Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces

Abstract

We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt (5) and SL 2(5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non‐abelian finite quasisimple groups except for the alternating group Alt (5) are Beauville groups.

Beauville surfaces and finite groups

“…According to a recent preprint by Fairbairn, Magaard and Parker [10], this conjecture is true. Similarly, Theorem 5.1 raises the question of which other quasisimple groups admit strongly real Beauville structures.…”

“…Garion, Larsen and Lubotzky [12] have shown that all but finitely many non-abelian finite simple groups are Beauville groups. Subsequently Guralnick and Malle [15] have extended this to all such groups except A 5 , thus proving the conjecture of Bauer, Catanese and Grunewald, while Fairbairn, Magaard and Parker [10] have extended this further to all finite quasisimple groups except A 5 ∼ = L 2 (5) and its central cover SL 2 (5).…”

Deformation theory and finite simple quotients of triangle groups I

“…This is for the positive side. On the negative side we essentially prove a conjecture of Marion proposed in [21]: in that paper he studied (a, b, c)-generation of finite quasisimple (2,4,6), (2,6,6), (2, 6, 10) (3,4,4), (3,6,6), (4,6,12) 5,7,8,9,10,11,13,15,16,17,19,22,23,25,29,31, 37, 43} (r ≥ 4) (2,3,8) r ∈ {4, 5, 7, 9, 10, 11, 13, 17, 19, 25} (2,3,9) r ∈ {4, 5, 7, 10, 11, 13, 19} (2,3,10) r ∈ {4, 5, 7, 11, 13} (2,3,11) r ∈ {4, 5, 7, 13} (2,3,12) r ∈ {4, 5, 7, 13} (2, 3, c), c ≥ 13 r ∈ {4, 5, 7} (2,4,…”

“…Turning to general hyperbolic triples of integers, any finite simple group, being 2-generated, is a quotient of some triangle group T and in fact can be so realized in many independent ways. See for example [10,9,12] and the references therein establishing that every finite simple group other than Alt 5 admits an (unmixed) Beauville structure.…”

Corrigendum: Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces

“…Shortly after publication of our article, it was brought to our attention that the first six results related to the generation of classical groups by two primitive prime divisor elements in [1, Section 3] can already be found in the much earlier 1998 work of Niemeyer and Praeger [2]. The connections are as follows: [2] should be cited in place of [1] for all of these statements.…”