×

Braid groups and Hodge theory. (English) Zbl 1290.30050

A configuration of distinct points \((b_1,\ldots, b_n)\) in the complex plane gives rise to a compact Riemann surface which is a branched covering of \(\mathbb{P}^1\) whose equation is \(y^d=(x-b_1)\ldots (x-d_n)\). The author studies the unitary representations of the braid group and the geometric structures on moduli space that arise from such branched coverings, developing new connections between the braid group and arithmetic groups, ergodic theory, complex reflection groups, Teichmüller curves, moduli spaces of 1-forms, period maps, plane polgyons, hypergeometric functions and Thurston’s work on shapes of polyhedra. The approach uses the classification of certain arithmetic subgroups of \(U(r,s)\) which envelop the image of the braid groups. The paper is amazingly rich for the connections made between several theories. The author also mentions related open problems in surface topology.

MSC:

30F10 Compact Riemann surfaces and uniformization
14H15 Families, moduli of curves (analytic)
14H45 Special algebraic curves and curves of low genus
14H10 Families, moduli of curves (algebraic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] A’Campo N.: Tresses, monodromie et le groupe symplectique. Comment. Math. Helvetici 54, 318–327 (1979) · Zbl 0441.32004 · doi:10.1007/BF02566275
[2] Allcock D.: Braid pictures for Artin groups. Trans. Am. Math. Soc. 354, 3455–3474 (2002) · Zbl 1059.20032 · doi:10.1090/S0002-9947-02-02944-6
[3] Allcock D.: The moduli space of cubic threefolds. J. Algebraic Geom. 12, 201–223 (2003) · Zbl 1080.14531 · doi:10.1090/S1056-3911-02-00313-2
[4] Allcock D., Carlson J.A., Toledo D.: The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geom. 11, 659–724 (2002) · Zbl 1080.14532 · doi:10.1090/S1056-3911-02-00314-4
[5] Allcock, D., Carlson, J.A., Toledo, D.: The moduli space of cubic threefolds as a ball quotient. Mem. Am. Math. Soc. 209 (2011) · Zbl 1211.14002
[6] Andersen J.E., Masbaum G., Ueno K.: Topological quantum field theory and the Nielsen-Thurston classification of M(0, 4). Math. Proc. Cambridge Philos. Soc. 141, 477–488 (2006) · Zbl 1110.57009 · doi:10.1017/S0305004106009698
[7] Arnoux P.: Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore. Bull. Soc. Math. France 116, 489–500 (1988) · Zbl 0703.58045
[8] Arnoux P., Yoccoz J.-C.: Construction de diffeomorphisme pseudo-Anosov. C. R. Acad. Sci. Paris 292, 75–78 (1981) · Zbl 0478.58023
[9] Atiyah, M.F.: Representations of braid groups. In: Geometry of Low-Dimensional Manifolds, 2 (Durham, 1989). London Math. Soc. Lecture Note Ser., vol. 151, pp. 115–122. Cambridge University Press, Cambridge (1990)
[10] Bigelow S.J.: The Burau representation is not faithful for n = 5. Geom. Topol. 3, 397–404 (1999) · Zbl 0942.20017 · doi:10.2140/gt.1999.3.397
[11] Bigelow S.J.: Braid groups are linear. J. Am. Math. Soc. 14, 471–486 (2001) · Zbl 0988.20021 · doi:10.1090/S0894-0347-00-00361-1
[12] Birman, J.S.: Braids, links and mapping-class groups. In: Annals of Math. Studies, vol. 82. Princeton University Press (1974)
[13] Borel A.: Introduction aux groups arithmétiques. Hermann, Paris (1969)
[14] Borel A.: Linear Algebraic Groups, 2nd edn. Springer, Berlin (1991) · Zbl 0726.20030
[15] Borel A., Borel A.: Arithmetic subgroups of algebraic groups. Ann. Math. 75, 485–535 (1962) · Zbl 0107.14804 · doi:10.2307/1970210
[16] Broué M., Malle G., Rouqier R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190 (1998) · Zbl 0921.20046
[17] Burde G., Zieschang H.: Knots. Walter de Gruyter & Co, Berlin (1985)
[18] Burkhardt H.: Grundzüge einer allgemeinen Systematik der hyperelliptischen Functionen I. Ordnung. Math. Ann. 35, 198–296 (1890) · JFM 21.0496.01 · doi:10.1007/BF01443877
[19] Carlson J., Müller-Stach S., Peters C.: Period Mappings and Period Domains. Cambridge University Press, Cambridge (2003) · Zbl 1030.14004
[20] Chevalley, C., Weil, A.: [1934a] Über das Verhalten der Integral erster Gattung bei Automorphismen des Funktionenkörpers. In: Weil, A. (ed.) Oeuvres Scient., vol. I, pp. 68–71. Springer, Berlin (1980) · Zbl 0009.16001
