then π 1 ( A ) is non-residully finite. The key ingredient in the proof of Theorem 11 is nontrivil endomorphism of π 1 ( A ) tht is trivil on ll sttes or ll letters. The existence of such endomorphism follows from the fct tht the utomton group G A contins suutomton nontrivilly isomorphic to the originl utomton A. For exmple, for the Wise complex W the mp φ :,, c c,

4 x y y x 1, x 1 y 1, y 1 y x y, y y 1, y 1 x y x y y x 1 x 1 x x x x x 1 x 1 x x x y 1, y x, y 1 y 1 1 x 1 y, y 1 x 1, y y 1 Figure 3: A smllest CSC D with non-residully finite fundmentl group nd the ssocited utomton with G A = C3 C 3 nd G A = F2 extends to n endomorphism of π 1 ( W ) with Fix(φ) =,,c. Since π 1 ( W ) is not,,c -seprle, it is non-residully finite; the element (0 1 1) 4 elongs to the intersection of finite index sugroups of π 1 ( W ). Interestingly, tht the utomton ssocited to the Wise complex is well known in the theory of utomton groups. This utomton ws constructed y Aleshin in [1] in the first ttempt to generte free non-elin group y (initil) utomt. The proof ws considered not complete, nd the prolem remined open for mny yers. The first utomton reliztion of free non-elin groups ws mde y Glsner nd Mozes in [8] sed on the connection with squre complexes nd Burger-Mozes groups. The smllest utomtongenertingfreegroupconstructedyglsnerndmozeshs6sttesover14- letter lphet. Finlly, Voroets nd Voroets in [26] proved tht the Aleshin utomton genertes the free group of rnk three. Surprisingly, mong the hundreds of utomt with three sttes over the inry lphet only the Aleshin utomton genertes free nonelin group (see [3]). Automton reliztions of free groups nd free products of cyclic groups of order two re constructed in [24, 25, 27]. In [13] Jnzen nd Wise proved tht CSCs with t most three squres, five or seven squres hve residully finite fundmentl group. Burger nd Mozes in [5] proved tht for every n 109 nd m 150 there exists complete VH complex with n nd m elements in the verticl-horizontl decomposition hving non-residully finite (even virtully simple) fundmentl group. The smllest open cses were CSCs with four squres nd directed complete VH complexes with six squres. There re no ireversile utomt with less thn three sttes over the inry lphet generting n infinite group, nd only two ireversile utomt with three sttes: the Aleshin utomton nd the Bellterr utomton (see [3]). By Theorems 9 nd 11 these two utomt produce the smllest possile complete directed VH squre complexes with non-residully finite fundmentl groups. 4

5 The developed technique lso works for non-directed VH complexes. Using computtions with GAP, sed on Rttggi computtions from [22], we hve checked tht there re only two complete VH complexes D nd S with four squres nd one vertex tht could hve non-residully finite group (for ll other complexes the ssocited utomt generte finite groups). These two complexes re shown in Figures 3 nd 4 together with the ssocited utomt. Both of these complexes were studied y Rttggi, who conjectured tht they hve residully finite fundmentl groups (see [21, Section 4.10]). Jnzen nd Wise in [13] proved tht D dmits n nti-torus nd, therefore, π 1 ( D ) is smllest irreducile lttice in the direct product of two trees. The question ws rised whether π 1 ( D ) is residully finite nd in the theorem elow we nswer this question negtively. Interestingly, the corresponding utomt provide new simple utomton representtions of the free group F 2 nd the first utomton representtion of C 3 C 3. Theorem 15. Let D e the squre complex given y the four squres in Figure 3 nd A e the ssocited ireversile utomton. Then: 1. π 1 ( D ) is non-residully finite; 2. G A = C3 C 3 nd G A = F2. Since the universl cover of D is the direct product of two regulr trees of degree four, π 1 ( D ) is non-residully finite group isometric to F 2 F 2. Therefore, the full group C -lger of π 1 ( D ) is not residully finite dimensionl; this my e interesting in view of n open question whether the C -lger of F 2 F 2 is residully finite dimensionl, which is equivlent to the Connes emedding conjecture. In [13] Jnzen nd Wise suggested tht D is not unique exmple of complex with four squres tht produce n irreducile lttice. We confirm this y proving tht π 1 ( S ) is irreducile s well, however, we do not know whether π 1 ( S ) is residully finite. The freeness of utomton groups G A nd G A strongly suggest tht ll nontrivil norml sugroups of π 1 ( S ) hve finite index, wht ws conjectured in [21, Conjecture 23]. Moreover, the ssocited utomton A is smllest self-dul utomton tht genertes free group. Theorem 16. Let S e the squre complex given y the four squres in Figure 4 nd A e the ssocited ireversile utomton. Then: 1. G A = F2 nd G A = F2 ; 2.,x,,y,,x nd,y form nti-tori in π 1 ( S ). The pper is orgnized s follows. In Section 2 we recll sic fcts out utomton groups (see [9, 10, 20] for more detils). In Section 3 we descrie the connection etween utomton groups nd squre complexes. The residul properties of π 1 ( A ) re studied in Section 4. In Section 5 we prove Theorems 15 nd 16. In the lst section, following the pproch of Rttggi [22], we construct finitely presented torsion-free simple groups which decompose into n mlgmted free product F 7 F49 F 7. 5

