TY - JOUR

T1 - Fuchsian groups, finite simple groups and representation varieties

AU - Liebeck, Martin W.

AU - Shalev, Aner

PY - 2005/2

Y1 - 2005/2

N2 - 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 GLn(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) = Σχ(1)-s, where the sum is over all irreducible complex characters χ of G.

AB - 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 GLn(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) = Σχ(1)-s, where the sum is over all irreducible complex characters χ of G.

UR - http://www.scopus.com/inward/record.url?scp=12844273818&partnerID=8YFLogxK

U2 - 10.1007/s00222-004-0390-3

DO - 10.1007/s00222-004-0390-3

M3 - Article

AN - SCOPUS:12844273818

SN - 0020-9910

VL - 159

SP - 317

EP - 367

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 2

ER -