Fuchsian groups, subgroup growth, random quotients and random walks

Aner Shalev (Hebrew University, Jerusalem)

Fuchsian groups (acting as isometries of the hyperbolic plane) occur naturally in geometry, combinatorial group theory, and other contexts. They include surface groups, triangle groups, the modular group, free groups, etc. We use character-theoretic and probabilistic methods to study spaces of homomorphisms from Fuchsian groups to symmetric groups and to finite simple 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 mixing times of random walks on symmetric groups and simple groups. In particular we show that almost all homomorphisms from a Fuchsian group to alternating groups An are surjective, and this implies Higman's conjecture that every Fuchsian group surjects onto all large enough alternating groups.

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