6 x x y x y y x y x x 1 x 1 x y x 1, x y 1 y 1 x, x 1 y y 1 y y y 1 y 1 y y y 1 x 1 y 1, y x y 1 x 1, x y 1 x 1 x 1 x x 1 Figure 4: The squre complex S nd the ssocited utomton with G A = G A = F2 Acknowledgment. The first uthor would like to thnk D Aniele Dngeli, Rostislv Grigorchuk, Dmytro Svchuk, nd Yroslv Voroets for fruitful discussions. Also the uthors would like to thnk the developers of the progrm pckge AutomGrp [19] which hs een used to perform mny of the computtions descried in this pper. 2 Automt nd utomton groups 2.1 Automt Let X e nonempty set nd X the free monoid over X. The elements of X re finite words x 1 x 2...x n, where x i X nd n N, together with the empty word. The length of word v = x 1 x 2...x n is v = n. The set X n consists of ll words of length n. By X 1 we denote the set of forml elements x 1 for x X. Let X ± e the set of ll (non-reduced) words over X X 1. In contrst to generl elements of X ±, the nonempty elements of X re clled positive words over X. The free group generted y X is denoted y F X. We consider complete deterministic utomt-trnsducers (Mely utomt) with the sme input nd output lphets. Hence, in this rticle: Definition 1. An utomton is triple A = (S,X,λ), where S nd X re nonempty sets nd λ : S X X S is n ritrry mp. The set X is the lphet of input nd output letters, the set S is the set of sttes of A. Finite utomt hve finitely mny sttes nd finite lphet. We identify n utomton A = (S,X,λ) with directed leled grph on the vertex set S with the following edges: s x y t whenever λ(s,x) = (y,t). 6

7 Note tht for every s S nd every x X there exists unique rrow pssing from s nd leled y x y for some y X. The existence of such rrows indictes the completeness of A, while the uniqueness tht A is deterministic. In terms of utomt theory, n rrow s x y t mens tht if the utomton is initilized t stte s nd reds the letter x, then it outputs the letter y nd chnges its ctive stte to t. Two utomt A = (S A,X A,λ A ) nd B = (S B,X B,λ B ) re isomorphic if there exist ijections φ : S A S B nd ψ : X A X B such tht λ A (s,x) = (y,t) if nd only if λ B (φ(s),ψ(x)) = (ψ(y),φ(t)). If A nd B shre the sme lphet X = X A = X B, then A nd B re cll X-isomorphic if there exists n isomorphism tht is trivil on X. There re two stndrd opertions over utomt: tking dul utomton nd tking inverse utomton for invertile utomt. The dul of n utomton A = (S,X,λ) is the utomton (A) = (X 1,S 1,δ), where or in the grphicl representtion: δ(x 1,s 1 ) = (t 1,y 1 ) if λ(s,x) = (y,t), x 1 s 1 t 1 y 1 in (A) if s x y t in A. The reson to put inverse sign will e cler in the next section. Since A is complete nd deterministic, the dul (A) is complete nd deterministic too, so it is lwys well-defined utomton. Note tht ( (A)) = A. We sy tht A is self-dul if (A) is isomorphic to A. The inverse of n utomton A = (S,X,λ) is the tuple i(a) = (S 1,X,δ), where or in the grphicl representtion: δ(s 1,y) = (x,t 1 ) if λ(s,x) = (y,t), 1 y x s t 1 in i(a) if s x y t in A. The i(a) my e not deterministic/complete. The utomton A is clled invertile if i(a) is well-defined, which is equivlent to the following property: for every s S nd y X there exists n rrow s x y t in A for some x X nd t S. Note tht i(i(a)) = A. Any utomton A = (S,X,λ) cn e nturlly extended to n utomton A = (S,X,λ ) y consequently pplying the rule: s x 1 y 1 x 2 y 2 x 1 x 2 y 1 y 2 p nd p t produce s t, which in utomt theory corresponds to the consecutive processing of input strings of letters. Also, we cn extend A to n utomton A = (S,X,λ ) y consequently pplying the rule: s 2 y z t 2 nd s 1 x y t 1 produce s 2 s 1 x z t 2 t 1, 7

8 which is utomt theory corresponds to the (left) composition of utomt. By pplying consequently these rules we get well-defined utomton A = (S,X,λ ). If A is invertile, then we cn consider the utomton ± A = (A i(a)) with the set of sttes S ± over the lphet X. Note tht the set of reduced words over S spns suutomton, which produces n utomton structure on the free group F S. 2.2 Automton groups Every utomton A = (S,X,λ) produces two semigroup ctions left ction S X nd right ction S X defined y the rule: for g,h S nd u,v X, g(u) = v nd (g)u = h if g u v h in A, which is well-defined, ecuse A is complete nd deterministic. In other words, for ech stte s S, we put s(x 1 x 2...x n ) = y 1 y 2...y n for pth in A of the form s x 1 y 1 x 2 y 2 x 3 y 3 x n y n s2 s3... sn+1, nd the trnsformtion of X defined y word over S is exctly the left composition of the respective trnsformtions defined y the sttes. Similrly, the trnsformtion of S defined y word over X is the right composition of the trnsformtions defined y the letters. The trnsformtions of X defined y the sttes of A re invertile if nd only if A is invertile. The trnsformtion defined y stte s 1 of i(a) is inverse to the trnsformtion defined y the stte s of A. The utomton ± A defines n ction of S ± on X, nd the trnsformtion of X defined y word w S ± is exctly the composition of trnsformtions defined the sttes of A nd their inverses. Hence, every invertile utomton produces nturl ction of the free group F S on X. Definition 2. Let A = (S,X,λ) e finite invertile utomton. The quotient of F S y the kernel of its ction on X is clled the utomton group G A. From nother point of view, the utomton group G A is the group generted y the trnsformtions of X defined y the sttes of A under composition. The dul utomton (A) = (X 1,S 1,δ) produces left ction (X 1 ) (S 1 ) nd right ction (X 1 ) (S 1 ). By tking forml inverses, these two ctions re trnslted to the right ction S X nd the left ction S X, respectively, ssocited to A: g u v h in A if nd only if u 1 g 1 h 1 v 1 in (A). The dul utomton (A) is invertile if nd only if the right ction S X is invertile. In this cse, there is nturl utomton structure on F X over the lphet S nd the ction of F X on the spce S. The utomton group G (A) is the quotient of F X y the kernel of its ction on S. The group G (A) is generted y the trnsformtion of S defined y the letters in X under composition. 8

