Doubly Hurwitz Beauville groups

TL;DR

This paper investigates which finite simple groups can act as Beauville groups with a Hurwitz property, identifying specific families that satisfy the conditions and excluding many sporadic and small Lie type groups.

Contribution

It characterizes the simple groups that admit a Beauville structure with two generating triples of type (2,3,7), expanding understanding of Hurwitz Beauville groups.

Findings

Alternating groups $A_n$ satisfy the property for large $n$

Double covers $2.A_n$ also satisfy the property for large $n$

Certain classical groups do not satisfy the property, especially small Lie rank 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 $SL_n(q)$ if $n$ is sufficiently large, but by no sporadic simple groups or simple groups $L_n(q)$ ($n\le 7$), ${}^2G_2(3^e)$, ${}^2F_4(2^e)$, ${}^2F_4(2)'$, $G_2(q)$ or ${}^3D_4(q)$ of small Lie rank.