Mathematical Sciences: Geometric Structures and Discrete Groups

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This continuing investigation of geometric structures on manifolds and discrete groups will concentrate on complex hyperbolic geometry, spherical CR geometry, flat Lorentzian geometry, conformal 3-dimensional geometry, and real projective geometry. These geometries are intimately interrelated, yet each displays its own characteristic personality. Direct constructions of discrete groups by fundamental polyhedra (in the spirit of Poincare) in geometries in which totally geodesic hypersurfaces either do not exist or are not sufficiently flexible to build fundamental polyhedra, have led to alternate notions of half-spaces and hypersurfaces. The investigator has analyzed how such hypersurfaces intersect and the possible combinatorial types for polyhedra built from these objects. He applies this theory to the geometry and topology of moduli spaces of geometric structures on a manifold, these spaces being closely related to the better known moduli space of representations of a manifold's fundamental group, a structure profitably studied using gauge-theoretic techniques. The prototype of such moduli spaces is Teichmueller space, and one objective of this project is to see how the geometry of Teichmueller space extends to these more general moduli spaces. Despite the fact that we live in a 3-dimensional manifold, our intuition quickly fails us as a guide to the truth or falsity of fairly simple questions about their general geometric and topological properties. Nor is it terribly helpful in telling us where to look for answers. An elaborate theory has been developed which is growing by leaps and bounds and has been far more successful than raw intuition, one of the most surprising features of the theory being the prominent role that a non-Euclidean geometry, so-called hyperbolic geometry, occupies in this theory. Other special geometric structures also have been exploited in similar ways, somewhat as a scaffolding is used in constructing a building: One asks a topological question about a manifold, something intrinsically independent of geometry. One then imposes a geometric structure, deduces properties that seem to depend upon this geometric structure (and some of which, as intermediate steps, actually do depend upon it), and finally in a sense kicks away the scaffolding to reveal the desired answer to the original question. By this I mean that one observes that one has deduced a result that does not mention the geometric structure explicitly, and that one then succeeds in showing that any other choice for the geometric structure would have led to the same final result. This is only one intriguing aspect of such research, for it is true that the geometric structures involved have now attracted a considerable amount of interest in their own right.