9 Every utomton group G dmits nturl sequence of finite index sugroups G = St G (0) St G (1) St G (2)..., where St G (n) = {g G : g(v) = v for ll v X n } is the stilizer of words of length n. Since their intersection is trivil, every utomton group is residully finite. Another property of ll utomton groups is tht they hve solvle word prolem. 3 Automt nd squre complexes In this section we descrie the connection etween utomt nd squre complexes discovered y Glsner nd Mozes in[8]. We give somewht different presenttion with emphsis on comintoril nd lgeric properties. 3.1 Squre complexes A squre complex is comintoril 2-complex whose 2-cells re squres, i.e., they re ttched y comintoril pths of length four. We re interested in specil clss of squre complexes complete VH squre complexes, introduced in [28]. A squre complex is clled VH complex if its 1-cells cn e prtitioned into two clsses V nd H such tht the ttching mp of ech 2-cell lterntes etween the edges of V nd H. If the ttching mp of ech 2-cell preserves the orienttion of the edges of V nd H, then the VH complex is clled directed. The Gromov link condition implies tht VH complex is non-positively curved if there re no doule edges in the links of vertices. A squre complex is clled complete squre complex (CSC for short), if the link of ech vertex is complete iprtite grph. A nturl exmple of complete VH complex is direct product of two grphs. Moreover, squre complex is complete if nd only if its universl cover is direct product of two trees (see [28, Theorem 1.10]), therefore, the fundmentl group of compct CSC cts freely nd cocompctly on CAT(0) spce. 3.2 Squre complexes ssocited to utomt Let A = (S,X,λ) e finite utomton. We ssocite to A set of Wng tiles, unit squres with leled edges, s follows: x W A = s t for ech rrow s x y t in A. y The set W A contins #S #X squres, whose horizontl sides re leled y letters from X, while the verticl sides re leled y sttes from S. In ddition, ll horizontl sides re oriented from left to right, nd ll verticl edges re oriented from ottom to up. 9

10 The squre complex A ssocited to A hs one vertex, directed loop for every s S nd every x X, nd 2-cell for every squre in W A. The complex A is directed VH squre complex. The fundmentl group of A hs finite presenttion π 1 ( A ) = S,X sx = yt for ech rrow s x y t in A. It is direct to see tht n rrow g v u h in A implies tht W A dmits tiling of finite rectngle such tht its left side is leled y g, the top side y v, the ottom side y u, nd the right side y h, which produces the reltion gv = uh in π 1 ( A ). The next sttement follows from the fct tht we consider complete utomt (see similr sttements in [22, Section 4.1] nd [6]). Proposition 1. For ny finite utomton A the set of Wng tiles W A dmits periodic tiling of the plne nd the group π 1 ( A ) contins Z 2 s sugroup. Therefore, π 1 ( A ) is never Gromov-hyperolic. Proof. Let us construct the directed grph Γ on the vertex set S X, where we put the rrow (s,x) (t,y) if s x y t in A. Since A is finite, the grph Γ contins directed cycle (s 1,x 1 ) (s 2,x 2 )... (s n,x n ) (s 1,x 1 ). Therefore, we hve the trnsitions in the utomton A : x 1 x 1 x 2 x 2 s n...s 2 s 1 s1 s n...s 2 s2 s 1...s 3... xn xn s n...s 2 s 1. Put w = s n...s 2 s 1 nd u = x 1 x 2...x n. Then A contins loop t w leled y u u, which corresponds to the reltion wu = uw in π 1 ( A ). It follows tht W A dmits tiling of rectngle with left/right lels w nd top/ottom lels u, which extends to periodic tiling of the plne. Now we show tht the sugroup w,u of π 1 ( A ) is isomorphic to Z 2. Since A is directed VH complex, there exists nturl surjective homomorphism φ : π 1 ( A ) Z Z which extends s (1,0) for ll s S nd x (0,1) for ll x X. Since we lredy know tht w,u is elin nd φ(w) = (n,0), φ(u) = (0,n), the sttement follows. It seems to e n interesting prolem to develop method which, given finite utomton A, descries ll periodic tilings for the tileset W A. These periodic tilings correspond to loops in the utomton A leled y v v for some v X. However, we do not see nice finite description of ll possile periodic tilings. For generl squre complexes the following prolem remins open. Prolem 1. Is it true tht if the fundmentl group of squre complex is CAT(0) ut not Gromov-hyperolic, then it contins Z 2 s sugroup? 10

11 A : s x y t (A) : x 1 s 1 t 1 y 1 i(a) : 1 y x s t 1 i(a) : 1 s t y x 1 i (A) : t x 1 y 1 s i (A) : x t 1 s 1 y i i(a) : t 1 y 1 x 1 s 1 i i(a) : y t s x Figure 5: Arrows in the eight utomt otined from A y pssing to the dul nd inverse utomt This prolem is specil cse of fmous Gromov s question on whether ech CAT(0) group which is not Gromov-hyperolic contins Z 2 s sugroup. One of the pproches to get negtive nswer to these prolems would e to construct set of Wng tiles which dmits only non-periodic tilings of the plne with strong restrictions on side lels (see discussion in [14, Section 4], where the first 4-wy deterministic periodic tileset is constructed). 3.3 Bireversile utomt nd complete squre complexes Independently in utomt theory, tiling theory nd in the study of squre complexes people cme up to similr clsses of ojects with strong deterministic properties. In utomt theory this property is clled ireversiility. It ws introduced in [18] in reltion to commensurtors of free groups. By pplying the inverse nd dul opertions to ny utomton A we get eight (not necessry deterministic nd complete) utomt: A, (A), i(a), i (A), i(a), i (A), i i(a), i i (A) = i i(a). (1) TherrowsinechoftheseutomtcorrespondingtonrrowinAreshowninFigure5. Definition 3. A finite utomton A is clled ireversile if ll eight utomt in (1) re well-defined, i.e., complete nd deterministic. Since we strt with complete nd deterministic utomton A, ll eight utomt in(1) re complete if nd only if ll of them re deterministic. Actully, since the complete nd deterministic properties re preserved under pssing to the dul utomton, n utomton A is ireversile if nd only if i(a), i (A), nd i i(a) re deterministic (equiv., complete), or in other words, A, A nd i(a) re invertile. Notice tht ll the rrows in Figure 5 produce the sme reltion in the fundmentl groups of the corresponding squre complexes: sx = yt, s 1 y = xt 1, x 1 t = sy 1, y 1 t 1 = s 1 x 1. Therefore, the trivil mp on S nd X extends to n isomorphism etween the squre complexes ssocited to the eight utomt. 11

