Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks.
In this long and detailed paper, the authors consider Fuchsian groups occurring in many branches of mathematics as they mention. They use character-theoretic and probabilistic methods to study the space of homomorphisms from Fuchsian groups to symmetric groups. They obtain several applications such as counting the branched coverings of Riemann surfaces, subgroup growth and random finite quotients of Fuchsian groups, and random walks on symmetric groups. By showing that nearly all homomorphisms from a Fuchsian group to alternating groups are onto, they prove Higman's conjecture. They also establish a conjecture similar for symmetric groups to Higman's conjecture.
scientific article; zbMATH DE number 2116873
scientific article