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Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks. (English) Zbl 1068.20052

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.

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20C30 Representations of finite symmetric groups
20E07 Subgroup theorems; subgroup growth
20E26 Residual properties and generalizations; residually finite groups
20P05 Probabilistic methods in group theory
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
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