12 The ireversiility of n utomton A cn e checked using finite iprtite grph Γ A ssocite to A. The vertex set of Γ A will e the disjoint union (S S 1 ) (X X 1 ). Ech rrow in A contriutes four edges in Γ A : s x y t in A (s,x), (s 1,y), (t,x 1 ), (t 1,y 1 ) in Γ A. The four edges represent corresponding rrows in A, i(a), i (A), nd i i(a). Since we wnt ech of these utomt to e complete, there should y edges (s,x), (s 1,y), (t,x 1 ), nd (t 1,y 1 ) for ll s,t S nd x,y X. Therefore, A is ireversile if nd only if Γ A is complete iprtite grph. Note tht Γ A is the link of the unique vertex of A. Bireversiility hs nice interprettion in terms of the tileset W A. Note tht the deterministic property of A implies tht the colors of two edges djcent to the top left corner uniquely determines tile from W A. The other three corners re responsile for deterministic properties of i(a), i (A), nd i i(a). Therefore, the ireversiility of utomt corresponds to the 4-wy deterministic property of Wng tilesets (this property mens tht the colors of ny two djcent edges uniquely determine Wng tile). Proposition 2. Let A e finite utomton. The following sttements re equivlent: 1. A is ireversile; 2. Γ A is complete iprtite grph; 3. W A is 4-wy deterministic; 4. A is complete squre complex; 5. A is non-positively curved; 6. the universl cover of A is the direct product of two trees. Proof. The equivlence of items 1, 2, 3 nd 4 is explined ove. The equivlence of items 1, 5 nd 6 is proved in [8], nd the equivlence of items 4, 5, 6 follows from [28]. Every ireversile utomton A = (S,X,λ) cn e extended to n utomton A ± with thestteset S ±1 ndthelphet X ±1,inwhich therrowsregiven ythefirst columnof Figure 5. Bsiclly, A ± is the union of A, i(a), i (A), nd i i(a), while its dul (A ± ) is the union of (A), i(a), i (A), nd i i(a). Note tht the utomton A ± is ireversile s well. Then we cn nturlly extend the stte set of A ± to words over S ±1 nd the lphet to words over X ±1 s in Section 2.1 nd construct n utomton ± A ± = (S ±,X ±,λ ). The left group ction F S X ssocited to A is extended to the left ction F S X ± ssocited to A ±, while the right group ction S F X is extended to the right ction S ± F X. Note tht the sets of reduced words F X X ± nd F S S ± re invrint under these ctions (they induce suutomton in ± A ± ). 12

13 3.4 The fundmentl group of A for ireversile utomt The next sttement contins some sic properties of the fundmentl groups of complexes A known for ll complete VH squre complexes with one vertex (see [28]). Theorem 3. Let A = (S,X,λ) e ireversile utomton. 1. The group π 1 ( A ) is torsion-free nd CAT(0). 2. The sugroups of π 1 ( A ) generted y S nd X re free of rnk #S nd #X respectively. 3. (Norml forms) The group π 1 ( A ) dmits n exct fctoriztion y its free sugroups S nd X. In prticulr, every element γ π 1 ( A ) cn e uniquely written in the form γ = gv nd in the form γ = uh for g,h S nd v,u X. There is nice direct connection etween the trnsitions in ireversile utomton A nd the two norml forms in π 1 ( A ) given in Theorem 3. The completeness of the iprtite grph Γ A mens tht the genertors of π 1 ( A ) stisfy S ±1 X ±1 = X ±1 S ±1, i.e., for ny s S ±1 nd x X ±1 there exists unique pir y X ±1 nd t S ±1 such tht sx = yt in π 1 ( A ). This explins the norml form in π 1 ( A ): given ny word γ over genertors, we cn move every s S ±1 to the left nd every x X ±1 to the right in order to find the representtion γ = gv for g S nd v X. We cn lso move every s S ±1 to the right nd every x X ±1 to the left, nd otin nother norml form γ = uh for u X nd h S. Every permuttion of genertors sx = yt corresponds to trnsition in A ±. Therefore, for reduced words g,h F S nd v,u F X we hve gv = uh in π 1 ( A ) if nd only if g v u h in ± A ±. (2) In prticulr, we will frequently use the following reltion etween the ction of F S on X ± nd the group π 1 ( A ): for g F S nd v F X, g(v) = v if nd only if v 1 gv F S. 3.5 Automton groups generted y ireversile utomt Every ireversile utomton A = (S, X, λ) gives rise to eight invertile utomt A, (A), i(a), i (A), i(a), i (A), i i(a), i i (A) = i i(a), n utomton A ±, nd two group ctions: left ction F S X ± nd right ction S ± F X. The utomton group G A ± is the quotient of F S y the kernel of its ction on X ±. The susets of positive words X nd negtive words (X 1 ) re invrint under 13

