Fuchsian groups, finite simple groups and representation varieties

• 成果类型: Article
• 署名作者: Liebeck, MW; Shalev, A
• 署名单位: Imperial College London; Hebrew University of Jerusalem
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0390-3
• 发表日期: 2005
• 页码: 317-367
• 关键词: maximal-subgroups; projective-representations; probabilistic methods; exceptional groups; fundamental-groups; number; generation; GROWTH; ORDERS

摘要

Let Gamma be a Fuchsian group of genus at least 2 (at least 3 if Gamma is non-oriented). We study the spaces of homomorphisms from Gamma to finite simple groups G, and derive a number of applications concerning random generation and representation varieties. Precise asymptotic estimates for |Hom(Gamma, G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G|(mu(Gamma)+ 1+ o(1)), where mu(Gamma) is the measure of G. We then prove that a randomly chosen homomorphism from Gamma to G is surjective with probability tending to 1 as | G| --> infinity. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties Hom(Gamma, (G) over bar), where (G) over bar 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' zeta(G)(s) = Sigmachi(1)(-s), where the sum is over all irreducible complex characters chi of G.