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Surface group representations with maximal Toledo invariant. (English) Zbl 1208.32014
The framework of this paper is the so-called higher Teichmüller theory, that is, the extension of Teichmüller theory to representations of fundamental groups of surfaces to Lie groups that are more general than \(\mathrm{PSL}(2,\mathbb{R})\). The authors consider the case where \(G\) is a Lie group of Hermitian type.
Given a compact connected oriented surface \(\Sigma\), possibly with boundary, the authors define and study a bounded integer-valued function on the representation variety of \(\pi_1(\Sigma)\) into \(G\), which they call the Toledo invariant. The name was chosen because D. Toledo, in 1989, studied such a function in the particular setting of representations of fundamental groups of surfaces in complex hyperbolic spaces. The authors call maximal representations those whose Toledo invariant has maximal value. They develop the theory of maximal representations in this setting of a group \(G\) of Hermitian type.
The Toledo invariant is reminiscent of the Euler number, which is an integer-valued function defined on the representation variety \(\mathrm{Hom}(\pi_1(S), \mathrm{PSL}(2,\mathbb{R}))/\mathrm{PSL}(2,\mathbb{R})\) and which is constant on the connected components of the representation variety. The classical Teichmüller space is precisely the connected component with maximal Euler number (Goldman’s thesis, 1980).
In the setting of Lie groups of Hermitian type, the level set of the maximal value of the modulus of the Toledo invariant is a union of connected components of representations that the authors call maximal. The space of maximal representations is an instance of a higher Teichmüller space. Global properties of the components of the representation variety that have maximal Toledo invariant were studied from another point of view, in the case of closed surfaces, by Bradlow, García-Prada and Gothen, using Higgs bundles. In the paper under review, the authors establish properties of the Toledo invariant function on the representation variety such as continuity, uniform boundedness, additivity under connected sums of surfaces and congruence relations mod \(\mathbb{Z}\). They obtain information about the representation variety and geometric properties of maximal representations. They prove a structure theorem for such representations. They associate to maximal representations boundary maps with monotonicity (positivity) properties expressed in terms of maximal triples of points in the Shilov boundary of the symmetric space associated to the Lie group \(G\), and they give a formula for the Toledo invariant in terms of the bounded Euler class of a group action on a circle. The latter generalizes the notion of rotation number for circle homeomorphisms which was shown by Poincaré to be a semi-conjugacy invariant of oriention-preserving homeomorphisms of the circle.

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M50 General geometric structures on low-dimensional manifolds
22E40 Discrete subgroups of Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:
[1] J. Barge and É. Ghys, ”Cocycles d’Euler et de Maslov,” Math. Ann., vol. 294, iss. 2, pp. 235-265, 1992. · Zbl 0894.55006 · doi:10.1007/BF01934324 · eudml:165000