14 the ction of F S. The corresponding restricted ctions produce G A = G i(a) nd G i (A) = G i i (A). Similrly, the dul utomton group G (A ± ) is the quotient of F X y the kernel of its ction on S ±, while G (A) = G i (A) nd G i(a) = G i i(a) re the quotients of the corresponding ctions on S nd (S 1 ). The next sttement shows tht in this wy we get just two groups G A nd G (A) nd descries how to recover them from π 1 ( A ). Theorem 4. Let A = (S,X,λ) e ireversile utomton. Then G A = G i(a) = G i (A) = G i i (A) = GA ± = F S /K, where K is the mximl norml sugroup of π 1 ( A ) tht is contined in F S = S, nd G (A) = G i (A) = G i(a) = G i i(a) = G (A ± ) = F X /K. where K is the mximl norml sugroup of π 1 ( A ) tht is contined in F X = X. Proof. We show tht the ction of G A ± on X is fithful. Let g F S ct trivilly on X, nd let us show tht g cts trivilly on X ±. For ny v X there exists unique g 1 F S such tht gv = vg 1 in π 1 ( A ) nd g 1 cts trivilly on X. We cn repet this process nd construct sequence of elements g 1,g 2,... in F S such tht which corresponds to the directed pth gv = vg 1, g 1 v = vg 2, g 2 v = vg 3,... in π 1 ( A ), g v v g 1 v v g 2 v v g 3 v v... in ± A ±. Sincell g i hvethesmelength, thissequence iseventully periodic.moreover, thenorml form in π 1 ( A ) (or the deterministic properties of ireversile utomt) implies tht this sequence is periodic nd there exists n N such tht g n = g. Hence, gv n = v n g in π 1 ( A ), which implies gv n = v n g nd g(v 1 ) = v 1. Therefore, g cts trivilly on (X 1 ) s well nd hence on X ±. It follows tht G A = GA ±. The other cses re nlogous. Let K < π 1 ( A ) e the kernel of the ction of F S X ± so tht G A ± = F S /K. Then K is preserved under conjugtion y elements of F S. Since every element of K cts trivilly on F X, K is preserved under conjugtion y elements of F X. Therefore, K is norml sugroup of π 1 ( A ). It is mximl mong norml sugroups tht re contined in F S, ecuse ll elements of such sugroups ct trivilly on words over X ±1. The dul cse is nlogous. The following sttement is comintion of [24, Proposition 2.2] nd [5, Proposition 1.2]. A group π 1 ( A ) is clled reducile if it contins finite index sugroup of the form K H for K < F S nd H < F X. Corollry 5. Let A = (S,X,λ) e ireversile utomton. The following sttements re equivlent: 14

15 1. G A is finite; 2. G A is finite; 3. π 1 ( A ) is reducile. Proof. If G A is finite, then every orit of the ction F S X contins t most #G A elements. Therefore, every element v X ± elongs to suutomton of ± ( A) with t most #G A sttes. Since there re only finitely mny different utomt with fixed numer of sttes, the group G A is finite. Hence, the items 1 nd 2 re equivlent. If G A nd G A re finite, then K K hs finite index in π 1 ( A ). Conversely, if π 1 ( A ) contins finite index sugroup K 1 K 2 with K 1 < F S nd K 2 < F X, then the kernels K > K 1 nd K > K 2 hve finite index in F S nd F X respectively. Therefore, G A nd G A re finite. Theorem 4 suggests two wys to generte free groups y utomt. Remrk 1. If for some ireversile utomton A we hd n infinite group G A nd just-infinite 1 group π 1 ( A ), then G A nd G A would e free groups freely generted y S nd X respectively. However, π 1 ( A ) cnnot e just-infinite, ecuse it projects onto Z 2. Nevertheless, this pproch works for some non-directed VH complexes, where just-infinite exmples exist (see [5]) nd produce free utomton groups. Remrk 2. It follows from the theorem tht if {g 1,g 2,...,g n } is n orit of the ction S ± F X nd some g i represents nontrivil element of G A, then ll g 1,...,g n represent nontrivil elements of G A. In prticulr, if G A is infinite nd F X cts trnsitively on ll reduced words of length n for ech n N, then G A is free group freely generted y S. However, F X cnnot hve this trnsitivity property, ecuse positive words over S re invrint under the ction. Nevertheless, s in the previous remrk, this pproch works for some non-directed VH complexes nd ws used y Glsner nd Mozes in [8] to construct the first exmples of utomt generting free groups. Besides Corollry 5, it is not cler wht is the reltion etween the groups G A nd G (A). For ll exmples tht we know, the groups G A nd G (A) re either oth finitely presented or oth infinitely presented. Question 1. Which ireversile utomt generte finitely presented groups? Is it true for ireversile utomton A tht G A is virtully free if nd only if G A is virtully free? Is it true tht groups generted y ireversile utomt re liner? For these questions it seems useful to consider the quotient of π 1 ( A ) y the kernels of the ction of S on X nd of the ction of X on S. We get group T A = π 1 ( A )/K K = G A G (A), 1 An infinite group is clled just-infinite if every nontrivil norml sugroup hs finite index 15

16 c Figure 6: A self-inverse self-dul utomton generting the lmplighter group Z 3 Z which dmits n exct fctoriztion y oth utomton groups. The group T A rings informtion out G A nd G (A), nd their interconnection in π 1 ( A ). It follows from the results of Y. Voroets (privte communiction) tht the group T A for the Aleshin nd Bellterr utomt is finitely presented. The first exmple of ireversile utomton with non-finitely presented groups G A nd G (A) is shown in Figure 6; it is self-dul nd genertes the lmplighter group Z 3 Z (see [2]). Question 2. For which ireversile utomt A is the group T A finitely presented? 3.6 Anti-tori nd tilings of the plne Let A e ireversile utomton nd W A ± the set of Wng tiles ssocited to A ±. Since W A ± is complete nd 4-wy deterministic, ll possile tilings of the plne y W A ± cn e descried s follows: for every pir of sequences (s i ) i Z, s i S ±1 nd (x i ) i Z, x i X ±1 there exists unique tiling t : Z 2 W A ± of the plne such tht the sequence (s i ) i Z is red long the verticl xes, while the sequence (x i ) i Z is red long the horizontl xes. For reduced sequences this tiling corresponds to plne in the universl cover of A. Let F S nd F X. Let us consider the tiling of the plne with the verticl xes leled y () i Z nd the horizontl xes leled y () i Z. If this tiling is periodic in oth verticl ndhorizontl directions, thenthereexist nonzero n,m Zsuch tht n m = m n in π 1 ( A ), which mens tht there is torus in A. If this tiling is not periodic, then we come to the following definition. Definition 4. For F S nd F X, the sugroup, is clled n nti-torus in π 1 ( A ) if n m m n for ll n,m Z\{0}. 16

