Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. We use character-theoretic and probabilistic methods to study the spaces of homomorphisms from Fuchsian groups to symmetric groups. We obtain a wide variety of applications, ranging from counting branched coverings of Riemann surfaces, to subgroup growth and random finite quotients of Fuchsian groups, as well as random walks on symmetric groups. In particular, we show that, in some sense, almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higmanʹs conjecture that every Fuchsian group surjects onto all large enough alternating groups. As a very special case, we obtain a random Hurwitz generation of An, namely random generation by two elements of orders 2 and 3 whose product has order 7. We also establish the analogue of Higmanʹs conjecture for symmetric groups. We apply these results to branched coverings of Riemann surfaces, showing that under some assumptions on the ramification types, their monodromy group is almost always Sn or An. Another application concerns subgroup growth. We show that a Fuchsian group Γ has (n!)μ+o(1) index n subgroups, where μ is the measure of Γ, and derive similar estimates for so-called Eisenstein numbers of coverings of Riemann surfaces. A final application concerns random walks on alternating and symmetric groups. We give necessary and sufficient conditions for a collection of ‘almost homogeneous’ conjugacy classes in An to have product equal to An almost uniformly pointwise. Our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups.