An infinite family of 2-groups with mixed Beauville structures

We construct an infinite family of triples $(G_k,H_k,T_k)$, where $G_k$ are 2-groups of increasing order, $H_k$ are index-2 subgroups of $G_k$, and $T_k$ are pairs of generators of $H_k$. We show that the triples $u_k = (G_k,H_k,T_k)$ are mixed Beauville structures if $k$ is not a power of 2. This is the first known infinite family of 2-groups admitting mixed Beauville structures. Moreover, the associated Beauville surface $S(u_3)$ is real and, for $k>3$ not a power of 2, the Beauville surface $S(u_k)$ is not biholomorphic to $\bar{S(u_k)}$.


Introduction
In this article we construct infinitely many 2-groups G k and show that they admit mixed Beauville structures if k is not a power of 2.
It was mentioned in [BCG1] that it is rather difficult to find a finite group admitting a mixed Beauville structure.Computer calculations show that there are no such groups of order < 2 8 (see [BCG2,Remark 4.2]).By the definition, if a p-group admits a mixed Beauville structure, then p = 2. Until now, only finitely many 2-groups admitting mixed Beauville structures are known.There are two examples of order 2 8 in [BCG2], and five more of orders 2 14 , 2 16 , 2 19 , 2 24 , 2 27 in [BBPV].The family in this paper is the first known infinite family of 2-groups admitting mixed Beauville structures.
A mixed Beauville structure of a finite group G is a triple (G, H, T ), where H is an index 2 subgroup of G and T = (h 1 , h 2 ) is a pair of elements h 1 , h 2 ∈ H generating H with particular properties.
Since so little is known about groups admitting mixed Beauville structures it is generally assumed that they are very rare.Clearly no simple group can admit a mixed Beauville structure.B. Fairbairn proved that the same holds true for all almost simple groups G whose derived groups [G, G] are sporadic (see [F,Theorem 8]).The only other known construction of groups admitting mixed Beauville structures was given in [BCG1].These groups are of the form K [4] = (K × K) ⋊ (Z/4Z), where K is a group with particular properties listed in [BCG1,Lemma 4.5].The nature of these other mixed Beauville structures (K [4] , K [2] , T = (a, c)) is very different from our family of 2-groups.For example, ν(T ) = ord(a)ord(c)ord(ac) contains necessarily two different primes.Since, for 2-groups, ν(T ) is necessarily a power of 2, this other construction cannot provide examples of 2-groups admitting mixed Beauville structures.
Our groups G k are 2-quotients of a just infinite group G with seven generators x 0 , . . ., x 6 , acting simply transitively on the vertices of an A 2 -building.This infinite group first appeared in [EH], and then again in [CMSZ] in connection with buildings.In [PV], we observed that G has an index 2 subgroup H, generated by x 0 , x 1 , and we used the corresponding index 2 quotients H k ✁ G k for explicit Cayley graph expander constructions.The considerations in [PV] showed that |G 3 | = 2 8 and, for k ≥ 3, For simplicity of notation, we use the same symbols x i for the generators of G and their images in the finite quotients G k .Any mixed Beauville structure u = (G, H, T ) gives rise to a Beauville surface S(u) ∼ = (C T × C T )/G of mixed type.A natural question is whether this Beauville surface S(u) is real.An algebraic surface S is called real if there is a biholomorphism σ : S → S with σ 2 = id.For the details we refer, e.g., to the papers [BCG1] and [BCG2].
Let us now formulate the main result of this paper.
Theorem 1.Let k ≥ 3 be not a power of 2 and The mixed Beauville surface S(u 3 ) is real.
(ii) For every k > 3 not a power of 2, the Beauville surface S(u k ) is not biholomorphic to its complex conjugate S(u k ).
For the proof, we realise G as a group of (finite band) upper triangular infinite Toeplitz matrices.The 2-quotients G k are obtained via truncations of these matrices at their (k + 1)-th upper diagonal, and they have a certain nilpotency structure.Our proof exploits this nilpotency structure as well as subtle periodicity properties of these matrices.It also becomes transparent via these periodicity properties why, in the above theorem, k ≥ 3 must necessarily avoid the powers of 2.
