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

[1]  G. Michler,et al.  Character table and blocks of finite simple triality groups , 1987 .

[2]  A. Lubotzky,et al.  Varieties of Representations of Finitely Generated Groups , 1985 .

[3]  Jason Fulman,et al.  Derangements in simple and primitive groups , 2002 .

[4]  Marston Conder,et al.  Hurwitz groups: A brief survey , 1990 .

[5]  E. Witten.  On quantum gauge theories in two dimensions , 1991 .

[6]  László Pyber,et al.  Upper Bounds for the Number of Conjugacy Classes of a Finite Group , 1997 .

[7]  Aner Shalev,et al.  Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks , 2004 .

[8]  Marc Perret,et al.  NUMBER OF POINTS OF PRYM VARIETIES OVER FINITE FIELDS , 2006, Glasgow Mathematical Journal.

[9]  Robert Steinberg,et al.  The Representations of GL(3,q), GL(4,q), PGL(3,q), and PGL(4,q) , 1951, Canadian Journal of Mathematics.

[10]  Gary M. Seitz,et al.  Fixed point ratios in actions of finite exceptional groups of lie type. , 2002 .

[11]  The asymptotics of $e^{P\left( z \right)}$ and the number of elements of each order in $S_n$ , 1986 .

[12]  Gerhard Hiss,et al.  Low-Dimensional Representations of Quasi-Simple Groups , 2001, LMS J. Comput. Math..

[13]  Ernest E. Shult,et al.  On a class of doubly transitive groups , 1972 .

[14]  D. Gluck.  Sharper Character Value Estimates for Groups of Lie Type , 1995 .

[15]  M. Liebeck.  On the Orders of Maximal Subgroups of the Finite Classical Groups , 1985 .

[16]  V. Chernousov,et al.  Representation varieties of the fundamental groups of compact orientable surfaces , 1996 .

[17]  Harold N. Ward,et al.  On Ree’s series of simple groups , 1966 .

[18]  Kenneth D. Johnson.  The Structure of Parabolic Subgroups , 2004 .

[19]  M. Liebeck,et al.  Reductive subgroups of exceptional algebraic groups , 1996 .

[20]  Aner Shalev,et al.  Classical groups, probabilistic methods, and the (2,3)-generation problem , 1996 .

[21]  T. A. Springer,et al.  Seminar on Algebraic Groups and Related Finite Groups , 1970 .

[22]  G. Seitz,et al.  On the minimal degrees of projective representations of the finite Chevalley groups , 1974 .

[23]  T. Müller,et al.  Character Theory of Symmetric Groups and Subgroup Growth of Surface Groups , 2002 .

[24]  Aner Shalev,et al.  Simple groups, permutation groups, and probability , 1999 .

[25]  Persi Diaconis,et al.  Random walks on the symmetric group generated by conjugacy classes , 1996 .

[26]  Ascher Wagner,et al.  An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field , 1977 .

[27]  M. Liebeck,et al.  Random Generation of Finite Simple Groups , 1999 .

[28]  Michael Aschbacher,et al.  On the maximal subgroups of the finite classical groups , 1984 .

[29]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[30]  Aner Shalev,et al.  Simple Groups, Probabilistic Methods, and a Conjecture of Kantor and Lubotzky , 1996 .

[31]  R. Lawther.  Elements of specified order in simple algebraic groups , 2004 .

[32]  Meinolf Geck,et al.  Group representation theory , 2007 .

[33]  R. Lyndon.  The equation $a^2b^2=c^2$ in free groups. , 1959 .

[34]  G. E. Wall.  On the conjugacy classes in the unitary, symplectic and orthogonal groups , 1963, Journal of the Australian Mathematical Society.

[35]  N. Iwahori,et al.  The conjugacy classes of the finite Ree groups of type$(F_4)$ , 1975 .

[36]  William M. Kantor,et al.  The probability of generating a finite classical group , 1990 .

[37]  William M. Kantor,et al.  The probability of generating a classical group , 1994 .

[38]  A. Mednykh.  On the number of subgroups in the fundamental group of a closed surface , 1988 .

[39]  Representation varieties of the fundamental groups of non-orientable surfaces , 1997 .

[40]  Volume of representation varieties , 2002, math/0212012.

[41]  W. Goldman.  Topological components of spaces of representations , 1988 .

[42]  M. Liebeck,et al.  On the Orders of Maximal Subgroups of the Finite Exceptional Groups of Lie Type , 1987 .

[43]  Gunter Malle,et al.  Generation of classical groups , 1994 .

[44]  Alan Williamson,et al.  The Probability of Generating the Symmetric Group , 1978 .

[45]  Frank Lübeck.  Small Degree Representations of Finite Chevalley Groups in Defining Characteristic , 2001, LMS J. Comput. Math..

[46]  Aner Shalev,et al.  Random (r, s)–Generation of Finite Classical Groups , 2002 .

[47]  Gary M. Seitz,et al.  The maximal subgroups of positive dimension in exceptional algebraic groups , 2004 .

[48]  S. Lang,et al.  NUMBER OF POINTS OF VARIETIES IN FINITE FIELDS. , 1954 .

[49]  P. Tiep,et al.  Minimal characters of the finite classical groups , 1996 .

[50]  M. Liebeck,et al.  Character Degrees and Random Walks in Finite Groups of Lie Type , 2005 .