Finite covers of hyperbolic 3-manifolds

University Of Texas At Austin, Austin TX

Investigators

Abstract

Motivated by considerations in topology, geometry, arithmetic and group theory the proposal intends to study real, complex and quaternionic hyperbolic manifolds. This will involve the study of discrete groups, their connections with number theory, and expanding graphs. We will also explore various interconnections between these topics with a view to better understanding the topology of hyperbolic 3-manifolds. Three dimensional manifolds are locally like the space we live in and understanding these objects have been one of the central themes of research in the last 30 years. The importance of these objects extends far beyond their intrinsic interest, since their study connects to mathematical physics, mathematical biology and computer science. One of the aims of this proposal is to explore some of these connections via families of "expanding graphs". These graphs are well-known in computer science because of their importance in building efficient networks. Remarkably, this is connected to the main study of the proposal, "finite covers of 3-manifolds". By their nature, many of the problems in the proposal are cross-disciplinary, and hence progress will have a broader impact. In addition, these have proved to be fertile grounds for the training of graduate students.