p-Basilica Groups

Mediterr. J. Math. (2022) 19:275 https://doi.org/10.1007/s00009-022-02187-z 1660-5446/22/060001-28 published online October 31, 2022 c The Author(s) 2022 p-Basilica Groups Elena Di Domenico, Gustavo A. Fern´andez-Alcober, Marialaura Noce and Anitha Thillaisundaram Abstract. We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups. Mathematics Subject Classification. 20E08, 20E18, 28A78, 20E28. Keywords. Groups acting on rooted trees, weakly branch groups, congruence subgroup properties, Hausdorff dimension, maximal subgroups. 1. Introduction Let pbe a prime and let Tbe the p-adic tree. Groups acting on p-adic trees have been well studied over the past decades, owing to their nice structure, their importance in the theory of just infinite groups, and the fact that many such groups have exotic properties; see [6] for a good introduction. Lots of the interesting examples of such groups share one common property: that of being branch or weakly branch, where branchness is a measure of how close The first three authors are supported by the Spanish Government grant MTM2017-86802P, partly with FEDER funds, and by the Basque Government grant IT974-16. The first and third authors are also partially supported by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA—INdAM). The first author acknowledges support from the Department of Mathematics of the University of Trento. The third author acknowledges financial support from a London Mathematical Society Joint Research Groups in the UK (Scheme 3) grant. The fourth author acknowledges support from EPSRC, grant EP/T005068/1. 275 Page 2 of 28 E. Di Domenico et al. MJOM the structure of the group resembles the structure of the full automorphism group of the tree T; see Sect. 2for precise definitions. An example of such a group is the Basilica group. This group acts on the binary tree, is weakly branch but not branch, is torsion-free, is of exponential growth, is generated by a finite automaton, and is not subexponentially amenable [22]. The Basilica group is generated by two elements, aand b, which are recursively defined as follows: a=(1,b)andb=(1,a)σ, where σis the cyclic permutation (1 2), which swaps the two maximal subtrees, and the notation (x, y) indicates the independent actions on the respective maximal subtrees, for xand yautomorphisms of the binary tree. The Basilica group is also the iterated monodromy group of the complex polynomial z2−1, and is a notable example in Nekrashevych’s theory which links automata groups to complex dynamics; see [27, Section 6.12.1]. Furthermore, the Julia set of z2−1, which is the set of accumulation points of the backward iterations of an arbitrary point in the complex plane under z2−1, can be approximated by a sequence of finite Schreier graphs obtained by the action of the Basilica group at each level of the binary tree; see [12], where all limits of finite Schreier graphs of the Basilica group were classified up to isomorphism. The Basilica group has also been studied in other contexts in group theory, for example in [13] it has been proved that the Basilica group has no nontrivial Engel elements. In this paper, we are interested in a natural generalisation of the Basilica group, which we call the p-Basilica group, that acts on the p-adic tree, for p any prime. Such a group Gis generated by the following 2 elements: a=(1,p−1 ...,1,b)andb=(1,p−1 ...,1,a)σ, where σis the cyclic permutation (1 2 ··· p). Clearly the 2-Basilica group coincides with the Basilica group. This generalisation of the Basilica group mirrors Sidki and Silva’s generalisation of the Brunner–Sidki–Vieira group; see [32] and [11]. A different generalisation of the Basilica group to the p-adic tree, with pgenerators, was first investigated by Sasse in her Master thesis [30], and Sasse’s work has been recently developed further by Petschick and Rajeev [29]. As seen below, our 2-generator p-Basilica groups, also known in [29] as Basilica groups of level 2, are more similar to the Basilica group. The generalisations of the Basilica groups considered by Sasse, Petschick and Rajeev include the Basilica groups of levels strictly greater than 2, and they differ more significantly from the Basilica group. We prove the following in Sects. 3and 4; see Theorem 3.3, Theorem 3.5 and Lemma 4.10. Theorem A. Let Gbe a p-Basilica group, for pa prime. Then Gis not branch, but it is weakly regular branch over G. Furthermore: (i) G/G∼ =Z×Z. (ii) G/γ3(G)∼ =Z. (iii) G/G ∼ =Z2p−1. (iv) γ3(G)/G ∼ =Z2p−2. MJOM p-Basilica Groups Page 3 of 28 275 Additionally in Sects. 