We prove a conjecture of Boston that if , all p-central quotients of the free group on two generators and of the free product of two cyclic groups of order p are Beauville groups. In the case of the free product, we also determine Beauville structures in p-central quotients when . As a consequence, we give an infinite family of Beauville 3-groups, which is different from the ones that were known up to date.
A Beauville surface of unmixed type is a compact complex surface isomorphic to , where and are algebraic curves of genus at least 2 and G is a finite group acting freely on and faithfully on the factors such that and the covering map is ramified over three points for . Then the group G is said to be a Beauville group.
It is easy to formulate the condition for a finite group G to be a Beauville group in purely group-theoretical terms. For a couple of elements , we define
that is, the union of all subgroups of G which are conjugate to , to or to . Then G is a Beauville group if and only if the following conditions hold:
G is a 2-generator group.
There exists a pair of generating sets and of G such that .
Then and are said to form a Beauville structure for G. We call the triple associated to for . The signature of a triple is the tuple of orders of the elements in the triple.
In 2000, Catanese  proved that a finite abelian group is a Beauville group if and only if it is isomorphic to , where and . On the other hand, all finite quasisimple groups other than and are Beauville groups [8, 9] (see also  and ).
If p is a prime, Barker, Boston and Fairbairn  have shown that the smallest non-abelian Beauville p-groups for and are of order and , respectively. They have also proved that there are non-abelian Beauville p-groups of order for every and every . The existence of infinitely many Beauville 2-groups and 3-groups has been settled in the affirmative in , and in  and , respectively. In particular, by [17, Theorem 2], there are quotients of the ordinary triangle group which are Beauville 3-groups of every order greater than or equal to . Among them, one can find the 3-central quotients for all . In all these groups, the signature of one of the triples of the Beauville structure takes the constant value . On the other hand, by considering quotients of the Nottingham group over , Fernández-Alcober and Gül  have recently given an infinite family of Beauville 3-groups, for all orders at least , and in this case the signatures of the triples are not bounded.
In , Boston conjectured that if and F is either the free group on two generators or the free product of two cyclic groups of order p, then its p-central quotients are Beauville groups. The goal of this paper is to prove Boston’s conjecture. In fact, in the case of the free product, we extend the result to .
The main results of this paper are as follows.
Let be the free group on two generators. Then a p-central quotient is a Beauville group if and only if and .
Let be the free product of two cyclic groups of order p. Then a p-central quotient is a Beauville group if and only if and or and .
We will see that the signatures of the triples in the Beauville structures arising from Theorem B are unbounded as n goes to infinity. As a corollary, we get examples of Beauville 3-groups of every order greater than or equal to which are completely different from the ones given by Stix and Vdovina. We also compare these examples with the Beauville quotients of the Nottingham group over given in , and we show that the two infinite families only coincide at the group of order , see Theorem 3.7.
We use standard notation in group theory. If G is a group, then we denote by the conjugacy class of the element . Also, if p is a prime, then we write for the subgroup generated by all powers as g runs over G and for the subgroup generated by the elements of G of order at most . The exponent of G, denoted by , is the maximum of the orders of all elements of G.
2 The free group on two generators
In this section, we give the proof of Theorem A. We begin by recalling the definition of p-central series for the convenience of the reader.
For any group G, the normal series
given by for is called the p-central series of G.
Then a quotient group is said to be a p-central quotient of G. To prove the main theorems, we need the following properties of the subgroups (see [15, Definition 1.4 and Theorem 1.8, respectively]): we have
and any element of can be written in the form
Also observe that if , then , since for any we have .
Let G be a group and . For , we have
By the Hall–Petrescu formula (see [15, Lemma 1.1]), we have
Now the result follows, since by (2.1), and for we have
Note that if in Lemma 2.2, then
Before we proceed to prove Theorem A, we will need to introduce a lemma.
Let be the free group on two generators. Notice that for , coincides with , and thus elements outside are potential generators in . In order to determine Beauville structures in the quotients , it is fundamental to control nd powers of elements outside in these quotients groups.
Let be the free group on two generators. Then and are linearly independent modulo for .
We argue by way of contradiction. Suppose that
It follows from (2.2) that
and then we have
Write for some and some . Then
On the other hand, an element of the free group F belongs to if and only if the exponent sum of both generators is zero. Hence we get , which is a contradiction. ∎
As a consequence of Lemma 2.3, x and y have order modulo .
and since and are linearly independent modulo by Lemma 2.3, the following lemma is straightforward.
If , the power subgroups are all different and of order p in , as M runs over the maximal subgroups of G. In particular, all elements in are of order .
After these preliminaries, we can now prove Theorem A.
A p-central quotient is a Beauville group if and only if and .
