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

Vincent Koziarz (1) , Julien Maubon (1)

1 IECN - Institut Élie Cartan de Nancy

Let G be either SU(p,2) with p>=2, Sp(2,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 uniform lattices of SU(m,1), m>1, 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)xU(2)), on which it acts cocompactly.

Mots clés: representation; lattice; complex hyperbolic; Toledo invariant; Higgs bundles; Hermitian symmetric spaces

Domaines: Géométrie différentielle [math.DG]

Vincent Koziarz, Julien Maubon. Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type. Geometriae Dedicata, 2008, pp.85-111. ⟨hal-00135135v2⟩