3and 4we establish other basic properties of the p-Basilica groups G, such as being torsion-free (Theorem 3.6), contracting (Theorem 3.8), just non-solvable (Corollary 4.3), and having its automorphism group Aut(G) equal the normaliser of Gin Aut(T) (Corollary 4.6). We also show that the groups are super strongly fractal (Theorem 4.5), which means for any n∈N, the projection of the nth level stabiliser StG(n)atany nth level vertex is the whole of G; see Section 2for the precise definition. This yields the first examples of finitely generated weakly branch, but not branch, groups that are super strongly fractal. Now, one of the main properties concerning the p-Basilica groups that we investigate is the congruence subgroup property, where we say that G≤ Aut(T) has the congruence subgroup property if every finite-index subgroup of Gcontains a level stabiliser StG(n) for some n∈N. Equivalently, the group Ghas the congruence subgroup property if the profinite completion of Gequals its closure in Aut(T). Garrido and Uria-Albizuri [19] introduced a weaker version of the congruence subgroup property: a group G≤Aut(T) is said to have the pcongruence subgroup property if every normal subgroup of p-power index contains some level stabiliser. In [19], examples of weakly branch, but not branch, groups with the p-congruence subgroup property and not the congruence subgroup property were provided. For podd, their examples were the Grigorchuk–Gupta–Sidki (GGS-)groups defined by the constant vector, and for p= 2, their example was the Basilica group. In Sect. 5, we extend this result to p-Basilica groups, for all odd primes p: Theorem B. Let Gbe a p-Basilica group, for pa prime. Then Ghas the pcongruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. We recall that a group G≤Aut(T) has the weak congruence subgroup property if every finite-index subgroup contains the derived subgroup of some level stabiliser; cf. [31]. The p-Basilica groups are the first examples of weakly regular branch groups with the p-congruence subgroup property but not the weak congruence subgroup property. In Subsect. 5.2, we compute the orders of the congruence quotients G/ StG(n) for all n∈N,forap-Basilica group G. This enables us to compute the Hausdorff dimension of the closure of the p-Basilica group Gin the group Γ of p-adic automorphisms of T. We recall that Γ∼ =lim ←− n∈N Cpn ···Cp is a Sylow pro-psubgroup of Aut(T) corresponding to the p-cycle (1 2 ··· p). For a subgroup Gof Γ, the Hausdorff dimension of the closure of Gin Γ is given by hdimΓ(G) = lim n→∞ log |G :St G(n)| log |Γ:St Γ(n)|∈[0,1],(1.1) 275 Page 4 of 28 E. Di Domenico et al. MJOM where lim represents the lower limit. The Hausdorff dimension of Gis a measure of how dense Gis in Γ. This concept was first applied by Abercrombie [1] and by Barnea and Shalev [2] in the more general setting of profinite groups. We note that the Hausdorff dimension of the closures of several prominent weakly branch groups, such as the first [20] and second [28] Grigorchuk groups, the siblings of the first Grigorchuk group [35], the GGS-groups [15], the branch path groups [14], and generalisations of the Hanoi tower groups [33], have been computed. Theorem C. Let Gbe a p-Basilica group, for pa prime. Then (i) The orders of the congruence quotients of Gare given by logp|G:St G(n)|=pn−1+pn−3+···+p3+p+n 2for neven, pn−1+pn−3+···+p4+p2+n+1 2for nodd. (ii) The Hausdorff dimension of the closure of Gin Γis hdimΓ(G)= p p+1. In Sect. 6, we give a recursive presentation, a so-called L-presentation, for the p-Basilica groups (Proposition 6.3), plus we show that the p-Basilica groups are amenable but not elementary subexponentially amenable (Lemma 6.2), and have exponential growth (Theorem 6.1); we refer to Sect. 6 for the definitions. To the best of our knowledge, the only other infinite family of weakly branch groups that are amenable but not elementary subexponentially amenable is the family of p-generator Basilica groups acting on the p-adic tree; see [30]. Francoeur [17, Thm. 4.28] proved that the Basilica group does not possess maximal subgroups of infinite index, thus providing the first example of a weakly branch but not branch group without maximal subgroups of infinite index. Also, the Basilica group has non-normal maximal subgroups [16, Cor. 8.3.2]. In Subsect. 6.3, we extend these results to p-Basilica groups for all primes p, likewise giving another infinite family of weakly branch groups with such properties. Note that the first infinite family of weakly branch, but not branch, groups without maximal subgroups of infinite index was given by Francoeur and Thillaisundaram in [18], namely the GGS-groups defined by the constant vector. Theorem D. Let Gbe a p-Basilica group, for pa prime. Then all maximal subgroups of Ghave finite index, and Ghas infinitely many non-normal maximal subgroups. Notation. Throughout, we use left-normed commutators, for example, [x, y, z]=[[x, y],z]. For a group G, a subgroup H≤Gand g∈G, we write [H,g]=[h, g]|h∈H.AlsoifNGthen we write g≡Nhto mean that the images of gand hin G/N coincide. For Ga group and pa prime, we write Wp(G) for the wreath product of Gwith a cyclic group of order p. MJOM p-Basilica Groups Page 5 of 28 275 2. Preliminaries 2.1. The Group Aut(T) Let pbe a prime and let Tbe the p-adic tree, i.e. the rooted tree having pdescendants at every vertex. If we choose an alphabet Xwith pletters, Tcan be represented as the graph whose vertices are the elements of the free monoid X∗, the root is the empty word ∅,andwis a descendant of u provided that w=ux with x∈X. For a given vertex u, the set of vertices uv with v∈X∗are said to succeed u.TheyformatreeTurooted at u, which is isomorphic to T.We denote by |u|the length of uas a word. For every