Probabilistic group theory and fuchsian groups

In this paper we survey recent advances in probabilistic group theory and related topics, with particular emphasis on Fuchsian groups. Combining a character-theoretic approach with probabilistic ideas we study the spaces of homomorphisms from Fuchsian groups to symmetric groups and to finite simple groups, and obtain a whole range of applications. In particular we show that a random homomorphism from a Fuchsian group to the alternating group A_n is surjective with probability tending to 1 as n → ∞, and this implies Higman’s conjecture that every Fuchsian group surjects to all large enough alternating groups. We also show that a random homomorphism from a Fuchsian group of genus ≥ 2 to any finite simple group G is surjective with probability tending to 1 as |G| → ∞. These results can be viewed as far-reaching extensions of Dixon’s conjecture, establishing a similar result for free groups. Other applications concern counting branched coverings of Riemann surfaces, studying the subgroup growth of Fuchsian groups, and computing the dimensions of representation varieties of Fuchsian groups over fields of arbitrary characteristic. A main tool in our proofs are results of independent interest which we obtain on the representation growth of symmetric groups and groups of Lie type. These character-theoretic results also enable us to analyze random walks in symmetric groups and in finite simple groups, with certain conjugacy classes as generating sets, and to determine the precise mixing time. Most of the results outlined here were proved in recent joint works by Liebeck and myself, but we shall also describe related results by other authors.