MULTI-GGS-GROUPS HAVE THE CONGRUENCE SUBGROUP PROPERTY ALEJANDRAGARRIDOAND JONEURIA-ALBIZURI Abstract. We generalize the result about the congruence subgroup 7 property for GGS-groups in [AGU] to the family of multi-GGS-groups; 1 thatis,allmulti-GGS-groupsexcepttheonedefinedbytheconstantvec- 0 torhavethecongruence subgroup property. Even if theresult remains, 2 new ideas are needed in order to generalize theproof. n a J 2 1. Introduction ] R Groups acting on regular rooted trees have been studied since the 1980s G as examples of groups with exotic properties. For instance, the Gupta– Sidki group ([GS83]) is well known as a particularly uncomplicated answer . h to the General Burnside Problem. It acts on the ternary rooted tree and t a is two-generated by a rooted automorphism and a directed one (see [AGU] m for terminology and notation used here). A similar example, acting on the [ 4-regular rooted tree, was introduced by Grigorchuk in [Gri80] and is now 1 sometimes known as the “second Grigorchuk group”. A generalization of v these two examples is provided by the family of Grigorchuk–Gupta–Sidki 0 (GGS) groups1. Each group in this family acts on the p-regular tree, where 4 4 p ≥ 3 (but here, as in most papers where they are studied, we assume p 0 prime) and is also generated by a rooted and a directed automorphism. The 0 latter generator is defined according to a vector in Fp−1. . p 1 The classical congruence subgroup property for linear algebraic groups 0 has a natural analogue for groups acting on regular rooted trees. In this 7 1 case, the principal congruence subgroups are the level stabilizers, stG(n) for : each n ∈ N. Thus the congruence subgroup property is satisfied if every v i subgroup of finite index contains a principal congruence subgroup. In other X words, if the topology given by the finite index subgroups (the profinite r a topology) coincides with the topology given by the level stabilizers. In[AKT],afamilyfurthergeneralizingGGS-groupswasdefinedandstud- ied. They were called multi-edge spinal groups, but we prefer the term multi-GGS-groups. Each group in the family is defined like a GGS-group, except that there are more directed generators. In [AGU] we proved that 2010 Mathematics Subject Classification. Primary 20E08. A.GarridoissupportedbytheAlexandervonHumboldtFoundation. J.Uria-Albizuri acknowledges financial support from the Spanish Government, grants MTM2011-28229- C02 and MTM2014-53810-C2-2-P, from the Basque Government, grants IT753-13 and IT974-16 and also from theBasque Goverment predoctoral grant PRE-2014-1-347. 1ThenamewascoinedbyBaumslagandisnottobeconfusedwithGeneralizedGolod– Shafarevichgroups,whichalsoprovidenegativeanswerstotheGeneralBurnsideProblem, but are of a very different nature. 1 2 A.GARRIDOANDJ.URIA-ALBIZURI a GGS-group has the congruence subgroup property if and only if it is not defined by the constant vector. In this note we show that the result is still true for the whole family of multi-GGS-groups. Theorem A. Let G be a multi-GGS-group. Then G is just infinite and has the congruence subgroup property if and only if G is not the GGS-group defined by the constant vector. Acknowledgements. WearegratefultoG-A.Ferna´ndez-Alcoberforuseful comments. 