Let us explain the difference between the results in [BBPV] and in this article: In [BBPV], we used the computational algebra system Magma to check that the first six groups of an infinite family of 2groups admit mixed Beauville structures, which led us to conjecture that this holds true for the full infinite family.In this paper, we provide a rigorous theoretical proof that an infinite family of 2-groups admit mixed Beauville structures.In view of the final Remark 7.1, it is very surprising that all our groups (except for G 2 j with j ∈ N 0 ) admit mixed Beauville structures.Moreover, there is overwhelming evidence that the families of groups in both papers agree, and it has been verified computationally for the first 100 groups in both families that they are pairwise isomorphic (see [PV,Conjecture 1]).
Let us finish our introduction with the following question: Acknowledgement: The first author likes to thank Uzi Vishne for useful correspondences.The research of Nigel Boston is supported by the NSA Grant MSN115460.

Mixed Beauville structures and associated surfaces
The following presentation follows [BCG1] closely.Let G be a finite group and H ⊂ G be a subgroup of index 2.For x ∈ H let i.e., Σ(x) is the union of all conjugates of the cyclic subgroup generated by x.For T = (x 0 , x 1 ) ∈ H × H, we define Σ(T ) := Σ(x 0 ) ∪ Σ(x 1 ) ∪ Σ((x 0 x 1 ) −1 ).
Next, we explain how to construct the Beauville surface S = S(u) associated to a mixed Beauville structure u = (G, H, T = (x 0 , x 1 )).Let P 0 , P 1 , P 2 ∈ P 1 be a sequence of points ordered counterclockwise around a base point O ∈ P 1 and, for 0 ≤ i ≤ 2, let γ i ∈ π 1 (P 1 \{P 0 , P 1 , P 2 }, O) be represented by a simple counterclockwise loop around P i such that γ 0 γ 1 γ 2 = id.By Riemann's existence theorem, there exists a surjective homomorphism Φ : π 1 (P 1 \{P 0 , P 1 , P 2 }, O) → H with Φ(γ 0 ) = x 0 and Φ(γ 1 ) = x 1 , and a Galois covering λ T : C T → P 1 , ramified only in {P 0 , P 1 , P 2 }, with ramification indices equal to the orders of the elements x 0 , x 1 , x 0 x 1 .These data induce a well defined action of H on the curve C T , and by the Riemann-Hurwitz formula, we have for all x ∈ H and (z 1 , z 2 ) ∈ C T ×C T .This action is fixed point free, and the quotient (C T × C T )/G is the associated mixed Beauville surface S.
By the Theorem of Zeuthen-Segre, we have for the topological Euler number as well as the relations (see [Cat,Theorem 3.4]), where K S 2 is the self-intersection number of the canonical divisor and χ(S) = 1+p g (S)−q(S) is the holomorphic Euler-Poincaré characteristic of S.
Let us briefly indicate how we prove the reality statements (i),(ii) for the mixed Beauville surfaces in Theorem 1: For T = (c, a) ∈ H × H let T −1 = (c −1 , a −1 ).Every mixed Beauville structure u = (G, H, T ) gives rise to another mixed Beauville structure ι(u) = (G, H, T −1 ), and we have S(ι(u)) = S(u) (see [BCG1,(39) Moreover, in accordance with [BCG1,(11) and (32)], let σ 3 , σ 4 be maps on M(G), defined by and A M (G) be the group generated by the maps σ ψ (ψ ∈ Aut(G)) and σ 3 , σ 4 .Then we have the following facts (see [BCG1,Prop. 4.7]): Choosing the mixed Beauville structures u k from Theorem 1, we find an automorphism ψ : ) and σ ψ (ι(u 3 )) = u 3 , and it follows from (b) that S(u 3 ) is real.On the other hand, for k > 3 and not a power of 2, we show that there is no homomorphism ψ : [BCG1,Lemma 2.4] and the criterion (a) above, this implies that S(u k ) cannot be biholomorphic to S(u k ).(Note that our pair (x 0 , x 1 ) corresponds, in the notation of [BCG1], to the pair (c, a).)

The 2-groups G k and H k
Let K be the simplicial complex constructed from the following 7 triangles by identifying sides with the same labels x i .
x 4 x 4 x 5 x 5 x 5 x 6 x 6 x 6 Figure 1.Labeling scheme for the simplicial complex K It is easily checked that the vertices of all triangles are identified, and that the fundamental group π 1 (K) is isomorphic to the infinite abstract group (1) G = x 0 , . . ., x 6 | x i x i+1 x i+3 = id for i = 0, . . ., 6 , where i, i + 1 and i + 3 are taken modulo 7. Realising the triangles as equilateral Euclidean triangles, we can view the universal covering of K as a thick Euclidean building of type A 2 , on which G acts via covering transformations.
