Doubly Hurwitz Beauville groups

Abstract

If S is a Beauville surface (C 1 × C 2 )/G, then the Hurwitz bound implies that |G| ≤ 1764 χ(S), with equality if and only if the Beauville group G acts as a Hurwitz group on both curves C i . Equivalently, G has two generating triples of type (2, 3, 7), such that no generator in one triple is conjugate to a power of a generator in the other. We show that this property is satisfied by alternating groups A n , their double covers 2.A n , and special linear groups SL n (q) if n is sufficiently large, but by no sporadic simple groups or simple groups L n (q) (n ≤ 7), 2 G 2 (3 e ), 2 F 4 (2 e ), 2 F 4 (2) ′ , G 2 (q) or 3 D 4 (q) of small Lie rank.

Joining dessins together

“…It is a 3-sheeted regular covering of A, branched over the two points corresponding to the fixed points a and b of x, so that A ∼ = G/C 3 where the group C 3 of automorphisms of G permutes the three copies of A cyclically. It follows from Lemma 5.2 that, like A, this dessin G also has monodromy group PSL 2 (13), but now in its imprimitive representation of degree 42 on the cosets of a dihedral subgroup…”

“…. , N are all drawn in [13], whereas only A, B, C, F and G are used in this paper; their basic properties are listed in the Appendix.…”

“…Note that A has been redrawn from Figure 1 so that the free edges are now in the outer face; the reason for this will become clear in the next section, when we consider joining operations (see Figure 5). The monodromy group of A is PSL 2 (13) in its natural representation of degree 14 on the projective line P 1 (F 13 ); the point-stabilisers are the Sylow 13-normalisers H ∼ = C 13 ⋊ C 6 . One can take x : t → 1/(1 − t), y : t → −t and z : t → (t + 1)/t as standard generators.…”

“…This dessin corresponds to a conjugacy class of subgroups M of index 14 in ∆; since A has genus 0, and x and y each have two fixed points while z has none, these subgroups M are quadrilateral groups, with signature (0; 2, 2, 3, 3). We have A ∼ = A/H where A is the Galois cover of A, corresponding to the core K of M in ∆; this is the unique reflexible dessin in the Galois orbit of three Macbeath-Hurwitz dessins of genus 14 with automorphism group PSL 2 (13). (See [16] for further details of this Galois orbit and the corresponding quotient dessins.…”

“…The dessins S(1)S and S(1)S in the first row, clearly mirror images of each other, are also isomorphic under a rotation of order 2, and are each invariant under the antipodal isometry of the sphere. Although the monodromy group of S and S is PGL 3 (2) ∼ = PSL 2 (7), and for S(1)S and S(1)S it is a covering of this group, this dessin has monodromy group PSL 2 (13), in its natural representation. It is, in fact, one of a Galois orbit of three dessins corresponding to this action; the others are A, shown in Figure 3, and the dessin S(2)S ∼ = S(2)S, also invariant under the antipodal isometry, shown in the second row of Figure 12.…”