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 A are surjective, and
this implies Higman's conjecture that every Fuchsian group surjects
onto all large enough alternating groups.
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