The investigator will continue research on the arithmetic of hyperbolic 3-manifolds begun under DMS-9108050. His main emphasis is on the study of incompressible surfaces in hyperbolic 3-manifolds, representations of the fundamental groups of such 3-manifolds, and the geometry and structure of closed geodesics in hyperbolic 3-manifolds. Since the space of everyday life is three dimensional, understanding the structure of objects of this dimension is likely to be especially relevant to the real world. This world is certainly locally Euclidean, but on a cosmological scale it may well be hyperbolic -- at issue is the behavior of light paths of enormous length. In any event the techniques invented to answer questions about hyperbolic manifolds have surprisingly many applications elswehere in mathematics and theoretical physics.