17 It seems to e n interesting nd difficult prolem, given ireversile utomton A, descrie ll nti-tori in π 1 ( A ). This prolem ws studied in [23] for certin Burger-Mozes groups, where nti-tori correspond to non-commuting pirs of Hmilton quternions. Proposition 6. Let A = (S,X,λ) e ireversile utomton. If π 1 ( A ) dmits n ntitorus, then G A is infinite. Proof. Let F S nd F X generte n nti-torus. We will prove tht represents n element of infinite order in G A. Let us ssume tht n (v) = v for ll v X ±. Then we hve cycle in the utomton n n (see the proof of Theorem 4). This mens tht n m = m n in π 1 ( A ) for some m N, which contrdicts the ssumption of the sttement. It is interesting whether the converse holds: Question 3. Is it true tht if G A is infinite then π 1 ( A ) contins n nti-torus? This question is relted to the next one studied in the theory of utomton groups. Question 4. Cn ireversile utomton A generte n infinite torsion group G A? By Proposition 6, positive nswer to Question 4 implies negtive nswer to Question 3. However, we expect negtive nswer to Question 4 nd positive nswer to Question 3. It ws shown in [16] tht ireversile utomton with two nd three sttes cnnot generte n infinite torsion group. Note, however, tht there re fmous exmples of infinite torsion groups generted y non-ireversile utomt like the Grigorchuk group. 3.7 Automt from squre complexes Let e complete directed VH squre complex with one vertex. Let S nd X e the loops t the corresponding verticl-horizontl decomposition of. Since is complete nd directed, for ny s S nd x X there exists unique pir y X nd t S such tht sx = yt in π 1 ( ). This property defines n utomton A over the lphet X with the set of sttes S. This utomton is ireversile nd the squre complex A is exctly. The sme construction works for non-directed complexes. Let e complete (not necessry directed) VH squre complex with one vertex. Let S nd X e the loops t the corresponding verticl-horizontl decomposition of. The completeness of implies tht for ny s S ±1 nd x X ±1 there exists unique pir y X ±1 nd t S ±1 such tht sx = yt in π 1 ( ). As ove, these reltions produce ireversile utomton A over the lphet X X 1 nd with the set of sttes S S 1. Two exmples of non-directed squre complexes nd the ssocited utomt re shown in Figures 3 nd 4. Note tht the squre complex A ssocited to A is not, ecuse when we construct squre complex from utomton we tret every stte s n independent color nd do not tke into ccount inverse elements. If we strt with directed complex = A ssocited to ireversile utomton A, then A is exctly the utomton A ± constructed in Section

18 4 Residul finiteness of π 1 ( A ) In this section we study the residul properties of the group π 1 ( A ) for ireversile utomt A. Our nlysis follows the pproch of Wise from [28] nd relies on the following sttement. Let H e sugroup of group G. Then G is clled H-seprle if H is the intersection of finite index sugroup of G. A group G is residully finite if nd only if it is seprle with respect to the trivil sugroup. Theorem 7 ([30, Theorem 7.2]). Let φ e n endomorphism of finitely generted residully finite group G, nd let Fix(φ) e the sugroup of elements fixed y φ. Then G is F ix(φ)-seprle. Proof. We include the proof for completeness. Let G \ Fix(φ). Then 1 φ() e nd 1 φ() N for some sugroup N of finite index n. We cn ssume tht N is fully invrint y pssing to the intersection of ll sugroups of index n. Then Fix(φ)N hs finite index nd does not contin. Corollry 8 ([30, Corollry 7.3]). Let D = G H G e the doule of group G long its sugroup H. If D is residully finite, then G is H-seprle. In [28] Wise proved tht π 1 ( W ) is not,,c -seprle using the fct tht it contins n nti-torus. We do not know which utomt dmit n nti-torus (see discussion in Section 3.6); insted, we re using the following lemm, which relies on Zelmnov s solution of the restricted Burnside prolem. Note tht the conclusion of the lemm immeditely follows from the existence of n nti-torus. Let A = (X,S,λ) e ireversile utomton. For m N, let P m e the set of ll pirs (x,y) of different letters x,y X for which there exist g S nd u X such tht g m (ux) = uy (this mens there is reltion g m ux = uyh in the group π 1 ( A ) for some h S ). Note tht P m is empty only when g m is trivil in G A for every g S. Lemm 1. Let A = (X,S,λ) e ireversile utomton. If G A is infinite, then m N P m is nonempty. Proof. Let us ssume tht this intersection is empty. Then, for ech pir of different letters (x,y), there exists m xy N such tht (x,y) / P mxy. Let m e the product of ll these numers m xy. Note tht if n divides m, then P m P n, ecuse we cn rewrite the equlity g m (ux) = uy s (g m/n ) n (ux) = uy. Since m xy divides m for ll pirs (x,y), the set P m is empty. It follows tht the element g m is trivil in G A for every g S. Hence the group G A hs finite exponent. Since G A is finitely generted nd residully finite, it should e finite y the solution of the restricted Burnside prolem. We get contrdiction. Remrk 3. For utomt over the inry lphet X = {0,1} it is strightforwrd to see tht the infiniteness of G A implies tht m N P m = {(0,1),(1,0)}, nd we do not need to rely on the restricted Burnside prolem. 18

19 Theorem 9. Let A = (X,S,λ) e ireversile utomton. 1. If G A is finite, then π 1 ( A ) is virtully direct product of two free groups nd therefore residully finite. 2. If G A is infinite, then π 1 ( A ) is not S -seprle nd not X -seprle. Proof. The first item follows from Corollry 5. Let G A e infinite. If π 1 ( A ) is S -seprle, then for ech g π 1 ( A ) \ S there exists sugroup H < π 1 ( A ) of finite index such tht S < H nd g / H. Then there exists norml sugroup N π 1 ( A ) of finite index such tht g / S N. Let (x,y) m N P m. We re going to prove tht the element x 1 y of the group π 1 ( A ) elongs to S N for every norml sugroup N of finite index in π 1 ( A ). Since x 1 y S y Theorem 3, it will follow tht π 1 ( A ) is not S -seprle. Let N < π 1 ( A ) e norml sugroup of index n. Then for every g S the element g n elongs to N. Since (x,y) P n, there exist words g,h S S nd u X such tht g n ux = uyh in π 1 ( A ). Therefore which completes the proof. x 1 y = h 1 (hx 1 u 1 g n uxh 1 ) S N, Corollry 10. Let A = (X,S,λ) e ireversile utomton. The doule D A = π 1 ( A ) S π 1 ( A ) of π 1 ( A ) long S is residully finite if nd only if G A is finite. Proof. If G A is infinite, then D A is not residully finite y Theorem 9 nd Corollry 8. If G A is finite, then G A i (A) = GA is finite. The group D A is the fundmentl group of the complex A i (A), which is residully finite y Theorem 9. Remrk 4. It follows from the proof of Theorem 7 tht for ech (x,y) m N P m the nontrivil element (x 1 y) 1 φ(x 1 y) = y 1 xx 1 y of π 1 ( A ) S π 1 ( A ) elongs to the intersection of finite index sugroups. Remrk 5. Theorem 9 holds for the fundmentl group of non-directed complete VH complexes with one vertex nd the ssocited ireversile utomt, ecuse the element x 1 y from the proof remins nontrivil in π 1 ( ). The ove theorem does not tell us when the group π 1 ( A ) is residully finite, nd the next question seems to e difficult: Question 5. For which ireversile utomt A is the group π 1 ( A ) residully finite? We nswer this question in some cses. Theorem 11. Let A = (X,S,λ) e ireversile utomton over n lphet with two letters or with two sttes. If G A is infinite, then π 1 ( A ) is non-residully finite. 19

