DISCRETE GROUPS · STRUCTURES OF SPACES OF COMPLEX HYPERBOLIC DISCRETE GROUPS
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2004-10-01 → 2006-09-30
- EU contribution
- €223,657
- Participants
- 1
- Scheme
- EIF
Lines connect the coordinator with its partners.
Results in brief
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.
Data: CORDIS, © European Union
Project objective
The study of discrete subgroups of complex hyperbolic isometries has been a growing research area for more than a decade although its origins can be found in the works of mathematicians of the 19th century. Many leading mathematicians were led to work in this area due to its fascinating and challenging research problems. This project is a part of a wider research programme whose goal is to understand the space of discrete representations of surface groups into the group of complex hyperbolic isometries. This goal is ambitious and will probably be achieved over a longer time scale than this project's duration. Therefore the objective of this programme is to make significant contributions to that wider plan and to answer natural questions arising, such as how can discrete representations be characterised or if the collection of discrete representations has an analytic structure.At present there exist only a few techniques for constructing spaces of discrete groups of complex hyperbolic isometries and most of them just produce a method of constructing a fundamental domain. The missing ingredient for a further comprehensive treatment is the use of analytic tools, which will fill the gap to classify these representations, which are discrete and produce results analogous to those in the real hyperbolic case, or, determine where such techniques break down. John Parker, the scientist in charge, is an expert on discrete groups of complex hyperbolic isometries and loannis Platis, the participant researcher, is an expert on analytic techniques for describing spaces of discrete groups of real hyperbolic isometries. The potential of this project rests on their synergy in these techniques. This project, extending Platis's previous expertise will give him a broader research perspective. Direct contact with people working in related fields will provide him with knowledge and skills, which he can take back to Greece to strengthen the area there.
Original text from CORDIS.
Participants
- UNIVERSITY OF DURHAM · DURHAMCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union