New Beauville surfaces and finite simple groups

Abstract

In this paper we construct new Beauville surfaces with group either PSL(2, p e), or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion. © 2013 Springer-Verlag Berlin Heidelberg

Beauville Surfaces and Probabilistic Group Theory

“…A similar theorem also applies for symmetric groups, see [19], and a similar conjecture was raised in [19], replacing A n by a finite simple classical group of Lie type of sufficiently large Lie rank, namely,…”

“…Concerning the simple alternating groups, it was established in [15] that A 5 is indeed the only one not admitting an unmixed Beauville structure. In [16,19], the conjecture is shown to hold for the projective special linear groups PSL 2 (q) (where q > 5), the Suzuki groups 2 B 2 (q) and the Ree groups 2 G 2 (q) as well as other families of finite simple groups of Lie type of small rank. More precisely, the projective special and unitary groups PSL 3 (q), PSU 3 (q), the simple groups G 2 (q) and the Steinberg triality groups 3 D 4 (q) are shown to admit an unmixed Beauville structure if q is large (and the characteristic p is greater than 3 for the simple exceptional groups of type G 2 or 3 D 4 ).…”

“…Moreover for the alternating and symmetric groups Garion and Penegini [19] proved another conjecture that Bauer, Catanese and Grunewald proposed in [1], that almost all of these groups admit a Beauville structure with fixed type, namely,…”

“…In this section we discuss the specific case of PSL 2 (q), and briefly sketch the proof of Garion and Penegini [19] for the following theorem, which is based on results of Macbeath [37]. Theorem 10.…”

“…In order to construct an unmixed Beauville structure for PSL 2 (q) one needs to find a quadruple (A 1 , B 1 ; A 2 , B 2 ) of elements of PSL 2 (q) satisfying the three conditions given in Definition 1. This can be done directly by finding specific elements in the group satisfying these conditions (see [16]) or indirectly by using the following results of Macbeath [37] (see [17,19]). This theorem immediately implies Condition (i).…”