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- Shelly Garion, Aner Shalev
- 2007

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… (More)

In memory of Fritz Grunewald, an inspiring mathematician and a dear friend Abstract. A Beauville surface is a rigid complex surface of the form (C 1 × C 2)/G, where C 1 and C 2 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,… (More)

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… (More)

In this paper 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… (More)

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… (More)

- SHELLY GARION
- 2007

The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph Γ k (G) whose vertices are the generating k-tuples of G (for a fixed k). We show that if G = PSL(2, q) or PGL(2, q), where q is a prime power, then Γ k (G) is connected for any k ≥ 4. This generalizes… (More)

- SHELLY GARION
- 2013

We characterize Beauville surfaces of unmixed type with group either PSL2(p e) or PGL2(p e), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = x a y b 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.

An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n ≥ 4 the group Out(Fn) = Aut(Fn)/ Inn(Fn) admits a faithful highly transitive action on a countable set.

- SHELLY GARION
- 2013

A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called " Beauville structure… (More)