n∈N∪{0}, the set Lnof all words of length nis called the nth layer of the tree. Automorphisms of TasagraphformagroupAut(T) under composition. Let ube a vertex of Tand let f∈Aut(T). We use exponential notation for images of automorphisms and, more generally, permutations. Thus the image of uunder f∈Aut(T)isuf. Note that automorphisms leave each layer invariant, so ufand ubelong to the same layer. The label of fat uis the permutation f(u) of the alphabet Xdefined by the rule (ux)f=ufxf(u),for every x∈X. The portrait of fis the set of all labels of f, and there is a one-to-one correspondence between automorphisms of Tand portraits. The support of fis the set of vertices with non-trivial label. We say that fis rooted if the support is contained in the root, and fis directed if the support is infinite and consists only of descendants of a given infinite path starting at the root. In a similar way, the section fuof fat uis the automorphism of T defined by (uv)f=ufvfu,for every v∈X∗. For all f,g ∈Aut(T)andu, v ∈X∗,wehave(fu)v=fuv,(fg)u=fuguf, (fg)ug=(gu)−1fuguf.(2.1) 2.2. Subgroups of Aut(T) For a vertex uof T,thevertex stabiliser St(u) is the subgroup consisting of all automorphisms of Tfixing u. The map ψu:f→ fuis a homomorphism from St(u) onto Aut(T). For every n∈N,thenth level stabiliser is St(n)=  u∈Ln St(u). Then St(n) is a normal subgroup of Aut(T)andAut(T) is isomorphic to the inverse limit of the finite groups Aut(T)/St(n). Hence Aut(T) is a profinite group with {St(n)}n∈Nas a basis of neighbourhoods of the identity. For every n∈N, we have an isomorphism ψn:St(n)−→ Aut(T)×pn ···×Aut(T) f−→ (fu)u∈Ln. 275 Page 6 of 28 E. Di Domenico et al. MJOM The map ψ1can be extended to an isomorphism ψ: Aut(T)−→ Aut(T)Sym(X) f−→ ψ1(f)τ, where τis the label of fat the root. Thus we can define automorphisms of Aut(T) by giving their image under ψ. Let Sym(X) be the symmetric group over the alphabet X.Ifσ∈ Sym(X) is a fixed p-cycle, we can obtain a Sylow pro-psubgroup Γ(σ)of Aut(T) by considering all automorphisms whose portrait only contains labels from σ;thatis, Γ(σ)=f∈Aut(T)|f(u)∈σfor all u∈X∗∼ =lim ←− n∈N Cpn ···Cp. As mentioned in the introduction, we also write Γ = Γ(1 2 ··· p). Now let Gbe a subgroup of Aut(T). We write StG(u) = St(u)∩G and StG(n) = St(n)∩G. The latter is a normal subgroup of Gand we set Gn=G/ StG(n), which is called the nth congruence quotient of G.Wesay that Gis •Level-transitive if it acts transitively on Lnfor every n∈N. •Self-similar if all sections of elements of Gat all vertices belong to G. •Fractal if it is self-similar and ψu(StG(u)) = Gfor every vertex uof the tree. •Super strongly fractal if it is self-similar and ψu(StG(n)) = Gfor every u∈Lnand every n∈N. Note that if Gis self-similar then ψn(StG(n)) ⊆G×pn ···×Gfor all n,and if furthermore Gis contained in a Sylow pro-psubgroup Γ(σ) then ψ(G)⊆ Gσ=Wp(G). The nth congruence quotient of Γ(σ) is isomorphic to the iterated wreath product of ncopies of Cpand Γ(σ) is the inverse limit of these finite p-groups. The rigid vertex stabiliser RistG(u)ofavertexuin Gis the subgroup consisting of all automorphisms in Gthat fix all vertices outside Tu. Then for every n∈Nwe define the rigid nth level stabiliser as RistG(n)=RistG(u)|u∈Ln=u∈Ln RistG(u)G. If Gis level-transitive we say that Gis a branch group if |G:Rist G(n)|<∞ for all n∈N, and that it is weakly branch if RistG(n)= 1 for all n.IfGis level-transitive and self-similar, and K×···×K⊆ψ(K) for some K≤G, we say that Gis regular branch over Kif |G:K|<∞and that Gis weakly regular branch over Kif K= 1. Observe that being (weakly) regular branch implies being (weakly) branch. 2.3. p-Basilica Groups Let X={x1,...,x p}and let Γ be the Sylow pro-psubgroup of Aut(T) corresponding to the p-cycle σ=(x1x2··· xp). The p-Basilica group is the subgroup Gof Γ generated by the automorphisms aand bgiven by ψ(a)=(1,...,1,b)andψ(b)=(1,...,1,a)σ. MJOM p-Basilica Groups Page 7 of 28 275 (a) (b) Figure 1. The portrait of the generators of a p-Basilica group Figure 2. The 3-Basilica automaton Note that the 2-Basilica group is the well-known Basilica group mentioned in the introduction. The portraits of aand bare described in Fig. 1. We recall that an automorphism f∈Aut(T) is called bounded if the sets {w∈Xn|fw=1}have uniformly bounded cardinalities over all n.A group G≤Aut(T) is said to be a bounded automata group if Gis finitely generated, self-similar and every element g∈Gis bounded and finite state (i.e. the set {gv|v∈X∗}is finite). For every prime pthe p-Basilica group is a group generated by a finite bounded automaton with set of states {Id; a;b}. In Fig. 2, there is the generating automaton in the case p=3. We remark that being an automata group, for any prime p,thep-Basilica group has solvable word problem, [27, Prop. 2.13.8]. 3. First Properties In this section, we prove some basic properties of the p-Basilica groups. We start with the following elementary but essential result. Lemma 3.1. Let Gbe a p-Basilica group, for a prime p.ThenGis fractal and level-transitive. Proof. By [36, Lem. 2.7], it suffices to show that Gacts transitively on the first layer and that ψx(StG(x)) = Gfor some x∈X(see also [21, Sec. 3]). 