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Fuchsian groups, finite simple groups and representation varieties. (English) Zbl 1134.20059

Summary: 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 \(|\operatorname{Hom}(\Gamma,G)|\) 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 \(\Gamma\). We then prove that a randomly chosen homomorphism from \(\Gamma\) to \(G\) is surjective with probability tending to 1 as \(|G|\to\infty\). Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties \(\operatorname{Hom}(\Gamma,\overline G)\), where \(\overline G\) is \(\text{GL}_n(K)\) 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’ \(\zeta^G(s)=\sum\chi(1)^{-s}\), where the sum is over all irreducible complex characters \(\chi\) of \(G\).

MSC:

20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20D06 Simple groups: alternating groups and groups of Lie type
20P05 Probabilistic methods in group theory
11M41 Other Dirichlet series and zeta functions
20C33 Representations of finite groups of Lie type
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
20E32 Simple groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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References:

[1] Aschbacher, No article title, Invent. Math., 76, 469 (1984) · Zbl 0537.20023 · doi:10.1007/BF01388470
[2] Azad, No article title, Commun. Algebra, 18, 551 (1990) · Zbl 0717.20029 · doi:10.1080/00927879008823931
[3] Benyash-Krivets, No article title, (Russian) Mat. Sb., 188, 47 (1997) · doi:10.4213/sm242
[4] Conder, No article title, Bull. Am. Math. Soc., 23, 359 (1990) · Zbl 0716.20015 · doi:10.1090/S0273-0979-1990-15933-6
[5] Deriziotis, No article title, Trans. Am. Math. Soc., 303, 39 (1987)
[6] Dixon, No article title, Math. Z., 110, 199 (1969) · Zbl 0176.29901 · doi:10.1007/BF01110210
[7] Dornhoff, L.: Group Representation Theory, Part A. Marcel Dekker 1971 · Zbl 0227.20002
[8] Fulman, J., Guralnick, R.: Derangements in simple and primitive groups. In: Ivanov, A., Liebeck, M.W., Saxl, J. (eds.), Groups, Combinatorics and Geometry: Durham, 2001. World Scientific 2003 · Zbl 1036.20002
[9] Fulman, J., Guralnick, R.: The number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. Preprint · Zbl 1256.20048
[10] Gorenstein, D., Lyons, R., Solomon, R.: The classification of the finite simple groups, Vol. 3. Math. Surv. Monogr., Vol. 40, No. 3. Am. Math. Soc. 1998 · Zbl 0890.20012
[11] Gluck, No article title, J. Algebra, 174, 229 (1995) · Zbl 0842.20014 · doi:10.1006/jabr.1995.1127
[12] Goldman, No article title, Invent. Math., 93, 557 (1988) · Zbl 0655.57019 · doi:10.1007/BF01410200
[13] Guralnick, No article title, Commun. Algebra, 22, 1395 (1994) · Zbl 0820.20022 · doi:10.1080/00927879408824912
[14] Guralnick, R., Lübeck, F., Shalev, A.: Zero-one generation laws for Chevalley groups. To appear · Zbl 1515.20323
[15] Guralnick, No article title, J. Algebra, 219, 345 (1999) · Zbl 0948.20052 · doi:10.1006/jabr.1999.7869
[16] Kantor, No article title, Geom. Dedicata, 36, 67 (1990) · Zbl 0718.20011 · doi:10.1007/BF00181465
[17] Kleidman, P.B., Liebeck, M.W.: The Subgroup Structure of the Finite Classical Groups. Lond. Math. Soc. Lect. Note Ser. 129. Cambridge: Cambridge University Press 1990 · Zbl 0697.20004
[18] Landazuri, No article title, J. Algebra, 32, 418 (1974) · Zbl 0325.20008 · doi:10.1016/0021-8693(74)90150-1
[19] Lang, No article title, Am. J. Math., 76, 819 (1954) · Zbl 0058.27202 · doi:10.2307/2372655
[20] Lawther, R.: Elements of specified order in simple algebraic groups. Trans. Am. Math. Soc. To appear · Zbl 1096.20038
[21] Lawther, No article title, Pac. J. Math., 205, 393 (2002) · Zbl 1058.20001 · doi:10.2140/pjm.2002.205.393