For simplicity let us call G the quotient group . We first show that if or 3, then G is not a Beauville group. By way of contradiction, suppose that and form a Beauville structure for G. Since G has maximal subgroups, we may assume that and are in the same maximal subgroup. Then by (2.3), we have
which is a contradiction.
Thus we assume that . First of all, notice that if , is a Beauville group, by Catanese’s criterion. So we will deal with the case . Let u and v be the images in G of x and y, respectively. We claim that and form a Beauville structure for G. If we have and , we need to show that
for all , , and . Observe that and lie in different maximal subgroups of G in every case, since u and v are linearly independent modulo and .
and again by Lemma 2.4, and lie in the same maximal subgroup of G, which is a contradiction. We thus complete the proof that G is a Beauville group. ∎
3 The free product of two cyclic groups of order p
Now we focus on the free product of two cyclic groups of order p. Notice that since has exponent p, we have for all .
We start with an easy lemma whose proof is left to the reader.
Let be a group homomorphism, let and , . If , the condition implies that for .
To prove the main theorem we also need a result of Easterfield  regarding the exponent of . More precisely, if G is a p-group, then for every , the condition implies that
A key ingredient of the proof of Theorem B will be based on p-groups of maximal class with some specific properties. Let , where s is of order p and . The action of s on A is via θ, where θ is defined by the companion matrix of the pth cyclotomic polynomial . Then G is the only infinite pro-p group of maximal class. Since and annihilates A, this implies that for every ,
Thus all elements in are of order p. An alternative construction of G can be given by using the ring of cyclotomic integers (see [16, Example 7.4.14]).
Let P be a finite quotient of G of order for . Let us call the abelian maximal subgroup of P and for . Then one can easily check that
and every element in is of order .
Now we can begin to determine which p-central quotients of F are Beauville groups. We first assume that . The free product F of two cyclic groups of order 2 is the infinite dihedral group . Then by [3, Lemma 3.7], no finite quotient of F is a Beauville group. In the remainder, we consider the case where p is an odd prime.
Let for . If u and v are the images of x and y in G, then for any all elements in the coset have order .
Let P be the p-group of maximal class of order which is mentioned above and let and . Since all elements in are of order p and , the map
is well-defined and an epimorphism. Set . Since ψ is an epimorphism, we have , where every element in the coset has the same order as , namely . Then for every , we have . On the other hand, . Then by (6), we get , and consequently . ∎
We deal separately with the cases and .
If , then the p-central quotient is a Beauville group for every .
For simplicity let us call G the quotient group . Observe that .
If , then is a Beauville group, by Catanese’s criterion. Thus we assume that . Let u and v be the images of x and y in G, respectively. We claim that and form a Beauville structure for G. Let and . Assume first that or v, which are elements of order p, and . If for some , then , and hence , which is a contradiction since . Next we assume that . Since , for every we have , which is of order p. Thus for all we have
Since , it then follows from Lemma 3.1 that . This completes the proof. ∎
In order to deal with the prime 3, we need the following lemmas.
Let G be a p-group which is not of maximal class such that . Then for every there exists .
Note that a p-group has maximal class if and only if it has an element with centralizer of order (see [14, III.14.23]). Thus for every we have , and hence
Since , there exists such that . ∎
Lemma 3.5 ([10, Lemma 3.8])
Let G be a finite p-group and let be an element of order p. If , then
Let . Then the following hold:
The p-central quotient is a Beauville group if and only if .
The series can be refined to a normal series of F such that two consecutive terms of the series have index p and for every term N of the series is a Beauville group.
Since the smallest Beauville 3-group is of order , the quotient can only be a Beauville group if . We first assume that . Now consider the group
where we have omitted all commutators between generators which are trivial. This is the smallest Beauville 3-group. Since , maps onto H. On the other hand, it is clear that and so . Consequently, is a Beauville group. Thus we assume that .
Now let us call G the quotient group . Consider the map defined in the proof of Theorem 3.3. Since ψ is an epimorphism, we have
Observe that the subgroup has index 3 in , since the subgroup is of order 3. Choose a normal subgroup N of F such that and . Then ψ induces an epimorphism from to P.
We will see that is a Beauville group, which simultaneously proves (i) and (ii). Let u and v be the images of x and y in L, respectively. Set . Then . On the other hand, since , we have , and consequently we get in L. Since the subgroup is not of maximal class, L is not of maximal class. Thus, by Lemma 3.4, there exist elements such that and . We claim that and form a Beauville structure for L. Let and .
If , which is of order 3, and or , we get for every , as in the proof of Theorem 3.3. When and or , the same argument applies. If we are in one of the following cases: and , or and , then the condition follows from Lemma 3.5.
Thus the quotients in Theorem 3.6 constitute an infinite family of Beauville 3-groups of order for all .