2. Multi-GGS-groups Definition. A multi-GGS-group G is a group of automorphisms of the p- regular rooted tree, where p is an odd prime. G is generated by the rooted automorphism a which acts on the first level of the tree by the cyclic per- mutation (1 ... p) and by some finite number r of directed automorphisms b1,...,br ∈ stG(1). Each bi is defined by a vector ei = (ei,1,...,ei,p−1) ∈ Fp−1: p ψ(b )= (aei,1,...,aei,p−1,b ) i i and we require that the defining vectors e ,...,e be linearly independent. 1 r We denote by G the group generated by aand b with b given by a 1 1 constant vector. (Note that all constant vectors yield the same group). Let us first mention some properties about multi-GGS-groups that will be useful in the proof of the main theorem. The first lemma is a collection of results from [AKT]. Lemma 1. Let G = ha,b ,...,b i be a multi-GGS-group with defining vec- 1 r tors e ,...e ∈ Fp−1. 1 r p (i) We may assume that e = 1 for all e with i = 1,...,r. i,1 i (ii) If G is not G then ψ(γ (st (1)) = γ (G)×.p..×γ (G). 3 G 3 3 (iii) G/G′ ∼= Cr+1. p Lemma 2. If the multi-GGS-group G is generated by r ≥ 2 directed gener- ators, then ψ(st (1)′)) = G′ ×.p..×G′. In particular, G is regular branch G ′ over its commutator subgroup G. Proof. Since ψ(st (1)) ≤ G×.p..×G, we need only show the ’≥’ inclusion G in the statement. Supposethat b1 has non-symmetricdefiningvector (e1,1,...,e1,p−1) (that is, e1,i 6= e1,p−i for some i). By (i) in Lemma1, we can supposethat e1,1 = 1 and e1,p−1 = m 6= 1. By the same argument as in [AZ14, Lemma 3.4], ψ([b ,ba][ba−1,b ]m...[ba,ba2]mp−1)≡ ([a,b ]1−m,1,...,1) 1 1 1 1 1 1 1 wherethecongruenceismoduloγ (G)×.p..×γ (G).Part(ii)ofLemma1im- 3 3 plies that ([a,b ]1−m,1,...,1) ∈ ψ(st (1)′) and therefore ([a,b ],1,...,1) ∈ 1 G 1 ′ ψ(st (1)) because m 6= 1. By a general fact about commutators, G [an,b ] = [a,b ]an−1[a,b ]an−2...[a,b ]a[a,b ] 1 1 1 1 1 MULTI-GGS-GROUPS HAVE THE CONGRUENCE SUBGROUP PROPERTY 3 for any n. From this and the facts that ψ(st (1)) ≤ G × .p.. × G is a subdirect G embedding and that st (1)′ E st (1), we obtain that ([an,b ],1,...,1) ∈ G G 1 ψ(st (1)′) for any n. Moreover, since st (1)′ E G, we may conjugate the G G above element by a suitable power of a to conclude that (1,...,1[an,b ]) ∈ 1 ′ ψ(st (1)) for any n. G For any other directed generator b , i ψ([b ,ba]) = ([a,b ],1...,1,[b ,aei,p−1]). 1 i i 1 ′ Therefore, by the above, ([a,b ],1...,1) ∈ ψ(st (1)). Thus (x,1,...,1) ∈ i G ′ ′ ψ(st (1)) for each normal generator x of G. Once again the subdirect G embedding ψ(st (1)) ≤ G×.p..×G allows us to conclude that 1×p.−.1.×1× G ′ ′ G ≤ ψ(st (1)) and since G acts transitively on the first level of the tree G we also have that G′×.p..