[21] Cohen P., Wolfart J.: Modular embeddings for some non-arithmetic Fuchsian groups. Acta Arith. 56, 93–110 (1990) · Zbl 0717.14014
[22] Cohen P., Wolfart J.: Algebraic Appell-Lauricella functions. Analysis 12, 359–376 (1992) · Zbl 0761.33007
[23] Cohen P., Wolfart J.: Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés ab éliennes. Ann. Sci. École Norm. Sup. 26, 665–690 (1993) · Zbl 0822.14014
[24] Coxeter H.S.M.: Finite groups generated by unitary reflections. Abh. Math. Sem. Univ. Hamburg 31, 125–135 (1967) · Zbl 0189.32302 · doi:10.1007/BF02992390
[25] Deligne P., Mostow G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. IHES 63, 5–89 (1986) · Zbl 0615.22008 · doi:10.1007/BF02831622
[26] Deligne, P., Mostow, G.D.: Commensurabilities among Lattices in PU(1, n). Annals of Math. Studies. Princeton University Press (1993) · Zbl 0826.22011
[27] Dolgachev, I.V., Kondo, S.: Moduli of K3 surfaces and complex ball quotients. In: Arithmetic and Geometry around Hypergeometric Functions. Progr. Math., vol. 260, pp. 43–100. Birkhäuser, Boston (2007) · Zbl 1124.14032
[28] Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton University Press, Princeton (2012) · Zbl 1245.57002
[29] Gross B.H.: On the centralizer of a regular, semi-simple, stable conjugacy class. Represent. Theory 9, 287–296 (2005) · Zbl 1107.20034 · doi:10.1090/S1088-4165-05-00283-9
[30] Howe R.E., Moore C.C.: Asymptotic properties of unitary representations. J. Funct. Anal. 32, 72–96 (1979) · Zbl 0404.22015 · doi:10.1016/0022-1236(79)90078-8
[31] Hurwitz A.: Über algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41, 403–442 (1893) · JFM 24.0380.02 · doi:10.1007/BF01443420
[32] Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987) · Zbl 0631.57005 · doi:10.2307/1971403
[33] Kapovich, M.: Periods of abelian differentials and dynamics. Preprint (2000) · Zbl 0953.20035
[34] Kapovich M., Millson J.J.: On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties. Publ. Math. IHES 88, 5–95 (1998) · Zbl 0982.20023 · doi:10.1007/BF02701766
[35] Kassel C., Turaev V.: Braid Groups. Springer, Berlin (2008) · Zbl 1208.20041
[36] Kerckhoff S.: The Nielsen realization problem. Ann. Math. 177, 235–265 (1983) · Zbl 0528.57008 · doi:10.2307/2007076
[37] Klein F.: Volesungen über die Hypergeometrische Funktion. Springer, Berlin (1939)
[38] Kondo S.: A complex hyperbolic structure for the moduli space of curves of genus three. J. Reine Angew. Math. 525, 219–232 (2000) · Zbl 0990.14007
[39] Kontsevich M., Zorich A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153, 631–678 (2003) · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x
[40] Krammer D.: Braid groups are linear. Ann. Math. 155, 131–156 (2002) · Zbl 1020.20025 · doi:10.2307/3062152
[41] Lawrence R.J.: Homological representations of the Hecke algebra. Commun. Math. Phys. 135, 141–191 (1990) · Zbl 0716.20022 · doi:10.1007/BF02097660
[42] Leininger C.J.: On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number. Geom. Topol. 8, 1301–1359 (2004) · Zbl 1088.57002 · doi:10.2140/gt.2004.8.1301
[43] Lochak P.: On arithmetic curves in the moduli space of curves. J. Inst. Math. Jussieu 4, 443–508 (2005) · Zbl 1094.14018 · doi:10.1017/S1474748005000101
[44] Looijenga E.: Prym representations of mapping class groups. Geom. Dedicata 64, 69–83 (1997) · Zbl 0872.57018 · doi:10.1023/A:1004909416648
[45] Looijenga E.: Artin groups and the fundamental groups of some moduli spaces. J. Topol. 1, 187–216 (2008) · Zbl 1172.14022 · doi:10.1112/jtopol/jtm009
[46] Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Wiley (1966) · Zbl 0138.25604
[47] Margulis G.A.: Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin (1991) · Zbl 0732.22008
[48] Masur H., Smillie J.: Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. 134, 455–543 (1991) · Zbl 0774.58024 · doi:10.2307/2944356
[49] McMullen C.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. 151, 327–357 (2000) · Zbl 0988.32012 · doi:10.2307/121120
[50] McMullen C.: Billiards and Teichmüller curves on Hilbert modular surfaces. J. Am. Math. Soc. 16, 857–885 (2003) · Zbl 1030.32012 · doi:10.1090/S0894-0347-03-00432-6
[51] McMullen C.: Teichmüller curves in genus two: discriminant and spin. Math. Ann. 333, 87–130 (2005) · Zbl 1086.14024 · doi:10.1007/s00208-005-0666-y
