Fuchsian groups, finite simple groups and representation varieties

Let Γ be a Fuchsian group of genus at least 2 (at least 3 if Γ is non-oriented). We study the spaces of homomorphisms from Γ to finite simple groups G, and derive a number of applications concerning random generation and representation varieties. Precise asymptotic estimates for |Hom(Γ, G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G|^{μ(Γ)+1+0(1)}, where μ(Γ) is the measure of Γ. We then prove that a randomly chosen homomorphism from Γ to G is surjective with probability tending to 1 as |G| → ∞. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties Hom(Γ, Ḡ), where Ḡ is GL_n(K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic. A key ingredient of our approach is character theory, involving the study of the 'zeta function' ζ_G(s) = Σχ(1)^-s, where the sum is over all irreducible complex characters χ of G.