[2] P. Blanc, ”Sur la cohomologie continue des groupes localement compacts,” Ann. Sci. École Norm. Sup., vol. 12, iss. 2, pp. 137-168, 1979. · Zbl 0429.57012 · numdam:ASENS_1979_4_12_2_137_0 · eudml:82033
[3] A. Borel, ”Class functions, conjugacy classes and commutators in semisimple Lie groups,” in Algebraic Groups and Lie Groups, Cambridge: Cambridge Univ. Press, 1997, vol. 9, pp. 1-19. · Zbl 0872.22011
[4] S. B. Bradlow, O. Garc’ia-Prada, and P. B. Gothen, ”Surface group representations and \({ U}(p,q)\)-Higgs bundles,” J. Differential Geom., vol. 64, iss. 1, pp. 111-170, 2003. · Zbl 1070.53054 · projecteuclid.org
[5] S. B. Bradlow, O. Garc’ia-Prada, and P. B. Gothen, ”Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces,” Geom. Dedicata, vol. 122, pp. 185-213, 2006. · Zbl 1132.14029 · doi:10.1007/s10711-007-9127-y · arxiv:math/0511415
[6] R. Brooks, ”Some remarks on bounded cohomology,” in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 97, pp. 53-63. · Zbl 0457.55002
[7] M. Burger, A. Iozzi, F. Labourie, and A. Wienhard, ”Maximal representations of surface groups: symplectic Anosov structures,” Pure Appl. Math. Q., vol. 1, iss. 3, Special Issue: In memory of Armand Borel. Part 2, pp. 543-590, 2005. · Zbl 1157.53025 · doi:10.4310/PAMQ.2005.v1.n3.a5 · arxiv:math/0506079
[8] M. Burger and A. Iozzi, A useful formula in bounded cohomology. · Zbl 1206.22006 · www.math.ethz.ch
[9] M. Burger and A. Iozzi, ”Boundary maps in bounded cohomology. Appendix to: “Continuous bounded cohomology and applications to rigidity theory” [Geom. Funct. Anal. 12 (2002), no. 2, 219-280; MR1911660 (2003d:53065a)] by M. Burger and N. Monod,” Geom. Funct. Anal., vol. 12, iss. 2, pp. 281-292, 2002. · Zbl 1006.22011 · doi:10.1007/s00039-002-8246-8
[10] M. Burger and A. Iozzi, ”Bounded Kähler class rigidity of actions on Hermitian symmetric spaces,” Ann. Sci. École Norm. Sup., vol. 37, iss. 1, pp. 77-103, 2004. · Zbl 1061.32016 · doi:10.1016/j.ansens.2003.09.001 · numdam:ASENS_2004_4_37_1_77_0 · eudml:82628
[11] M. Burger and A. Iozzi, ”Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity,” Geom. Dedicata, vol. 125, pp. 1-23, 2007. · Zbl 1134.53020 · doi:10.1007/s10711-006-9108-6
[12] M. Burger, A. Iozzi, and A. Wienhard, ”Tight homomorphisms and Hermitian symmetric spaces,” Geom. Funct. Anal., vol. 19, iss. 3, pp. 678-721, 2009. · Zbl 1188.53050 · doi:10.1007/s00039-009-0020-8 · arxiv:0710.5641
[13] M. Burger, A. Iozzi, and A. Wienhard, ”Surface group representations with maximal Toledo invariant,” C. R. Math. Acad. Sci. Paris, vol. 336, iss. 5, pp. 387-390, 2003. · Zbl 1035.32013 · doi:10.1016/S1631-073X(03)00065-7 · arxiv:math/0605656
[14] M. Burger, A. Iozzi, and A. Wienhard, ”Hermitian symmetric spaces and Kähler rigidity,” Transform. Groups, vol. 12, iss. 1, pp. 5-32, 2007. · Zbl 1138.32012 · doi:10.1007/s00031-005-1135-0
[15] M. Burger and N. Monod, ”Continuous bounded cohomology and applications to rigidity theory,” Geom. Funct. Anal., vol. 12, iss. 2, pp. 219-280, 2002. · Zbl 1006.22010 · doi:10.1007/s00039-002-8245-9
[16] J. Clerc, ”L’indice de Maslov généralisé,” J. Math. Pures Appl., vol. 83, iss. 1, pp. 99-114, 2004. · Zbl 1061.53056 · doi:10.1016/j.matpur.2003.09.010