20 Proof. We cn ssume tht X = {x,y}. There exists n N such tht (y 1 x) n nd (x 1 y) n ct trivilly on S S 1, i.e., s(y 1 x) ±n s 1 X for every s S S 1. Let us show tht the mp φ(x) = x(y 1 x) n, φ(y) = y(x 1 y) n nd φ(s) = s for ll s S extends to n endomorphism of π 1 ( A ). Since our lphet is inry, stte s S stilizes x if nd only if it stilizes y. Let S + e the set of ll sttes tht stilize x nd y, nd let S e the set of ll sttes tht stilize x 1 nd y 1. Every defining reltion for π 1 ( A ) of the form sx = xt or sy = yt implies tht s S + nd t S, while reltion of the form sx = yt or sy = xt is possile only when s S + nd t S. It follows tht every t S stisfies the reltions ty 1 x = y 1 xt 1 nd tx 1 y = x 1 yt 2 for some t 1,t 2 S, while every t S stisfies the reltions ty 1 x = x 1 yt 1 nd tx 1 y = y 1 xt 2 for some t 1,t 2 S. Since (y 1 x) ±n ct trivilly on S, we hve reltions t(y 1 x) ±n = (y 1 x) ±n t for every t S, t(y 1 x) ±n = (x 1 y) ±n t for every t S. Now for every defining reltion of the form sx = xt nd sy = yt (here s S + nd t,t S ), there re reltions sx(y 1 x) n = xt(y 1 x) n = x(y 1 x) n t nd sy(x 1 y) n = yt (x 1 y) n = y(x 1 y) n t. For defining reltions of the form sx = yt or sy = xt (here s S + nd t,t S ), there re reltions sx(y 1 x) n = yt(y 1 x) n = y(x 1 y) n t nd sy(x 1 y) n = xt (x 1 y) n = x(y 1 x) n t. Thus, φ preserves ll the defining reltions of π 1 ( A ) nd induces n endomorphism of π 1 ( A ). Since φ(z) egins nd ends on z for ech z {x,y,x 1,y 1 }, the imge φ(w) of reduced word w X is lso reduced. The length of φ(w) is greter thn the length of w; hence, no element in X cn e fixed y φ. Therefore, Fix(φ) = S nd π 1 ( A ) is non-residully finite y Theorems 7 nd 9. Remrk 6. The element (x 1 y) 1 φ(x 1 y) = (x 1 y) 1 (x(y 1 x) n ) 1 (y(x 1 y) n ) = (x 1 y) 2n elongs to the intersection of finite index sugroups of π 1 ( A ). The norml closure of (x 1 y) n nd (y 1 x) n hs infinite index in π 1 ( A ). Corollry 12. Let A = (X,S,λ) e ireversile utomton. Suppose tht ech connected component of A consists of two sttes nd the utomton group G A is infinite. Then π 1 ( A ) is non-residully finite. Proof. Since G A is infinite, y Theorem 9 we hve tht for some sttes, S (possily in different components) the element 1 is not X -seprle. By pplying rguments from the proof of Theorem 11, we cn construct nontrivil endomorphism φ i for ech 20

21 0 0 c , 1 0 Figure 7: The Bellterr utomton B generting G B = C2 C 2 C 2 connected component A i = (X,S i,λ) of A considering it s seprte utomton. By the construction, ll φ i mp elements of X to themselves. Hence, we could extend them to the unique endomorphism φ of the whole group π 1 ( A ). Similrly, the imge of reduced word w S is lso reduced nd is longer thn the initil word. So 1 is not fixed, nd therefore π 1 ( A ) is non-residully finite. Exmple 1. The smllest utomt tht stisfy the conditions of Theorem 11 re the Aleshin utomton A nd its friendly version the Bellterr utomton B shown in Figure 7. The corresponding endomorphism of π 1 ( A ) for the Aleshin utomton is shown in Introduction. The Bellterr utomton possesses n dditionl endomorphism. The sttes { 1, 1,c 1 } of B spn suutomton isomorphic to B. Hence, the mp φ : π 1 ( B ) π 1 ( B ), which replces ech letter from S S 1 y its inverse, is n utomorphism of π 1 ( B ) with Fix(φ) = X. The intersection of finite index sugroup of π 1 ( B ) contins, for exmple, the elements 2 2, 2 c 2, c 2 2. It is n open question whether the finiteness prolem for utomton groups is lgorithmiclly solvle. Proly not, ecuse it is unsolvle for utomton semigroups (see [7]). However, the finiteness prolem for utomton groups generted y ireversile utomt over the inry lphet (or with two sttes) is lgorithmiclly solvle (see [15]). This fct together with Theorems 9 nd 11 imply tht there is n lgorithm which for given complete VH complex with one vertex nd two edges in the verticl or horizontl prt verifies whether its fundmentl group π 1 ( ) is residully finite. Corollry 13. Let A e connected ireversile utomton generting n infinite utomton group. If there is nontrivil X-isomorphism etween A nd suutomton B of the utomton A, then π 1 ( A ) is non-residully finite. 21

