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Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity

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Abstract

We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1).

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Author notes

  1. Alessandra Iozzi

    Present address: Department of Mathematics, ETH Zentrum, Raemistrasse 101, CH-8092, Zuerich, Switzerland

Authors and Affiliations

  1. FIM, ETH Zentrum, Rämistrasse 101, CH-8092, Zürich, Switzerland

    Marc Burger

  2. Institut für Mathematik, Universität Basel, Rheinsprungstrasse 21, CH-4051, Basel, Switzerland

    Alessandra Iozzi

  3. Départment de Mathématiques, Université de Strasbourg, 7, Rue René Descartes, F-67084, Strasbourg Cedex, France

    Alessandra Iozzi

Authors
  1. Marc Burger
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  2. Alessandra Iozzi
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Correspondence to Alessandra Iozzi.

Additional information

Alessandra Iozzi was partially supported by FNS grant PP002-102765

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Burger, M., Iozzi, A. Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity. Geom Dedicata 125, 1–23 (2007). https://doi.org/10.1007/s10711-006-9108-6

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  • DOI: https://doi.org/10.1007/s10711-006-9108-6

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