Strongly Real Beauville Groups

Abstract

Abstract. A strongly real Beauville group is a Beauville group that defines a real Beauville surface. Here we discuss efforts to find examples of these groups, emphasising on the one extreme finite simple groups and on the other abelian and nilpotent groups. We will also discuss the case of characteristically simple groups and almost simple groups. En route we shall discuss several questions, open problems and conjectures as well as giving several new examples of infinite families of strongly real Beauville groups.

Coxeter groups as Beauville groups

“…For the remaining claim note that the Beauville structures constructed in Sections 3 and 4 as well as the Beauville structures previously constructed by the author in [10,Lemma 19] in the type A n case were constructed in such a way that not all of the elements x 1 , y 1 , x 2 and y 2 lie outside the derived subgroup and in particular these structures have the property that {{(x 1 , y −1 2 ), (y 1 , x − 2 1)}, {(x 2 , y −1 1 ), (y 2 , x −1 1 )}} will generate the whole of K 1 × K 2 . (Where possible we took the x i s from outside the derived subgroup and the y i s from inside, indeed only in the H 4 case did it seems difficult to find such a structure, but even in that case only x 2 lies inside the derived subgroup, so the elements of the product still lie in different cosets and thus generate the whole group.)…”

More on Strongly Real Beauville Groups

“…The question of which characteristically simple Beauville groups are strongly real was first investigated by the author in [22,Section 3]. More specifically the following conjecture was investigated.…”

“…In [26] Fuertes and González-Diez considered which of the symmetric groups are strongly real Beauville groups. The first place the more general question of which almost simple groups are (strongly real) Beauville groups was the author's discussion given in [22,Section 5] where the following conjecture is asserted. There are multiple 'warning shots' to be fired here -there are infinitely many almost simple groups that are not even 2-generated, let alone Beauville groups, the smallest example being PSL 4 (9) whose outer automorphism group is 2 × D 8 (and more generally, if p is an odd prime and r is an even positive integer then Aut(PSL 4 (p r )) is not 2-generated).…”

“…In [22,Lemma 7] the author proves that for n ≥ 5 the groups S n × S n are strongly real. Note that since the abelianisation of S n × S n × S n for any integer n > 1 is elementary abelian of order eight, which is clearly not 2-generated, it follows that S n × S n × S n cannot be 2-generated.…”

“…Whilst this article is largely expository in nature we also report incremental progress on various different problems that will appear here. Indeed, this work can be naturally viewed as a sequel to the author's earlier article [22], though the reader will lose little if they have neither read nor have a copy of [22] to hand.…”