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Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

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Abstract

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, \({{\rm Sp}(2,\mathbb {R})}\) or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.

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Authors and Affiliations

  1. Institut Elie Cartan, Université Henri Poincaré, B. P. 239, 54506, Vandoeuvre-les-Nancy Cedex, France

    Vincent Koziarz & Julien Maubon

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  1. Vincent Koziarz
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  2. Julien Maubon
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Correspondence to Julien Maubon.

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Koziarz, V., Maubon, J. Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type. Geom Dedicata 137, 85–111 (2008). https://doi.org/10.1007/s10711-008-9288-3

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