×G′ ≤ ψ(st (1)′). G Now suppose that all bi-s are defined by symmetric vectors (ei,j = ei,p−j for every i,j ∈ {1,...,p−1}). Again, by (i) in Lemma 1 we may assume that e = 1 for i = 1,...,r. Replacing each e by e −e for i = 2,...,r we i,1 i i 1 obtain the same group. Thus we have e = (0,∗,...,∗,0) for i = 2,...,r. i Let e be the first non trivial entry in e . Then we can also replace e by 2,j 2 1 e −ke where e +ke = 0 so that e = 0. We thus obtain that 1 2 1,j 2,j 1,j ψ([b ,ba]) = ([a,b ],1,...,1), 1 i i for i = 2,...,r, and ψ([baj,b ]) = (1,...,1,[b ,ae2,j],1,...,1,[ae1,p−j,b ]) 1 2 1 2 = (1,...,1,[b ,ae2,j],1,...,1,1), 1 where the last equality follows because e1,p−j = e1,j = 0. Repeating the same argument as in the previous case we obtain the result. (cid:3) Because of the above lemma (and Proposition 2.4 of [AGU]), in order to show that a multi-GGS-group G as in the lemma has the congruence ′′ subgroupproperty, it suffices to show that G contains some level stabilizer. This will be shown in Corollary 7. ′ Lemma 3. Let G be any multi-GGS-group. Then st (1) ≤ γ (G). G 3 Proof. Since st (1) is normally generated by b ,...,b (equivalently, gener- G 1 r ′ ated by the conjugates of b ,...,b by powers of a), we have that st (1) 1 r G is normally generated by commutators of the form [bam,ban] with i,j ∈ i j {1,...,r} and m,n ∈ F . Now notice that [bam,ban] = [b [b ,am],b [b ,an]] p i j i i j j which is congruent modulo γ (G) to [b ,b ]= 1. Thus all normal generators 3 i j of st (1)′ are contained in γ (G) E G, which proves our claim. (cid:3) G 3 Lemma 4. Let G 6= G be a multi-GGS-group. Then ψ(G′)≤ G×.p..×G. s Proof. Suppose that b is defined by a non-constant vector, so there exists 1 i ∈ {1,...,p − 1} such that e 6= e . Then ψ([b ,a] = b−1ba) has 1,i 1,i+1 1 1 1 ae1,ia−e1,i+1 6= 1 in the (i+1)th coordinate and therefore ψ([b1,a]a1−i) has ae1,ia−e1,i+1 in the first coordinate. So there exists an element x ∈ G′ such that ψ(x) has a in the first coordinate. For any i ∈ {1,...,r} the element 4 A.GARRIDOANDJ.URIA-ALBIZURI ψ([bi,a]) has a−e1,1bi in the first coordinate and so the first coordinate of ψ(xe1,1bi) is bi. Thus ϕ1(G′) = G. Conjugating by powers of a, we obtain that ϕ (G′) = Gfor each j ∈ {0,...,p} andtherefore G′ ≤ G×.p..×G. (cid:3) j s Lemma 5. Let G = ha,b ,...,b i be a multi-GGS-group with r ≥ 2. Then 1 r ψ (G′′) ≥ G′×.p.2.×G′. 2 ′ Proof. By Lemma 4 we know that there exist x,y ∈ G such that ψ(x) = i (a,∗,...,∗)andψ(y ) = (b ,∗,...,∗)foreachi∈ {1,...,r}(where∗denotes i i unknown, unimportant elements). On the other hand, by Lemma 2, for ′ ′ each h ∈ G there is some g ∈ G such that ψ(g) = (h,1,...,1). Thus ψ([x,g]) = ([a,h],1,...,1) and ψ([y ,g]) = ([b ,h],1,...,1) for i = 1,...,r. i i ′′ Now, [x,g],[y ,g] ∈ G implies that i ψ(G′′) ≥ γ (G)×.p..×γ (G). 3 3 Finally, Lemma 3 and another application of Lemma 2 yield the result. (cid:3) 3. Proof of the main Theorem Theorem 1 follows from Theorems A and B in [AGU] (the GGS-group case) and from the result in this section. Let us first establish some notation. Set G Gn b Gn G = , G = , G = . n st (n) n G′ n st (1)′ G n Gn Observe that in the same way in which ψ : st (1) → G×.p..×G holds, we G alsohave ψ(n) :stGn(1) → Gn−1×.p..×Gn−1. Denotingby πn theprojection from G to G , the following diagram commutes: n ψ st (1) G×.p..×G G πn πn−1×.p..×πn−1 ψ stGn(1) (n) Gn−1×.p..×Gn−1 Moreover, since ψ : st (1)′ −→ G′×.p..×G′ is an isomorphism, the map G ψb(n) : ssttGn((11))′ −→ Gn−1×.p..