Final Activity Report Summary - DISCRETE GROUPS (Structures of spaces of complex hyperbolic discrete groups)
Complex hyperbolic geometry is a generalisation of the more familiar (real) hyperbolic geometry and, although it has a long history, it has become an exciting field of research over the last two decades or so. The main focus of this project has been the study of discrete groups of complex hyperbolic geometry , in particular discrete, faithful, type-preserving representations of the fundamental group of a closed surface. Conjugacy classes of such representations comprise the complex hyperbolic quasi-Fuchsian space of this surface, generalising Teichmueller space.
Two major achievements of this project are, first, the construction of an open set of maximal dimension inside the complex hyperbolic quasi-Fuchsian space and, secondly, the construction of a real analytic structure on a subset of the representation space containing quasi-Fuchsian space. The first of these goals was achieved by constructing a flexible fundamental domain for groups in the neighbourhood of R-Fuchsian space. The second was achieved by exhibiting coordinates that generalise the classical Fenchel-Nielsen coordinates on Teichmueller space.
Two major achievements of this project are, first, the construction of an open set of maximal dimension inside the complex hyperbolic quasi-Fuchsian space and, secondly, the construction of a real analytic structure on a subset of the representation space containing quasi-Fuchsian space. The first of these goals was achieved by constructing a flexible fundamental domain for groups in the neighbourhood of R-Fuchsian space. The second was achieved by exhibiting coordinates that generalise the classical Fenchel-Nielsen coordinates on Teichmueller space.