[17] J. Clerc, ”An invariant for triples in the Shilov boundary of a bounded symmetric domain,” Comm. Anal. Geom., vol. 15, iss. 1, pp. 147-173, 2007. · Zbl 1123.32014 · doi:10.4310/CAG.2007.v15.n1.a5 · projecteuclid.org
[18] J. Clerc and K. Koufany, ”Primitive du cocycle de Maslov généralisé,” Math. Ann., vol. 337, iss. 1, pp. 91-138, 2007. · Zbl 1110.32009 · doi:10.1007/s00208-006-0028-4
[19] J. Clerc and K. Neeb, ”Orbits of triples in the Shilov boundary of a bounded symmetric domain,” Transform. Groups, vol. 11, iss. 3, pp. 387-426, 2006. · Zbl 1112.32010 · doi:10.1007/s00031-005-1117-2 · arxiv:math/0511259
[20] J. Clerc and B. Ørsted, ”The Maslov index revisited,” Transform. Groups, vol. 6, iss. 4, pp. 303-320, 2001. · Zbl 1078.53076 · doi:10.1007/BF01237249
[21] J. Clerc and B. Ørsted, ”The Gromov norm of the Kaehler class and the Maslov index,” Asian J. Math., vol. 7, iss. 2, pp. 269-295, 2003. · Zbl 1079.53120
[22] V. Fock and A. Goncharov, ”Moduli spaces of local systems and higher Teichmüller theory,” Publ. Math. Inst. Hautes Études Sci., iss. 103, pp. 1-211, 2006. · Zbl 1099.14025 · doi:10.1007/s10240-006-0039-4 · numdam:PMIHES_2006__103__1_0 · eudml:104216
[23] V. Fock and A. Goncharov, ”Moduli spaces of convex projective structures on surfaces,” Adv. Math., vol. 208, iss. 1, pp. 249-273, 2007. · Zbl 1111.32013 · doi:10.1016/j.aim.2006.02.007
[24] H. Furstenberg, ”A Poisson formula for semi-simple Lie groups,” Ann. of Math., vol. 77, pp. 335-386, 1963. · Zbl 0192.12704 · doi:10.2307/1970220
[25] &. Ghys, ”Groupes d’homéomorphismes du cercle et cohomologie bornée,” in The Lefschetz Centennial Conference, Part III, Providence, RI: Amer. Math. Soc., 1987, vol. 58, pp. 81-106. · Zbl 0617.58009
[26] W. M. Goldman, ”Discontinuous groups and the Euler class,” PhD Thesis , Univ. California Berkeley, 1980.
[27] W. M. Goldman, ”Topological components of spaces of representations,” Invent. Math., vol. 93, iss. 3, pp. 557-607, 1988. · Zbl 0655.57019 · doi:10.1007/BF01410200 · eudml:143609
[28] W. M. Goldman, ”The modular group action on real \({ SL}(2)\)-characters of a one-holed torus,” Geom. Topol., vol. 7, pp. 443-486, 2003. · Zbl 1037.57001 · doi:10.2140/gt.2003.7.443 · emis:journals/UW/gt/GTVol7/paper13.abs.html · eudml:123528 · arxiv:math/0305096
[29] P. B. Gothen, ”Components of spaces of representations and stable triples,” Topology, vol. 40, iss. 4, pp. 823-850, 2001. · Zbl 1066.14012 · doi:10.1016/S0040-9383(99)00086-5
[30] M. Gromov, ”Volume and bounded cohomology,” Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 5-99 (1983), 1982. · Zbl 0516.53046 · numdam:PMIHES_1982__56__5_0 · eudml:103988
[31] O. Guichard, ”Composantes de Hitchin et représentations hyperconvexes de groupes de surface,” J. Differential Geom., vol. 80, pp. 391-431, 2008. · Zbl 1223.57015
[32] L. Hernández, ”Maximal representations of surface groups in bounded symmetric domains,” Trans. Amer. Math. Soc., vol. 324, iss. 1, pp. 405-420, 1991. · Zbl 0733.32024 · doi:10.2307/2001515
[33] N. Hitchin, ”Lie groups and Teichmüller space,” Topology, vol. 31, iss. 3, pp. 449-473, 1992. · Zbl 0769.32008 · doi:10.1016/0040-9383(92)90044-I