×Gn−1 Gn is well defined. ′ Proposition 6. Let G = ha,b ,...,b i be a multi-GGS-group. Then G ≥ 1 r st (r+1). G Proof. Wewillprovebyinductiononn ∈Nthatd(G ) ≥ nforn = 2,...,r+ n 1. This implies in particular that G is generated by r+1 elements, and r+1 ′ ′ ′ then |G | = |G :G|, which implies that G = G st (r+1) and the result r+1 G follows. b p ′ Observe that d(G ) = d(G ) = d(G ), because G ≤ G and then n n n n n ′ ′ ′ Φ(G ) = G . Since G and st (1) are contained in Φ(G ), the minimum n n n Gn n number of generators does not change. Thecase n= 2 is obvious because if G is generated by one element, then 2 st (1) = st (2) and this cannot happen. Let us suppose the statement true G G MULTI-GGS-GROUPS HAVE THE CONGRUENCE SUBGROUP PROPERTY 5 forn≤ r,thatisd(G ) ≥ n. SinceG iselementary abelian, anditisgener- n n ated by the projections of the generators of G, we can choose a basis and we may assume that Gn = ha,b1,...,bn−1,...i where the first n generators are linearly independent in G . We want to prove the case n+1. Suppose for a n b contradiction that G can be generated by n elements. In order to gener- n+1 ate Gb , we need some element congruent to ba modulo st (1)′. On the n+1 Gn+1′ other hand, by the Burnside Basis Theorem, since st (1)/G has rank Gn+1 n+1 b at most n−1 because we assumed that d(G )≤ n, we can choose a basis n+1 b b ′ b1,...,bn−1 ofstGn+1(1)/stGn+1(1) (theseelements arelinearlyindependent because they map onto b1,...,bn−1, respectively, which are assumed to be linearly independent). We may thus suppose that Gbn+1 = hba,bb1,...,bbn−1i. Then bbn =bb1i1,0(bbba1)i1,1...(bbba1p−1)i1,p−1...bbnin−−11,0(bbban−1)in−1,1...(bbbanp−−11)in−1,p−1 b with i ∈ F . But then the images under ψ of the element on the j,k p (n) b b left hand side and right hand side must be equal in G . Since ψ (b ) = n (n) n (aen,1,...,aen,p−1,bn we are forced to have ij,k = 0 for k 6= 0. This means that en = i0,1e1 +···+i0,nen−1, which is impossible, because the defining b vectors are linearly independent. Thus, d(G ) ≥ n+1 and the theorem n+1 follows by induction. (cid:3) Corollary 7. Let G be as in Proposition 6. Then G has the congruence subgroup property. As remarked previously, it suffices to show, by Proposition 2.4 of [AGU] ′′ and Lemma 2, that G contains some level stabilizer. Lemma 5 and Propo- sition 6 yield that ψ (G′′) ≥ st (r+1)×.p.2.×st (r+1) ≥ ψ (st (r+3)). 2 G G 2 G ′′ Thus G ≥ st (r+3). G References [AGU] G.A.Ferna´ndez-Alcober,A.GarridoandJ.Uria-Albizuri.Onthecongruencesub- group propertyfor GGS-groups. To appear in Proc. Amer. Math. Soc. [AKT] T. Alexoudas, B. Klopsch and A. Thillaisundaram. Maximal subgroups of multi- edge spinal groups. Groups Geom. Dyn. 10 (2016), 619-648. [AZ14] G.A. Ferna´ndez-Alcober and A. Zugadi-Reizabal. GGS-groups: order of congru- encequotientsandHausdorffdimension.InTransactions oftheAmericanMathemat- ical Society, 366 (2014), 1993–2007. [Gri80] R.I. Grigorchuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1) (1980), 53–54. [GS83] N.Guptaand S.N.Sidki.OntheBurnsideproblem for periodic groups.In Math- ematische Zeitschrift, 182 (1983), 385–388. Mathematisches Institut, Heinrich-Heine-Universita¨t Du¨sseldorf, Univer- sita¨tsstr. 1, 40225, Du¨sseldorf, Germany E-mail address: [email protected] DepartmentofMathematics,UniversityoftheBasqueCountryUPV/EHU, 48080 Bilbao, Spain E-mail address: [email protected]