275 Page 8 of 28 E. Di Domenico et al. MJOM This is straightforward since bacts transitively on the first layer and since ψ(a)=(1,...,1,b)andψ(bp)=(a,...,a).  Next we consider the stabilisers in Gof the first two layers. Recall that Gn=G/ StG(n), and for convenience, we set A=aGand B=bG. Lemma 3.2. Let Gbe a p-Basilica group, for a prime p.Then: (i) StG(1) = Abp=a, ab,...,a bp−1,b pand G1=bStG(1)∼ =Cp. (ii) G2∼ =CpCpis a p-group of maximal class of order pp+1. Proof. (i) Observe that a∈StG(1), and that bn∈StG(1) if and only if p|n. Since StG(1) Gwe get Abp≤StG(1). Note that Abpis normal in Gsince G/A is cyclic. As a consequence, the inclusion Abp≤StG(1) is an equality, since G/Abphas order pand StG(1) is a proper subgroup of G. (ii) Since ψ(bp)=(a,...,a)anda∈StG(1), we have bp∈StG(2). By (i), if we consider G2=G/ StG(2), and we denote an element in the quotient using the bar notation in G2,wehave StG2(1) = a, ab,...,abp−1. Since ahas order pin G2and the tuples ψ(a)=(1,...,1,b)andψ(abi)=(1,i−1 ...,1,a −1ba, 1,...,1),for 1 ≤i≤p−1, commute with each other, StG2(1) is elementary abelian of order pp. Thus |G2|=pp+1. To complete the proof, observe that since G≤Γ, the quotient G2=G/ StG(2) embeds in Γ2=Γ/StΓ(2) ∼ =CpCp, and that the latter is a p-group of maximal class of order pp+1. Our next goal is to study the abelianisation of G. In the remainder, let A=aGand B=bGas before. Since G=a, b,wehaveA=aG, B=bG,andG=AB. Observe that ψ(a)=(1,...,1,b)andGbeing self-similar imply ψ(A)⊆B×···×B. (3.1) On the other hand, the map π:Wp(G)→G/Gsending (g1,...,g p)σito g1···gpGis clearly a group homomorphism. Since ψ(b)=(1,...,1,a)σ,it follows that (π◦ψ)(B)⊆A/G.(3.2) Theorem 3.3. Let Gbe a p-Basilica group, for a prime p.Then: (i) G/A =bAand G/B =aBare infinite cyclic. In particular, the elements aand bhave infinite order in G. (ii) G/G=aG×bG∼ =Z×Z. (iii) A∩B=G. Proof. Since G/G=A/G·B/G, with A/G=aGand B/G=bG, both (ii) and (iii) follow immediately from (i). We prove that G/A and G/B are infinite simultaneously. Assume for a contradiction that, for some n∈N,wehaveeitheran∈Bor bn∈A, and let MJOM p-Basilica Groups Page 9 of 28 275 us choose nas small as possible. If bn∈A⊆StG(1) then n=pm for some m, and consequently, ψ(bn)=(am,...,a m)∈ψ(A)⊆B×···×B, by (3.1). Hence, am∈B, which is impossible since m<n. On the other hand, if an∈Bthen bnG=(π◦ψ)(an)∈(π◦ψ)(B)⊆A/G by (3.2). Thus, bn∈A, which we just proved is not the case. This completes the proof.  Next we study rigid stabilisers and the branch structure of G. To this purpose, the following result is very useful. It is given in [15, Prop. 2.18] for GGS-groups, but the same proof works more generally for level-transitive fractal groups. We state this general version here for the convenience of the reader. Lemma 3.4. Let Gbe a level-transitive fractal subgroup of Aut(T),andlet Land Nbe two normal subgroups of G. Suppose that L=SGand that (1,...,1,s,1,...,1) ∈ψ(N)for every s∈S,wheresappears always at the same position in the tuple. Then L×···×L⊆ψ(N). Theorem 3.5. Let Gbe a p-Basilica group, for a prime p.Then (i) RistG(1) = Awith ψ(A)=B×···×B. In particular, the group Gis not branch. (ii) ψ(StG(1))=G×···×G. As a consequence, the group Gis weakly regular branch over G. Proof. (i) We already know from (3.1) that ψ(A)⊆B×···×B, and the reverse inclusion follows from Lemma 3.4, since ψ(a)=(1,...,1,b). Hence A≤RistG(1) ≤StG(1) = Abpand so RistG(1) = Abpnfor some n. Since ψ(Abpn)=(B×···×B)(an,...,a n) and ahas infinite order modulo Bby Theorem 3.3(i), it follows from the definition of the rigid stabiliser that n= 0. Hence RistG(1) = A, which has infinite index in G,soGis not branch. (ii) The inclusion ⊆is clear. For the reverse inclusion, observe that ψ([bp,a]) = (1,...,1,[a, b]). Since G=[a, b]G, the result follows from Lemma 3.4.  Theorem 3.6. Let Gbe a self-similar subgroup of Aut(T)and suppose that there exists a torsion-free quotient G/N with N≤StG(1).ThenGis torsionfree. In particular, the p-Basilica groups are torsion-free. Proof. For every n∈N∪{0},letPnstand for the set of torsion elements in St G(n)\St G(n+ 1). Then our goal is to prove that these sets are all empty. By way of contradiction, suppose that Pn=∅for some n, which we choose as small as possible. 275 Page 16 of 28 E. Di Domenico et al. MJOM which is contrary to (4.8). This proves the case n= 2. The general case follows in a similar fashion by induction on n. We close this section by determining the structure of the quotients G/γ3(G), G/G,andγ3(G)/G, which is key for Sect. 5. We need a couple of lemmas. Lemma 4.8. Let Gbe a p-Basilica group, for a prime p. For every n∈N,we have ψ(StG(n)) = StG(n−1) ×···×StG(n−1)Cpβ(n−1) . Proof. The inclusion ⊇is obvious from Theorem 4.1(i) and Theorem 4.4(i). For the other direction, let g∈StG(n) and write ψ(g)=(w1,...,w p) (bi1,...,b ip), where the first factor is in G×···×Gand the second is in C.Fix an index j∈{1,...,p}. Since wjbij∈StG(n−1) we have bij∈GStG(n−1), and then pβ(n−1) divides ijby Theorem 4.4.Thus,bij∈StG(n−1) and it follows that also wj∈StG(n−1). This proves the result.  