Hyperbolic bundles: deformations and invariants

Abstract

A central question in Riemannian geometry is the following: when does a given manifold admit a certain geometric structure? We are particularly interested in the case of a disc bundle over a closed orientable surface equipped with complex hyperbolic structure. Several examples of such bundles were constructed by the advisor and his collaborators. The next natural step is to begin studying the classifying spaces of these complex hyperbolic bundles, i.e., the corresponding Teichmüller spaces. Hence, we intend to search for new discrete invariants of discrete faithful representations of surface groups into the group of holomorphic isometries of the complex hyperbolic plane. Moreover, we will try to develop a new local version of Poincaré's polyhedron theorem that admits parabolic cycles of vertices and, therefore, could be effectively applied to deformations of the mentioned complex hyperbolic structures. (AU)