Observe that as a consequence of Lemma 3.2, the signatures of the triples in the Beauville structures arising from Theorems 3.3 and 3.6 are unbounded as n goes to infinity. Consequently, these examples are different from those of Stix and Vdovina, since in their examples the signatures of one of the triples of the Beauville structures take the constant value.
We next compare the infinite family of Beauville 3-groups in Theorem 3.6 with the ones given in , by considering quotients of the Nottingham group over . We will show that these two infinite families of Beauville 3-groups only coincide at the group of order .
Before proceeding we recall the definition of the Nottingham group and some of its properties. The Nottingham group over the field , for odd p, is the (topological) group of normalised automorphisms of the ring of formal power series. For any positive integer k, the automorphisms such that form an open normal subgroup of of index . The lower central series of is given by
(see [5, Remark 1 and Theorem 6, respectively]).
Also, each non-trivial normal subgroup of lies between some and (see [5, Remark 1 and Proposition 2]).
By [10, Theorem 3.10], if , a quotient is a Beauville group if and only if and for all , where . Furthermore, by [10, Theorem 3.11], for there exists a normal subgroup between and such that is a Beauville group. This gives quotients of which are Beauville groups of every order with .
Let be a normal subgroup of F such that is a Beauville group. Then is not isomorphic to any quotient of which is a Beauville group. On the other hand, is isomorphic to .
Since there is only one Beauville group of order , it follows that is isomorphic to . Now suppose that , where for and is a Beauville group. Since is of class and lies between two consecutive terms of the lower central series, we have . Note that if then
and so . If , then . Consequently, the isomorphism implies that . We next show that this is not possible.
Note that by (3.1), we have and by (3.2), . Thus the exponent of is 3. On the other hand, as in the proof of Theorem 3.6, there is an epimorphism from to a p-group of maximal class P of order with . It follows that cannot be isomorphic to . ∎
I would like to thank G. Fernández-Alcober, N. Gavioli and C. M. Scoppola for helpful comments and suggestions. Also, I would like to thank the Department of Mathematics at the University of the Basque Country for its hospitality while this paper was being written.
 Barker N., Boston N. and Fairbairn B., A note on Beauville p-groups, Exp. Math. 21 (2012), 298–306. Search in Google Scholar
 Barker N., Boston N., Peyerimhoff N. and Vdovina A., An infinite family of 2-groups with mixed Beauville structures, Int. Math. Res. Not. IMRN 2015 (2015), no. 11, 3598–3618. Search in Google Scholar
 Bauer I., Catanese F. and Grunewald F., Beauville surfaces without real structures, I, Geometric Methods in Algebra and Number Theory, Progr. Math. 235, Birkhäuser, Boston (2005), 1–42. Search in Google Scholar
 Boston N., A survey of Beauville p-groups, Beauville Surfaces and Groups, Springer Proc. Math. Stat. 123, Springer, Cham (2015), 35–40. Search in Google Scholar
 Camina R., The Nottingham group, New Horizons in Pro-p Groups, Progr. Math. 184, Birkhäuser, Boston (2000), 205–221. Search in Google Scholar
 Catanese F., Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), 1–44. Search in Google Scholar
 Easterfield T. E., The orders of products and commutators in prime power groups, Proc. Cambridge Phil. Soc. 36 (1940), 14–26. Search in Google Scholar
 Fairbairn B., Magaard K. and Parker C., Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces, Proc. Lond. Math. Soc. (3) 107 (2013), 744–798. Search in Google Scholar
 Fairbairn B., Magaard K. and Parker C., Corrigendum: Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces, Proc. Lond. Math. Soc. (3) 107 (2013), 1220–1220. Search in Google Scholar
 Garion S., Larsen M. and Lubotzky A., Beauville surfaces and finite simple groups, J. Reine Angew. Math. 666 (2012), 225–243. Search in Google Scholar
 González-Diez G. and Jaikin-Zapirain A., The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, Proc. Lond. Math. Soc. (3) 111 (2015), 775–796. Search in Google Scholar
 Guralnick R. and Malle G., Simple groups admit Beauville structures, J. Lond. Math. Soc. (2) 85 (2012), 649–721. Search in Google Scholar
 Huppert B., Endliche Gruppen I, Springer, Berlin, 1967. Search in Google Scholar
 Huppert B. and Blackburn N., Finite Groups II, Springer, Berlin, 1982. Search in Google Scholar
 Leedham-Green C. R. and McKay S., The Structure of Groups of Prime Power Order, Oxford University Press, New York, 2002. Search in Google Scholar
 Stix J. and Vdovina A., Series of p-groups with Beauville structure, Monatsh. Math. (2015), 10.1007/s00605-015-0805-9. Search in Google Scholar
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