Lemma 4.9. Let Gbe a p-Basilica group, for pa prime. Then the order of [a, b]modulo γ3(G)St G(n)is at least pβ(n−1) for every n∈N. Proof. Since [a, b] and [a, b−1] are inverse conjugate, we prove the result for the order of [a, b−1]. We use induction on n. The result is obvious if n=1,so we suppose that n≥2. If [a, b−1]pm∈γ3(G)St G(n), we want to show that m≥β(n−1). By way of contradiction, we assume that m<β(n−1). By applying ψto [a, b−1]pmand using Theorem 4.1(iii) and Lemma 4.8,weget c−pm 0=ydgc, (4.9) where y=yk0 0···ykp−2 p−2for some k0,...,k p−2∈Z,d∈D,g∈StG(n−1) × ···×StG(n−1), and c∈Cpβ(n−1) . If we reduce (4.9) modulo G×···×G and use that y0reduces to c−p 0,weget cpm−pk0+k1 0c−k1+k2 1···c−kp−3+kp−2 p−3c−kp−2 p−2∈Cpβ(n−1) .(4.10) Since c0,...,c p−2form a basis for the free abelian group C, it follows that all exponents in (4.10) are divisible by pβ(n−1) and, as a consequence, so is pm−pk0. Since m<β(n−1), it follows that the p-part of pk0is pm.Now since B=bGis abelian modulo γ3(G), the map τ:B×···×B−→ B/γ3(G) (g1,...,g p)−→ g1···gpγ3(G) is a group homomorphism. Observe that both Cand Dlie in the kernel of τ, and that τ(y0)=[a, b−1]γ3(G). Hence, by applying τto (4.9), we get [a, b−1]k0∈γ3(G)St G(n−1). Since the p-part of k0is pm−1, by the induction hypothesis we have m−1≥ β(n−2), and so m≥β(n−2) + 1 ≥β(n−1), contrary to our assumption. This completes the proof.  Theorem 4.10. Let Gbe a p-Basilica group, for pa prime. Then MJOM p-Basilica Groups Page 17 of 28 275 (i) G/γ3(G)∼ =Z. (ii) G/G ∼ =Z2p−1. (iii) γ3(G)/G ∼ =Z2p−2. Proof. (i) We observe that G/γ3(G) is cyclic and generated by the image of [a, b]. From Lemma 4.9, the order of [a, b] tends to infinity modulo γ3(G)St G(n)asngoes to infinity. Hence the statement immediately follows. (ii) The result follows from (i) and from Theorem 4.1 since we have G G ∼ =ψ(G) ψ(G)∼ =G γ3(G)×p ···× G γ3(G)×C ∼ =Z×p ···×Z×Zp−1. This completes the proof since (iii) is a straightforward consequence of (i) and (ii).  We recall the integral Heisenberg group H3(Z), which is the 2-generated group of 3×3 upper unitriangular matrices with integral entries, and has the presentation x, y, z |z=[x, y],xz=zx, yz =zy. Corollary 4.11. Let Gbe a p-Basilica group, for pa prime. Then G/γ3(G)is isomorphic to the integral Heisenberg group H3(Z). Proof. The proof is analogous to [19, Prop. 23] or to [17, Prop. 4.8], where it was proved for the case p= 2, using different approaches.  As noted in [17, Cor. 4.9], the previous result yields an alternative proof that the p-Basilica groups are not branch. As every proper quotient of a branch group is virtually abelian and the integral Heisenberg group is not virtually abelian, the result follows. 5. Congruence Subgroup Properties and Hausdorff Dimension 5.1. Congruence Subgroup Properties Let Gbe a p-Basilica group, for pa prime. Since G/G∼ =Z×Z, the group G does not have the congruence subgroup property as all quotients of Gby level stabilisers are p-groups. In this subsection we show that Ghas the pcongruence subgroup property (p-CSP for short) but not the weak congruence subgroup property. We recall that if Gis a subgroup of Aut(T)andNG, we say that G/N has the p-congruence subgroup property (or that Ghas the p-congruence subgroup property modulo N)ifeveryJG, such that G/J is a finite p-group and N≤J, contains some level stabiliser of G. According to [19, Lem. 6], if N≤Mare two normal subgroups of Gsuch that both G/M and M/N have the p-CSP then also G/N has the p-CSP. We need a couple of lemmas before proving that the p-Basilica groups have the p-CSP. 275 Page 18 of 28 E. Di Domenico et al. MJOM Lemma 5.1. Let Nand Gbe subgroups of Aut(T)with NGand G/N free abelian of rank r,forsomer∈N. Suppose that, for large enough n∈N,we have G/N StG(n)∼ =Cpλ1(n)×···×Cpλr(n),(5.1) with limn→∞ λi(n)=∞for 1≤i≤r.ThenG/N has the p-CSP. Proof. Let N≤JG, where |G :J|=pm, for some m∈N. Then Gpm≤J. Now choose an integer nsuch that λi(n)≥mfor 1 ≤i≤r.By(5.1), for large enough nwe have |G/N StG(n):(G/N StG(n))pm|=prm =|G/N :(G/N )pm|, which implies NStG(n)Gpm=NGpm, since by the third isomorphism theorem, G/N StG(n) (G/N StG(n))pm=G/N StG(n) NStG(n)Gpm/N StG(n)∼ =G NStG(n)Gpm and G/N (G/N )pm=G/N NGpm/N ∼ =G NGpm. Hence, StG(n)≤NGpm≤J,andG/N has the p-CSP.  The following lemma is a slight generalisation of [19, Thm. 1], which corresponds to the case when Nis chosen so that L≤K, and its proof is very similar. For convenience, we include the proof below. Lemma 5.2. Let GbeasubgroupofAut(T)that is weakly regular branch over a normal subgroup K.LetNbe a normal subgroup of Gsuch that: (i) K≤N≤K. (ii) If L=ψ−1(N×···×N)then G/N ,N/L,andN/Khave the p-CSP. Then Ghas the p-CSP. Proof. Set Lm=ψ−1 m(N×pm ···×N) for every m∈N. Note that LmG, since Gis self-similar and NG. We will prove by induction on mthat G/Lmhas the p-CSP. Since L1=Land both G/N and N/L have the p-CSP, the result is true for m= 1. Now we suppose that the result holds for mand we will show it for m+ 1. Observe that it suffices to prove that Lm/Lm+1 has the p-CSP. Recall that ψm(Lm)=N×pm ···×N and ψm(Lm+1)=L×pm ···×L, so ψminduces an isomorphism between Lm/Lm+1 and N/L ×pm ···×N/L. Since ψm(StLm(n)) = StN(n−m)×pm ···×StN(n−m) for every n≥m, MJOM p-Basilica Groups Page 19 of 28 275 and N/L has the p-CSP, it follows that N/L ×pm ···×N/L ∼ =(N×pm ···×N)/(L×pm ···×L)∼ =Lm/Lm+1 has the p-CSP. To conclude that Ghas the p-CSP, let JGbe such that G/J is a finite p-group. Since Gis level-transitive, we have RistG(m)≤Jfor some m∈N by [19, Lem. 4]. Define the subgroup Km≤RistG(m) by the condition ψm(Km)=K×pm ···×K. Since K≤Nit follows that K m≤Lm. Now taking into account that N/K has the p-CSP, the same argument as in the first paragraph of the proof yields that Lm/K mhas the p-CSP as well. Thus G/K mhas the p-CSP for every m∈N. Since K m≤RistG(m)≤J, this proves that Jcontains some level stabiliser in G, and consequently Ghas the p-CSP.  