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Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type
Authors:Vincent Koziarz  Julien Maubon
Affiliation:(1) Institut Elie Cartan, Université Henri Poincaré, B. P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
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})}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 K?hler 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.
Keywords:Complex hyperbolic space  Lattice  Hermitian symmetric space  Toledo invariant  Higgs bundles  Rigidity
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