Note that the presentation (1) is a presentation of G by 7 generators and 7 relations.It is easy to see that G is already generated by the three elements x 0 , x 1 , x 2 .Let H ⊂ G be the subgroup generated by the two elements x 0 , x 1 .Then H is an index-2 subgroup of G (see [PV,Prop. 2.1]).The groups G k and H k will be finite 2-quotients of these groups G and H.
We now recall the faithful representation of G by infinite upper triangular matrices given in [PV], where every element x ∈ G is represented as , and each entry a ij is a matrix in M(3, F 2 ) (and 0 and 1 stand for the zero and the identity matrix in M(3, F 2 )).Note that the matrix representation (2) has only finitely many non-zero upper diagonals.Moreover, the entries in every diagonal are repeating with period 3.
The proofs of the explicit formulas in the following Lemma are straightforward (see [PV]).Note that (b) is a refinement of [PV,Prop. 2.5].These formulas are crucial for our later considerations: Lemma 3.1.Note that in the following formulas all entries j in a 1/2 (j), b 1 (j), c 1/2 (j) are taken mod 3 and chosen to be in the range {1, 2, 3}.
(a) Let k, j ≥ 0 and Then both products M 1 M 2 and M 2 M 1 are of the form with c 1 (i) = a 1 (i)a 1 (k + i + 1) and Let G k and H k be the subgroups of all elements in G and H with vanishing first k upper diagonals (i.e., these elements are of the form M k (a 1 , . . .)).Then G k and H k are normal subgroups of G and H, and our groups G k and H k are the quotients G/G k and H/H k .We can think of G k and H k as truncations of the matrix groups G and H at their (k +1)-st upper diagonal.The finiteness of these quotients follows then easily from the 3-periodicity of the diagonals.
Remark 3.2.Another way to generate quotients of G (and H) is via the lower-exponent-2 series We conjecture (see [PV, Conjectures 1 and 2]) that these facts hold true for all k, which would mean that the group G has finite width 3 (see [KLGP] for definitions).

Powers of the generators
This and all the following sections are dedicated to the proof that the triple (G k , H k , T k ) satisfies the conditions (A), (B) and (C) of a mixed Beauville structure if k is not a power of 2. The triples are both spherical systems of generators of the group H (see, e.g., [BCG2] for this notion).A crucial step towards the proof of Theorem 1 is the explicit determination of the first two non-trivial diagonals of all powers of each of the elements x 0 , x 1 , x, y 0 , y 1 , y.By the first two non-trivial diagonals of a matrix M 0 (a 1 , a 2 , . . . ) = id we mean the pair a k , a k+1 with a 1 = • • • = a k−1 = 0 and a k = 0.Moreover, we call a k the leading diagonal of this matrix.In fact, it turns out that -in all considerations of this paper -only a good understanding of the first two non-trivial diagonals is needed and that the higher diagonals can be ignored.
Remark 4.2.The group G has more remarkable properties.In [PV,Prop. 2.6], we present a certain 3-periodicity of commutators.Another interesting property is that the subgroup generated by the squares x 2 0 , x 2 1 , . . ., x 2 6 of the seven generators is isomorphic to G (see [EH,p. 308]).
The next remark explains why the statement in Theorem 1 cannot hold for powers of 2: Remark 4.3.Notice in Proposition 4.1 above that the leading diagonals of the matrix representations of x 2 n and y 2 n agree for all n ≥ 0, since both elements are conjugate (see Lemma 3.1(c)).Let k = 2 n .Recall that we can think of the elements in H k as matrices truncated at their (k + 1)-st upper diagonal.Then the non-trivial group elements x k and y k agree in H k , since their leading diagonals coincide and are the kth upper diagonals.(To separate these two elements in H k , their first two non-trivial diagonals would have to survive under the truncation procedure.)This implies that Notice that condition (B) in the mixed Beauville structure implies the following property: To understand the first two non-trivial diagonals of all powers of x and y (not only the 2-powers), we consider the binary presentation of an arbitrary exponent n ∈ N: Then x n is equal to x t(n) multiplied with certain higher 2-powers of x (i.e., the powers x 2 α l with α l = 1 and l ≥ max{2, k + 1}).In view of Lemma 3.1(a), this multiplication does not change the first two nontrivial diagonals of x t(n) , which shows that the first two non-trivial diagonals of x t(n) and x n agree.Using the (easily computable) fact that The matrix representations of any power y n 1 (n ≥ 1) takes one of the following forms M 0 ([23, 224, 138], [33,146,501], . . .), M 0 ([23, 224, 138], [6,66,335], . . . ) [39,208,186], [0, 106, 247], . . .), M 2 even+2 −1 ([23, 224, 138], [26,26,26], . . .).