[22] Liebeck, No article title, Proc. Lond. Math. Soc., 50, 426 (1985) · Zbl 0591.20021 · doi:10.1112/plms/s3-50.3.426
[23] Liebeck, No article title, J. Algebra, 198, 538 (1997) · Zbl 0892.20017 · doi:10.1006/jabr.1997.7158
[24] Liebeck, No article title, Proc. Lond. Math. Soc., 55, 299 (1987) · doi:10.1093/plms/s3-55_2.299
[25] Liebeck, M.W., Seitz, G.M.: Reductive subgroups of exceptional algebraic groups. Mem. Am. Math. Soc., Vol. 121, No. 580. Providence, RI: Am. Math. Soc. 1996 · Zbl 0851.20045
[26] Liebeck, M.W., Seitz, G.M.: The maximal subgroups of positive dimension in exceptional algebraic groups. Mem. Am. Math. Soc., Vol. 169, No. 802, pp. 1-227. Providence, RI: Am. Math. Soc. 2004 · Zbl 1058.20040
[27] Liebeck, No article title, Geom. Dedicata, 56, 103 (1995) · Zbl 0836.20068 · doi:10.1007/BF01263616
[28] Liebeck, No article title, Ann. Math., 144, 77 (1996) · Zbl 0865.20020 · doi:10.2307/2118584
[29] Liebeck, No article title, J. Algebra, 184, 31 (1996) · Zbl 0870.20014 · doi:10.1006/jabr.1996.0248
[30] Liebeck, No article title, J. Am. Math. Soc., 12, 497 (1999) · Zbl 0916.20003 · doi:10.1090/S0894-0347-99-00288-X
[31] Liebeck, No article title, Bull. Lond. Math. Soc., 34, 185 (2002) · Zbl 1046.20046 · doi:10.1112/S0024609301008827
[32] Liebeck, No article title, J. Algebra, 276, 552 (2004) · Zbl 1068.20052 · doi:10.1016/S0021-8693(03)00515-5
[33] Liebeck, M.W., Shalev, A.: Character degrees of finite Chevalley groups. Proc. Lond. Math. Soc. To appear · Zbl 1077.20020
[34] Lübeck, No article title, LMS J. Comput. Math., 4, 22 (2001) · Zbl 1053.20008 · doi:10.1112/S1461157000000838
[35] Lubotzky, A., Magid, A.R.: Varieties of representations of finitely generated groups. Mem. Am. Math. Soc., Vol. 58, No. 336, pp. 1-117. Providence, RI: Am. Math. Soc. 1985 · Zbl 0598.14042
[36] Lulov, N.: Random walks on symmetric groups generated by conjugacy classes. Ph.D. Thesis, Harvard University 1996
[37] Lyndon, No article title, Mich. Math. J., 6, 155 (1959)
[38] Malle, No article title, Geom. Dedicata, 49, 85 (1994) · Zbl 0832.20029 · doi:10.1007/BF01263536
[39] Mednykh, No article title, Commun. Algebra, 16, 2137 (1988) · Zbl 0649.57001 · doi:10.1080/00927878808823684
[40] Mulase, M., Penkava, M.: Volume of representation varieties. Preprint · Zbl 1253.14030
[41] Müller, No article title, J. Lond. Math. Soc., 66, 623 (2002) · Zbl 1059.20021 · doi:10.1112/S0024610702003599
[42] Rapinchuk, No article title, Isr. J. Math., 93, 29 (1996) · Zbl 0857.14012 · doi:10.1007/BF02761093
[43] Shamash, No article title, Proc. Symp. Pure Math., 47, 283 (1987) · doi:10.1090/pspum/047.2/933418
[44] Shinoda, No article title, J. Fac. Sci., Univ. Tokyo, 22, 1 (1975)
[45] Springer, T.A., Steinberg, R.: Conjugacy classes. In: Borel, A., et al. (eds.) Seminar on algebraic groups and related topics. Lecture Notes Math., Vol. 131, pp. 168-266. Berlin: Springer 1970
[46] Steinberg, No article title, Can. J. Math., 3, 225 (1951) · Zbl 0042.25602 · doi:10.4153/CJM-1951-027-x
[47] Suzuki, No article title, Ann. Math., 75, 105 (1962) · Zbl 0106.24702 · doi:10.2307/1970423
[48] Tiep, No article title, Commun. Algebra, 24, 2093 (1996) · Zbl 0901.20031 · doi:10.1080/00927879608825690
[49] Wagner, No article title, Arch. Math., 29, 583 (1977) · Zbl 0383.20009 · doi:10.1007/BF01220457
[50] Wall, No article title, J. Aust. Math. Soc., 3, 1 (1965) · Zbl 0122.28102 · doi:10.1017/S1446788700027622
[51] Ward, No article title, Trans. Am. Math. Soc., 121, 62 (1966)
[52] Wilf, No article title, Bull. Am. Math. Soc., 15, 228 (1986) · Zbl 0613.05007 · doi:10.1090/S0273-0979-1986-15486-8
[53] Witten, No article title, Commun. Math. Phys., 141, 153 (1991) · Zbl 0762.53063 · doi:10.1007/BF02100009
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