Theorem 5.3. Let Gbe a p-Basilica group, for a prime p.ThenGhas the p-CSP. Proof. We apply Lemma 5.2 with K=N=G.ThusifL=ψ−1(G×···×G) then it suffices to prove that G/G,G/L and L/G have the p-CSP. Note that then also G/G has the p-CSP. First of all, the factor group G/Ghas the p-CSP by Lemma 5.1, since G/GStG(n)∼ =Cpβ(n−1) ×Cpβ(n)with β(n)=n/2, according to Theorem 4.4(ii). Next we deal with G/L, which is free abelian of rank p−1 by Theorem 4.1(i). By Lemma 4.8,wehaveψ(LStG(n)) = (G×···×G)Cpβ(n−1) and consequently G/L StG(n)∼ =C/Cpβ(n−1) . Hence, this case also follows from Lemma 5.1. Let us finally consider the case of L/G.Wehave L/G ∼ =(G×···×G)/(γ3(G)×···×γ3(G)) ∼ =Zp. Since ψ(G StL(n)) = γ3(G)St G(n−1) ×···×γ3(G)St G(n−1), it follows that L/G StL(n)∼ =G/(γ3(G)St G(n−1)) ×···×G/(γ3(G)St G(n−1)), and we can once again apply Lemma 5.1,bytakingintoaccountLemma4.9.  We note that in [29, Thm. 1.10] it was shown that s-generator Basilica groups, for s>2, have the p-CSP. It is worth mentioning that there are key structural differences between these groups and the p-Basilica groups; compare [29, Thm. 1.9]. Now we complete the proof of Theorem B. Recall that a group G≤ Aut(T) has the weak congruence subgroup property if every finite-index subgroup contains the derived subgroup of some level stabiliser. 275 Page 20 of 28 E. Di Domenico et al. MJOM Theorem 5.4. Let Gbe a p-Basilica group, for a prime p.ThenGdoes not have the weak congruence subgroup property. Proof. Let q=pbe a prime, and let N=aq,b q,[a, b]qγ3(G), which is normal and of finite index in G. By Corollary 4.11,wehaveG/N ∼ =H3(q). We claim that StG(n)≤ Nfor every odd n, which is enough to prove the theorem. Arguing by way of contradiction, since by Theorem 4.7 we have ψn(StG(n))=G×pn ···×G,andaccordingto(4.7) ψn([ap(n−1)/2,b p(n+1)/2]) = (1,pn−1 ... ,1,[b, a]) ∈G×pn ···×G for odd n, it follows that [ap(n−1)/2,b p(n+1)/2]∈N.Asγ3(G)≤N,weget [a, b]pn∈N. Since also [a, b]q∈N, we conclude that [a, b]∈N. This contradicts the fact that G/N ∼ =H3(q).  5.2. Hausdorff Dimension In this subsection, we determine the orders of the congruence quotients of the p-Basilica groups, and we compute their Hausdorff dimensions. That is, we prove Theorem C, which for convenience we recall here. Theorem C.LetGbe a p-Basilica group, for pa prime. Then: (i) The orders of the congruence quotients of Gare given by logp|G:St G(n)|=pn−1+pn−3+···+p3+p+n 2for neven, pn−1+pn−3+···+p4+p2+n+1 2for nodd. (ii) The Hausdorff dimension of the closure of Gin Γ is hdimΓ(G)= p p+1. Proof. (i) We argue by induction on n. The case n= 1 is clear, so we assume n≥2. Write n=2m+e, with e= 0 or 1. We need to establish that logp|G:St G(n)|=pn+1 −p1+e p2−1+m+e. Note that, by Theorem 4.4, |G:St G(n)|=|G:GStG(n)||GStG(n):St G(n)| =pn|G:St G(n)| and that |G:St G(n)|coincides with |ψ(G):ψ(StG(n))| =|(G×p ···×G)C:(St G(n−1) ×p ···×StG(n−1))Cpβ(n−1) | =p(p−1)β(n−1) |G:St G(n−1)|p =p(p−1)β(n−1)−p(n−1) |G:St G(n−1)|p, MJOM p-Basilica Groups Page 21 of 28 275 wherewehaveusedLemma4.8 and the fact that Cis free abelian of rank p−1. Here β(n−1) = (n−1)/2as before. Thus, logp|G:St G(n)| =plogp|G:St G(n−1)|+(p−1)(β(n−1) −n+1)+1 =plogp|G:St G(n−1)|−(p−1)(m+e−1) + 1, since β(n−1) = m. Now the result follows from the induction hypothesis. (ii) To get the Hausdorff dimension of Gin Γ, we just need to take into account formula (1.1) and the fact that logp|Γ:St Γ(n)|=1+p+···+pn−1=pn−1 p−1.  We remark that the Hausdorff dimension of the Basilica group was given by Bartholdi in [4]. Also the Hausdorff dimensions of the generalised Basilica groups were computed in [29, Thm. 1.7] using a very different approach. 6. Further Properties 6.1. Growth and Amenability Before proving the main results of this subsection, we need some preliminary definitions, namely the notions of growth of groups and amenability. Let Gbe a group generated by a finite symmetric subset S. The length function on Gis a metric on Gand therefore one can define the ball of radius n: B(n)={g∈G:|g|≤n}. We say that the map γ:N0−→ [0,∞) where γ(n)=|B(n)|,isthegrowth function of G. If we consider two growth functions γ1,γ 2, we say that γ2dominates γ1and we write γ1γ2if there exist C, α > 0 such that γ1(n)≤Cγ2(αn) for every n∈N.Ifγ1γ2and γ2γ1, we write γ1∼γ2.Itiseasytosee that this is an equivalence relation. Notice also that all growth functions of a finitely generated group are equivalent. If γ(n)nafor some a∈N, we say that Ghas polynomial growth. Instead Gis said to have exponential growth if limn→∞ γ(n)1/n >1 (notice that such a limit always exists). Finally γ(n)hasintermediate growth if γ(n) is equivalent to neither of the above. Notice that it is also common to say that a group Ghas subexponential growth if limn→∞ γ(n)1/n =1,orequivalently, if γ(n)=ef(n)for some (increasing) function f:N−→ R+satisfying limn→∞ f(n)/n =0. Next, we say that a group Gis amenable if there is a finitely additive left-invariant measure μon the subsets of Gsuch that μ(G) = 1. We denote the class of amenable groups by AG. The class EG of elementary amenable groups is the smallest class of groups containing all abelian groups and finite 275 Page 22 of 28 E. Di Domenico et al. MJOM groups and closed under quotients, subgroups, extensions and direct unions. We have EG ⊆AG, and this inclusion is strict. Furthermore the class SG of elementary subexponentially amenable groups is the smallest class of groups which contains all groups of subexponential growth and is closed under taking subgroups, quotients, extensions, and direct unions. Of course, the class SG contains the class EG. In the following we determine the growth of a p-Basilica group G, for pan odd prime, and we prove that Gis amenable but not elementary subexponentially amenable. The corresponding versions of Theorem 6.1 and Lemma 6.2 for p= 2 were proved in [22, Lem. 4 and Prop. 4] and in [24, Cor. 9], respectively. Theorem 6.1. Let G=a, bbe a p-Basilica group, for pan odd prime. Then the semigroup generated by aand bis free. Consequently, the group Gis of exponential growth. Proof. Let uand vbe two different words representing the same element in the semigroup generated by aand b, and with ρ= max(|u|,|v|) minimal. We note that |u|b≡p|v|b, where |u|bdenotes the b-length in u, which is equivalent to the number of occurrences of bin u. A direct check shows that ρ≥4. Suppose first that ucontains no b’s. Therefore, we have u=ai= ψ−1((1,...,1,b i)), for some i∈N. Since |v|bis a non-zero multiple of p, one deduces that v1is a non-empty word, where ψ(v)=(v1,...,v p). Indeed, every component of ψ(v) contains an a. Certainly |v1|<ρ, and this contradicts the minimality of ρ. So the number of occurrences of bin uis at least one. Suppose that both |u|b≡p|v|b≡p1. Apart from the possibilities amb, for m∈N, all sections of uand vwill have length strictly less than |u| and |v|, respectively. To not contradict the minimality of ρ,wemusthave u=am1b=ψ−1((1,...,1,b m1a)σ)andv=am2b=ψ−1((1,...,1,b m2a)σ) for some m1,m 2∈N. However, as noted above, all sections of bm1aand bm2a decrease in length. Hence, |u|b≡p|v|b≡pkfor k>1. Now in this case, all sections of uand vhave length strictly smaller than uand vrespectively. This again contradicts the minimality of ρ, and the proof is complete.  Lemma 6.2. For pa prime, the p-Basilica group Gis amenable but not elementary subexponentially amenable. In particular it is not elementary amenable. Proof. For p= 2, the result follows from [22, Prop. 13] and [9]. Hence we assume that pis odd. Since the p-Basilica group Gis a bounded automata group, from [7] it follows that Gis amenable. So it suffices to show that Gis not elementary subexponentially amenable. Since Gis weakly regular branch over G,from[24, Cor. 3] the result follows provided that ψu(StG(u)) contains Gfor some vertex u. We observe that ψ([b−1,a]p)=(1,p−2 ...,1,b −p,b p)andψ([a, bp]) = (1,p−1 ...,1,[b, a]), thus ψu([b−1,a]p)=aand ψu([a, bp]) = b, where u=xpxp. This completes the proof.  MJOM p-Basilica Groups Page 23 of 28 275 6.2. An L-Presentation For an alphabet S, we denote by FSthe free group on S. A group Ghas an L-presentation, also called endomorphic presentation, if there exists an alphabet S, sets Qand Rof reduced words in FS,andasetΦofgroup homomorphisms φ:FS→FSsuch that Gis isomorphic to a group with the following presentation: S|Q∪ φ∈Φ∗ φ(R), where Φ∗is the monoid generated by Φ; that is, the closure of {1}∪Φ under composition. An L-presentation is finite if S,Q,Rare finite and Φ = {φ}consists of just one element. Finite L-presentations have been computed for the first Grigorchuk group [26], the Brunner–Sidki–Vieira group [11], the Grigorchuk supergroup [5], the Fabrykowski–Gupta group [3], the Gupta–Sidki 3-group [3], and the twisted twin of the Grigorchuk group [8]. An L-presentation for the Basilica group is given in [22], and this Lpresentation is not finite, since the corresponding set Φ consists of more than one element. This is also the case for the L-presentations of the p-generator Basilica groups acting on the p-adic tree and generalised Basilica groups; see [30] and [29] respectively. Following the strategy in [22, Sec. 4], one can easily check that the p-Basilica groups have the following L-presentation. Theorem 6.3. Let Gbe a p-Basilica group, for a prime p. The group Ghas the presentation G=a, b |ξkθm([a, abl])=1for k,m ∈N∪{0}and l∈{1,...,p−1}, where ξ:a→ bpand θ:a→ abp+1 b→ ab→ b are endomorphisms of F{a,b}. 