Since these eight forms are all different, we conclude that g 2 ∈ Σ(T k ) for all g ∈ G k \H k .This shows that property (C) in the definition of a mixed Beauville structure is satisfied for all k ≥ 2.

Proof of property (B)
In this section, we prove that our triples (G k , H k , T k ) satisfy property (B) of a mixed Beauville structure with the choice g 0 = x 2 , for all k not a power of 2. Recall that x = (x 0 x 1 ) −1 and Σ(T ) = Σ(x 0 ) ∪ Σ(x 1 ) ∪ Σ(x), and x 2 Σ(T )x −1 2 = Σ(y 0 ) ∪ Σ(y 1 ) ∪ Σ(y).It follows immediately from inspection of the leading diagonals in Corollary 4.4 and Propositions 4.5 and 4.6 and the fact that these leading diagonals do not change under conjugation (see Lemma 3.1(c)) that, for the pair (x 0 , y 1 ), we have Σ(x 0 ) ∩ Σ(y 1 ) = {id}, and that the same trivial intersection holds also for all other pairs (x 0 , y), (x 1 , y 0 ), (x 1 , y), (x, y 0 ) and (x, y 1 ).So it only remains to prove the trivial intersection Σ(x) ∩ Σ(y) = {id}, and analogous trivial intersection results for the pairs (x 0 , y 0 ) and (x 1 , y 1 ).For this, the consideration of the leading diagonal is not sufficient and we have to study the behavior of the first two non-trivial diagonals under conjugation.From now on, let k be not a power of 2.
Next, let us look at the conjugation scheme for A 2 for any pair Again, this shows that we have (5) for any pair h, h ′ ∈ H k and every n, m with t(n) = t(m) = 2 r < k and odd r ≥ 1.Moreover, (5) also holds for any choice of n, m such that (i) one of t(n), t(m) is in {1, 3} and the other is of the form 2 r with odd r ≥ 1, or (ii) t(n) = 2 r 1 < k and t(m) = 2 r 2 < k with r 1 , r 2 ≥ 1 both odd and r 1 = r 2 , since then the number of first upper vanishing diagonals of hx n h −1 and h ′ y m (h ′ ) −1 do not agree.
Since we have |H k | ≥ 8192 for k > 3 not a power of 2, there cannot be a homomorphism ψ : H k → H k with ψ(x 0 ) = y 0 and ψ(x 1 ) = y 1 by the above criterion, showing that S(u k ) cannot be biholomorphic to S(u k ).This finishes the proof of Theorem 1.
Remark 7.1.The fact that the triples (G k , H k , T k ) are mixed Beauville structures (for k not a power of 2) is very remarkable.Let us reflect -by looking back at the proof of property (B) -why this result is so surprising: We know that for indices up to order k ≤ 100 we have which gives strong evidence that this should hold for all indices k ∈ N (see the finite width 3 conjecture in [PV, Conjecture 1]).This means that for any k ≤ 99, k ≡ 2 mod 3 and A 1 , there are at most four different choices A 2 such that (A 1 , A 2 ) represent the first two non-trivial diagonals of matrix representations M k (A 1 , A 2 , . . . ) of elements in G. On the other hand, it follows from the arguments in the proof of property (B) that we need at least four such possibilities to guarantee that Σ(x) ∩ Σ(y) = {id} (and to derive analogous results for the pairs (x 0 , y 0 ) and (x 1 , y 1 )).Moreover, these four possibilities must appear in the right combinations in all the conjugation schemes to guarantee the required trivial intersections.
For which 2-groups H does there exist a group G ⊃ H and a choice T ∈ H × H such that (G, H, T ) is a mixed Beauville structure?Both examples of groups of order 2 8 listed in [BCG2, Thm 0.1] and admitting mixed Beauville structures have the same index 2-subgroup which agrees with our group H 3 .The five other examples in [BBPV] agree with our examples H 5 , H 6 , H 7 , H 9 , H 10 .It would be interesting to know whether there are any other 2-groups H giving rise to mixed Beauville structures (G, H, T ), and which do not agree with one of our groups H k .