Final Activity Report Summary - ANALYSIS IN CHG (Applications of analysis in complex hyperbolic geometry)
Complex hyperbolic geometry is a generalisation of the more familiar (real) hyperbolic geometry and, though its origin goes back to the 19th century, it has become an exciting field of research just only over the last twenty years. This project is concerned with the study of geometrical and analytical study of the space QC of discrete representations of a surface group into the group of isometries of complex hyperbolic space (the complex hyperbolic quasi-Fuchsian space). It is the natural development of the Marie Curie project the participant Ioannis Platis carried out in Durham University under the guidance and collaboration of Dr J. R. Parker.
Its initial objective was to explore whether quasiconformal mapping theory of the Heisenberg group may be applied in the case of QC to produce analogous (or not) results of these obtained in the classical Teichmueller space case with the aid of ordinary quasiconformal mapping theory. Two major achievements of the project are, first, the extensive study of the cross-ratio variety and secondly the construction of a basic deformation of QC called the complex hyperbolic quakebending deformation. The first of these goals was essential for the further exploration of the analytical aspects of QC and the second is initiating the study of its syplectic geometry pointing clearly to its relation with the symplectic geometry of the classical Teichmueller space.
Its initial objective was to explore whether quasiconformal mapping theory of the Heisenberg group may be applied in the case of QC to produce analogous (or not) results of these obtained in the classical Teichmueller space case with the aid of ordinary quasiconformal mapping theory. Two major achievements of the project are, first, the extensive study of the cross-ratio variety and secondly the construction of a basic deformation of QC called the complex hyperbolic quakebending deformation. The first of these goals was essential for the further exploration of the analytical aspects of QC and the second is initiating the study of its syplectic geometry pointing clearly to its relation with the symplectic geometry of the classical Teichmueller space.