[34] S. Ihara, ”Holomorphic imbeddings of symmetric domains,” J. Math. Soc. Japan, vol. 19, pp. 261-302, 1967. · Zbl 0159.11102 · doi:10.2969/jmsj/01930261
[35] N. V. Ivanov, ”Foundations of the theory of bounded cohomology,” J. Soviet Math., vol. 37, pp. 1090-1115, 1987. · Zbl 0612.55006 · doi:10.1007/BF01086634
[36] V. A. Kaimanovich, ”SAT actions and ergodic properties of the horosphere foliation,” in Rigidity in Dynamics and Geometry, New York: Springer-Verlag, 2002, pp. 261-282. · Zbl 1054.37002
[37] A. Korányi and J. A. Wolf, ”Realization of hermitian symmetric spaces as generalized half-planes,” Ann. of Math., vol. 81, pp. 265-288, 1965. · Zbl 0137.27402 · doi:10.2307/1970616
[38] V. Koziarz and J. Maubon, ”Harmonic maps and representations of non-uniform lattices of \({ PU}(m,1)\),” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 58, iss. 2, pp. 507-558, 2008. · Zbl 1147.22009 · doi:10.5802/aif.2359 · numdam:AIF_2008__58_2_507_0 · eudml:10323 · arxiv:math/0309193
[39] F. Labourie, ”Cross ratios, Anosov representations and the energy functional on Teichmüller space,” Ann. Sci. École Norm. Sup., vol. 41, pp. 439-471, 2009. · Zbl 1160.37021 · smf4.emath.fr
[40] F. Labourie, ”Anosov flows, surface groups and curves in projective space,” Invent. Math., vol. 165, iss. 1, pp. 51-114, 2006. · Zbl 1103.32007 · doi:10.1007/s00222-005-0487-3
[41] C. Löh-Strohm, ”The proportionality principle of simplicial volume,” , preprint , 2005. · www.arXiv.org
[42] G. Lusztig, ”Total positivity in reductive groups,” in Lie Theory and Geometry, Boston, MA: Birkhäuser, 1994, vol. 123, pp. 531-568. · Zbl 0845.20034
[43] G. Lusztig, ”Total positivity in partial flag manifolds,” Represent. Theory, vol. 2, pp. 70-78, 1998. · Zbl 0895.14014 · doi:10.1090/S1088-4165-98-00046-6
[44] G. W. Mackey, ”Les ensembles boréliens et les extensions des groupes,” J. Math. Pures Appl., vol. 36, pp. 171-178, 1957. · Zbl 0080.02303
[45] J. Milnor, ”On the existence of a connection with curvature zero,” Comment. Math. Helv., vol. 32, pp. 215-223, 1958. · Zbl 0196.25101 · doi:10.1007/BF02564579 · eudml:139154
[46] N. Monod, Continuous Bounded Cohomology of Locally Compact Groups, New York: Springer-Verlag, 2001, vol. 1758. · Zbl 0967.22006 · doi:10.1007/b80626 · link.springer.de
[47] I. Satake, ”Holomorphic imbeddings of symmetric domains into a Siegel space,” Amer. J. Math., vol. 87, pp. 425-461, 1965. · Zbl 0144.08202 · doi:10.2307/2373012
[48] I. Satake, Algebraic Structures of Symmetric Domains, Tokyo: Iwanami Shoten, 1980, vol. 4. · Zbl 0483.32017
[49] D. Toledo, ”Representations of surface groups in complex hyperbolic space,” J. Differential Geom., vol. 29, iss. 1, pp. 125-133, 1989. · Zbl 0676.57012 · projecteuclid.org
[50] W. T. van Est, ”Group cohomology and Lie algebra cohomology in Lie groups, I, II, Nederl. Akad. Wetensch. Proc. Series A. 56,” Indag. Math., vol. 15, pp. 484-504, 1953. · Zbl 0051.26001
[51] A. Wienhard, ”The action of the mapping class group on maximal representations,” Geom. Dedicata, vol. 120, pp. 179-191, 2006. · Zbl 1175.32007 · doi:10.1007/s10711-006-9079-7 · arxiv:math/0605038
[52] D. Wigner, ”Algebraic cohomology of topological groups,” Trans. Amer. Math. Soc., vol. 178, pp. 83-93, 1973. · Zbl 0264.22001 · doi:10.2307/1996690
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