6.3. Virtually Nilpotent Quotients and Maximal Subgroups In this final subsection, we study nilpotency and virtual nilpotency of quotients of a p-Basilica group G, and we prove Theorem Dabout maximal subgroups of G. The following lemma will be useful for both purposes. Lemma 6.4. Let Gbe a p-Basilica group, for a prime p.ThenGhas a proper quotient isomorphic to Wp(Z). Proof. Let L=ψ−1(G×···×G). We have G=Ab, and on the other hand ψ(A)=B×···×Bby Theorem 3.5(i). Hence ψinduces an isomorphism between G/L and the semidirect product (B/G× ··· × B/G)ψ(b). Observe that ψ(b)=(1,...,1,a)σacts as σon the direct product of pcopies of B/G, and that ψ(bp)=(a,...,a) acts trivially. If we set N=Lbp then it is clear that NGand that G/N ∼ =Wp(Z), since B/G∼ =Zby Theorem 3.3(ii).  275 Page 24 of 28 E. Di Domenico et al. MJOM Recall from Corollary 4.3 that the p-Basilica groups are just non-solvable. In [16, Sec. 8.3] it was shown that the Basilica group is not just non-nilpotent. On the other hand, by [16, Lem. 8.3.5 and Prop. 8.3.6], all proper quotients of the Basilica group are virtually nilpotent. We extend these results to the p-Basilica groups for all primes p. Theorem 6.5. Let Gbe a p-Basilica group, for a prime p.Then: (i) The group Gis not just non-nilpotent. (ii) Every proper quotient of Gis virtually nilpotent, but Gitself is not virtually nilpotent. Proof. (i) By Lemma 6.4, the group Ghas a proper quotient isomorphic to Wp(Z). By the main result in [10], this wreath product is not nilpotent. Hence, Gis not just non-nilpotent. (ii) From Theorem 4.1(ii), the map ψinduces an embedding of G/G into the wreath product Wp(G/γ3(G)). Since the latter is virtually nilpotent, the quotient G/G is also. Now since Gis weakly regular branch over Gand G/G is virtually nilpotent, it follows that every proper quotient of Gis also virtually nilpotent by [17, Thm. 4.10]. On the other hand, the group Gis not virtually nilpotent by Gromov’s celebrated theorem [23], in light of Theorem 6.1. Let us now consider the maximal subgroups of G. We first prove that G does not possess maximal subgroups of infinite index. The proof is analogous to that of [17, Sec. 4.4], however with a necessary change to the end of [17, Prop. 4.27]. Due to the proof being so similar, we refer the reader to [17, Sec. 4.4], and only record here the part that needs to be changed. Recall that a subgroup Hof a group Gis prodense if HN =Gfor all non-trivial normal subgroups Nof G. Since a maximal subgroup of infinite index is a proper prodense subgroup, it suffices to show that there are no proper prodense subgroups in a p-Basilica group G.ForHa proper prodense subgroup of G,by[17, Lem. 3.1 and Thm. 3.2], for all vertices u∈T,the subgroup ψu(StH(u)) is a proper prodense subgroup of G. We consider a prodense subgroup Hof G, and seek a vertex usuch that ψu(StH(u)) = G, which then proves the theorem. As in [17, Prop. 4.27], there is a vertex vsuch that either ab, b−1a∈ ψv(StH(v)) or ba, b−1a∈ψv(StH(v)). In the former case, we obtain a2∈ ψv(StH(v)). Since pis an odd prime, it follows that b−1ap∈ψv(StH(v)). Now ψ((b−1ap)p)=(a−1bp,...,a −1bp)andψ(a−1bp)=(a,...,a,b −1a). Therefore, for u=vx1x1, we have either a, ab ∈ψu(StH(u)) or a, ba ∈ψu(StH(u)), and we are done. In the latter case, we have ba, b−1a∈ψv(StH(v)), and so b2∈ψv(StH(v)). As before, we obtain bpa=ψ−1((a,...,a,ab)) ∈ψv(StH(v)). Setting u=vx1, we see that a, ba ∈ψu(StH(u)) and the result follows. We conclude by showing the existence of non-normal maximal subgroups in the p-Basilica groups. MJOM p-Basilica Groups Page 25 of 28 275 Proposition 6.6. Let Gbe a p-Basilica group, for pan odd prime. Then for every prime qsuch that pdivides q−1, the group Ghas a non-normal subgroup of index q. Proof. By Lemma 6.4, the group Ghas a quotient isomorphic to Wp(Z), and so also a quotient isomorphic to Wp(Z/qZ). Thus it suffices to find a non-normal subgroup of index qin the latter group. Let V=Z/qZ×···×Z/qZbe the base group of Wp(Z/qZ). The characteristic polynomial corresponding to the action of σon Vis Xp−1, which by the condition that pdivides q−1, has pdifferent roots in Z/qZ.Letλ=1 be one of these roots, and let U=ube the eigenspace of λin V. Then we can write V=U×Kfor a suitable subgroup K.IfwesetH=Kσthen Hhas index qin Wp(Z/qZ). At the same time, His not a normal subgroup of Wp(Z/qZ), since otherwise [u, σ]=uλ−1= 1 belongs to U∩H=1.  Observe that there are actually infinitely many non-normal maximal subgroups in a p-Basilica group, due to Dirichlet’s theorem about primes in arithmetic progressions. We conclude by remarking that some of the results (except those related to the p-CSP) of this paper carry through to the more general m-Basilica groups, for m∈Z≥2. For this reason, we restrict ourselves to consider only prime numbers. As already pointed out in the introduction, we refer the reader for more generalisations of the Basilica group to [29, Sec. 5-8]. Acknowledgements We thank D. Francoeur, M. Petschick and K. Rajeev for helpful discussions. Furthermore, we are grateful to B. Klopsch for his useful comments and for pointing out Sasse’s work. We also thank the referee for suggesting valuable improvements to the exposition of the paper. Funding. Open access funding provided by Universit`a degli Studi di Salerno within the CRUI-CARE Agreement. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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