[52] McMullen C.: Prym varieties and Teichmüller curves. Duke Math. J. 133, 569–590 (2006) · Zbl 1099.14018 · doi:10.1215/S0012-7094-06-13335-5
[53] McMullen C.: Teichmüller curves in genus two: Torsion divisors and ratios of sines. Invent. Math. 165, 651–672 (2006) · Zbl 1103.14014 · doi:10.1007/s00222-006-0511-2
[54] McMullen C.: Foliations of Hilbert modular surfaces. Am. J. Math. 129, 183–215 (2007) · Zbl 1154.14020 · doi:10.1353/ajm.2007.0002
[55] Möller M.: Shimura- and Teichmüller curves. J. Mod. Dyn. 5, 1–32 (2011) · Zbl 1221.14033 · doi:10.3934/jmd.2011.5.1
[56] Mostow G.D.: On a remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math. 86, 171–276 (1980) · Zbl 0456.22012 · doi:10.2140/pjm.1980.86.171
[57] Mostow G.D.: Braids, hypergeometric functions, and lattices. Bull. Am. Math. Soc. (N.S.) 16, 225–246 (1987) · Zbl 0639.22005 · doi:10.1090/S0273-0979-1987-15510-8
[58] Parker, J.R.: Complex hyperbolic lattices. In: Discrete Groups and Geometric Structures, pp. 1–42. American Mathematical Society, Providence (2009) · Zbl 1200.22004
[59] Perron B.: A homotopic intersection theory on surfaces: applications to mapping class group and braids. Enseign. Math. 52, 159–186 (2006) · Zbl 1161.57009
[60] Picard E.: Sur des fonctions de deux variables indépendantes analogues aux fonctions modulaires. Acta Math. 2, 114–135 (1883) · JFM 15.0432.01 · doi:10.1007/BF02612158
[61] Ratner, M.: Interactions between ergodic theory, Lie groups and number theory. In: Proceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 156–182. Birkhaüser, Basel (1995) · Zbl 0923.22002
[62] Reeder M.: Torsion automorphisms of simple Lie algebras. Enseign. Math. 56, 3–47 (2010) · Zbl 1223.17020
[63] Sansone G., Gerretsen J.: Lectures on the Theory of Functions of a Complex Variable. P. Noordhoff, Groningen (1960) · Zbl 0093.26803
[64] Satake I.: Algebraic Structures of Symmetric Domains. Princeton University Press, Princeton (1980) · Zbl 0483.32017
[65] Sauter J.K.: Isomorphisms among monodromy groups and applications to lattices in PU(1, 2). Pac. J. Math. 146, 331–384 (1990) · Zbl 0759.22013 · doi:10.2140/pjm.1990.146.331
[66] Schwarz H.A.: Über diejenigen Fälle, in welchen die Gaußische hypergeomtrische Reihe eine algebraische Funktion ihres vierten Elements darstellt. J. Reine Angew. Math. 75, 292–335 (1873) · doi:10.1515/crll.1873.75.292
[67] Shephard G.C., Todd J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954) · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3
[68] Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1994) · Zbl 0872.11023
[69] Springer, T.A., Steinberg, R.: Conjugacy classes. In: Algebraic Groups and Related Finite Groups. Lecture Notes in Math., vol. 131, pp. 167–266. Springer, Berlin (1970) · Zbl 0249.20024
[70] Squier C.: The Burau representation is unitary. Proc. Am. Math. Soc. 90, 199–202 (1984) · Zbl 0542.20022 · doi:10.1090/S0002-9939-1984-0727232-8
[71] Takeuchi K.: Arithmetic triangle groups. J. Math. Soc. Jpn. 29, 91–106 (1977) · Zbl 0344.20035 · doi:10.2969/jmsj/02910091
[72] Thurston W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19, 417–431 (1988) · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
[73] Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift. Geom. Topol. Monogr., vol. 1, pp. 511–549. Geom. Topol. Publ. (1998) · Zbl 0931.57010
[74] Troyanov, M.: On the moduli space of singular euclidean surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory, vol. I, pp. 507–540. Eur. Math. Soc. (2007) · Zbl 1127.32009
[75] Veech W.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 553–583 (1989) · Zbl 0676.32006 · doi:10.1007/BF01388890
[76] Veech W.: Moduli spaces of quadratic differentials. J. Anal. Math. 55, 117–171 (1990) · Zbl 0722.30032 · doi:10.1007/BF02789200
[77] Veech W.: Flat surfaces. Am. J. Math. 115, 589–689 (1993) · Zbl 0803.30037 · doi:10.2307/2375075
[78] Whittaker E.T., Watson G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1952) · JFM 45.0433.02
[79] Yoshida M.: Hypergeometric Functions, My Love. Vieweg, Wiesbaden (1997) · Zbl 0889.33008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.