Geometry & Topology

A Milnor–Wood inequality for complex hyperbolic lattices in quaternionic space

Oscar García-Prada and Domingo Toledo

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We prove a Milnor–Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.

Article information

Geom. Topol., Volume 15, Number 2 (2011), 1013-1027.

Received: 14 October 2010
Accepted: 3 January 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Milnor–Wood inequality rigidity complex hyperbolic lattice


García-Prada, Oscar; Toledo, Domingo. A Milnor–Wood inequality for complex hyperbolic lattices in quaternionic space. Geom. Topol. 15 (2011), no. 2, 1013--1027. doi:10.2140/gt.2011.15.1013.

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  • A L Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Math. und ihrer Grenzgebiete 93, Springer, Berlin (1978) With appendices by D B A Epstein, J-P Bourguignon, L Bérard-Bergery, M Berger and J L Kazdan
  • G Besson, G Courtois, S Gallot, Inégalités de Milnor–Wood géométriques, Comment. Math. Helv. 82 (2007) 753–803
  • S B Bradlow, O García-Prada, P B Gothen, Surface group representations and ${\rm U}(p,q)$–Higgs bundles, J. Differential Geom. 64 (2003) 111–170
  • S B Bradlow, O García-Prada, P B Gothen, Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata 122 (2006) 185–213
  • M Bucher, T Gelander, Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes, C. R. Math. Acad. Sci. Paris 346 (2008) 661–666
  • M Burger, A Iozzi, A Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. $(2)$ 172 (2010) 517–566
  • J Carlson, S Müller-Stach, C Peters, Period mappings and period domains, Cambridge Studies in Advanced Math. 85, Cambridge Univ. Press (2003)
  • J A Carlson, D Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. (1989) 173–201
  • S S Chen, L Greenberg, Hyperbolic spaces, from: “Contributions to analysis (a collection of papers dedicated to Lipman Bers)”, (L V Ahlfors, I Kra, B Maskit, L Nirenberg, editors), Academic Press, New York (1974) 49–87
  • K Corlette, Flat $G$–bundles with canonical metrics, J. Differential Geom. 28 (1988) 361–382
  • S K Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. $(3)$ 55 (1987) 127–131
  • O García-Prada, P B Gothen, I Mundet i Riera, Higgs bundles and surface group representations in the real symplectic group
  • W M Goldman, J J Millson, Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987) 495–520
  • P Griffiths, W Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253–302
  • I Kim, P Pansu, Local rigidity in quaternionic hyperbolic space, J. Eur. Math. Soc. $($JEMS$)$ 11 (2009) 1141–1164
  • I Kim, P Pansu, B Klingler, Local quaternionic rigidity for complex hyperbolic lattices
  • B Klingler, Local rigidity for complex hyperbolic lattices and Hodge theory, to appear in Invent. Math. (2009)
  • S Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Math. 2, Marcel Dekker, New York (1970)
  • S Kobayashi, K Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Math. 15, Wiley Interscience, New York-London-Sydney (1969)
  • V Koziarz, J Maubon, Representations of complex hyperbolic lattices into rank $2$ classical Lie groups of Hermitian type, Geom. Dedicata 137 (2008) 85–111
  • V Koziarz, J Maubon, The Toledo invariant on smooth varieties of general type, J. Reine Angew. Math. 649 (2010) 207–230
  • J Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958) 215–223
  • D Toledo, Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989) 125–133
  • D Toledo, Maps between complex hyperbolic surfaces, Geom. Dedicata 97 (2003) 115–128 Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999)
  • J W Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971) 257–273