22 Proof. The isomorphism etween A nd B extends to n endomorphism φ : π 1 ( A ) π 1 ( B ) with fixed sugroup Fix(φ) contining X. By repeting the rguments from the proof of Theorem 9, one cn show tht there is pir of sttes, in A such tht 1 Fix(φ)N for every norml sugroup N F A of finite index. We just need to prove tht 1 Fix(φ). This will imply tht π 1 ( A ) is not Fix(φ)-seprle nd, therefore, it is not residully finite y Theorem 7. Let φ() = v nd φ() = u. Since B is connected, ll words representing the sttes of B hve the sme length, in prticulr, v = u. Therefore, the equlity φ( 1 ) = v 1 u = 1 is possile onlywhen v nduhve common eginning w so tht v = wndu = w. Since φ is nontrivil, the word w is nonempty. The first letter s of w is stte of A or A 1, nd we cn ssume s A. For ny stte t of A there is pth from s to t, which induces two pths from w nd w to two different words which strt with the letter t. It follows tht for ech stte of A there re t lest two sttes of B. This contrdiction completes the proof. An utomton is miniml if different sttes define different trnsformtions of words. Corollry 14. Let A e non-miniml connected ireversile utomton generting n infinite utomton group. Then π 1 ( A ) is non-residully finite. The first utomton for which we cnnot pply Corollry 13 is the self-dul utomton A with 3 sttes over 3-letter lphet shown in Figure 6. One cn show tht there is no suutomton in A tht is X-isomorphic to A. We do not know whether its π 1 ( A ) is residully finite. 5 Complete squre complexes with four squres Every utomton group G A possesses the following self-similrity property: for every g G A nd v X there exists unique element h G A such tht g(vw) = g(v)h(w) for ll w X. In the proof elow we use nottion h = g v. Theorem 15. Let D e the squre complex given y the four squres in Figure 3 nd A e the ssocited ireversile utomton. Then: 1. π 1 ( D ) is non-residully finite; 2. G A = C3 C 3 nd G A = F2. Proof. We hve X = {x,y,y 1,x 1 } nd S = {,, 1, 1 }. Since π 1 ( D ) contins n nti-torus (see [13]), the utomton group G A is infinite y Proposition 6, nd the group π 1 ( D ) is not X -seprle y Theorem 9. It is direct to check tht the mp defined y φ() = 4,φ() = 4 nd φ(x) = x,φ(y) = y 22

23 extends to n endomorphism of π 1 ( D ) with Fix(φ) = X. Therefore, π 1 ( D ) is nonresidully finite y Theorem 7. Further we indicte nontrivil elements in the intersection of its finite index sugroups. Let us descrie the group G A. One cn directly check the following crucil property: ny nontrivil orit of the ction of n element g St 1 (G A ) on X 2 consists of three elements, nmely, these orits re {xx,xy,xy 1 },{yx,yy,yx 1 },{y 1 x,y 1 y 1,y 1 x 1 },{x 1 y,x 1 y 1,x 1 x 1 }. It follows tht, for ny g G A nd v X with the property g(v) = v nd g v St 1 (G A ), if g(vz 1 z 2 ) vz 1 z 2 for some z 1,z 2 X, then the orit of vz 1 z 2 under g consists of three words vz 1 z 3 for z 3 X, z 3 z1 1. In prticulr, if g St n (G A ) nd g(vz) vz for some word v X n, z X (note tht such vz is necessry (freely) reduced word), then the orit of vz under g contins ll three reduced words of length n+1 with prefix v. We will show y induction on ntht the groupg A cts trnsitively onthe set of (freely) reduced words over X of length n for ech n N. Let us ssume tht the sttement holds for the words of length n. Since the group G A is infinite, there exists g G A such tht g St n (G A ) nd g(vz 1 ) vz 1 for some vz 1 X n+1. Note tht vz 1 nd g(vz 1 ) re reduced words. Then the orit of vz 1 under g consists of exctly three reduced words of length n+1 with prefix v. Now let wz 2 X n+1 e n ritrry reduced word. By induction hypothesis there exists h G A such tht h(w) = v. Then the word h(wz 2 ) is reduced nd hs prefix v; therefore, it elongs to the orit of vz 1 under g. This mens tht vz 1 nd wz 2 elong to the sme G A -orit. Our clim is proved. We re redy to show tht G A is freely generted y X. Let w e ny reduced word over X. The trnsitivity of the ction of G A on reduced words over X implies tht if w represents the trivil element of G A, then ll reduced word of length w represent the trivil element (see Remrk 2). However, this would men tht G A is finite, ut it is not. Similrly, we show tht G A = = C 3 C 3. One checks directly tht 3 = 3 = e in G A. We hve to prove tht every word of the form [ ±1 ] ±1 ±1... ±1 [ ±1 ] represents nontrivil element of G A. Note tht the genertors x nd y mp ±1 to ±1 nd ±1 to ±1, nd therefore G A preserves the set of lternting words. As ove, it is enough to show tht the dul group G A cts trnsitively on ll such lternting words of fixed length n for ech n N. The proof goes in the sme wy s ove nd relies on the following property of G A : ny nontrivil orit of the ction of n element g St 1 (G A ) on S 2 consists of two elements, nmely, these orits re {, 1 }, {, 1 }, { 1, 1 1 }, { 1, 1 1 }. (3) Then, for ny element g G A nd word v S with the property g(v) = v nd g v St 1 (G A ), if g(vz 1 z 2 ) vz 1 z 2 for some z 1,z 2 S, then g(vz 1 z 2 ) = vz 1 z2 1 nd g(vz 1 z2 1 ) = vz 1 z 2. Let us ssume y induction tht G A cts trnsitively on lternting words of length n. Since G A is infinite, there exist g St n (G A ) nd vz 1 S n+1 such tht g(vz 1 ) vz 1 ; therefore, g(vz 1 ) = vz1 1. Note tht vz 1 is necessry n lternting word, since otherwise vz 1 could e represented y shorter word nd would e fixed y g. Now 23