We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2, q) and SL(2, q). For SL(2, q), we show that this map is surjective onto the subset SL(2, q)\{−id} ⊂ SL(2, q) provided that q ≥ q 0 (n) is sufficiently large. Moreover, we give an estimate for q 0 (n). We also present examples demonstrating that this does not hold for all q.We conclude that the n-th Engel word map is surjective for the groups PSL(2, q) when q ≥ q 0 (n). By using the computer, we sharpen this result and show that for any n ≤ 4, the corresponding map is surjective for all the groups PSL(2, q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups.In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2, q), with q odd, answering another question of Shalev.Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2, q).

A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates.Comment: 20 page

Abstract. Let G be a finite simple group. We show that the commutator map α : G × G → G is almost equidistributed as |G| → ∞. This somewhat surprising result has many applications. It shows that a for a subset X ⊆ G we have α −1 (X)/|G| 2 = |X|/|G| + o(1), namely α is almost measure preserving. From this we deduce that almost all elements g ∈ G can be expressed as commutators g = [x, y] where x, y generate G.This enables us to solve some open problems regarding T -systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T -systems in G with two generators tends to infinity as |G| → ∞. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators.Some of our results apply for more general finite groups and more general word maps.Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function ζ G (s) = χ∈Irr(G) χ(1) −s plays a key role in the proofs.

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = xayb is surjective on the group PSL (2, q) (and SL (2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL (2, q) (and SL (2, q)). Our proof is based on the investigation of the trace map of positive words.

In this article we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either PSL(2, p), or an alternating group, or a symmetric group or an abelian group. We moreover extend these results to regular surfaces isogenous to a higher product of curves. © 2014 Copyright Taylor and Francis Group, LLC

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

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We characterize Beauville surfaces of unmixed type with group either PSL2(p e ) or PGL2(p e ), thus extending previous results of Bauer, Catanese