Doubly Hurwitz Beauville groups

Abstract

If $\mathcal{S}$ is a Beauville surface $({\mathcal{C}}_1\times {\mathcal{C}}_2)/G$, then the Hurwitz bound implies that $|G|\le 1764\chi (\mathcal{S})$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves ${\mathcal{C}}_i$. 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 $S{L}_n(q)$ if $n$ is sufficiently large, but by no sporadic simple groups or simple groups $L_n(q)$ ($n\le 7$), ${}^2{G}_2(3^e)$, ${}^2{F}_4(2^e)$, ${}^2{F}_4(2{)}^{\prime }$, $G_2(q)$ or ${}^3{D}_4(q)$ of small Lie rank.

37 pages, 12 figures