Complex hyperbolic manifolds homotopy equivalent to a Riemann surface

Abstract

We construct isometric actions of fundamental groups of closed Riemann surfaces on the complex hyperbolic plane, which realize all possible values of Toledo’s invariant τ . For integer values of τ these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over closed Riemann surfaces, whose topological type is described in terms of τ . We relate our geometric construction to arithmetic constructions and discuss integrality properties of τ .