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Fuchsian groups, finite simple groups and representation varieties

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Abstract

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+o(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 \(\text{Hom}(\Gamma,\bar G)\), where \(\bar G\) 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’ ζG(s)=∑χ(1)-s, where the sum is over all irreducible complex characters χ of G.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College, London, SW7 2BZ, England

    Martin W. Liebeck

  2. Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Aner Shalev

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  1. Martin W. Liebeck
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Liebeck, M., Shalev, A. Fuchsian groups, finite simple groups and representation varieties. Invent. math. 159, 317–367 (2005). https://doi.org/10.1007/s00222-004-0390-3

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