Topics in Discrete Groups and Geometry

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ABSTRACTTOPICS IN DISCRETE GROUPS AND GEOMETRYRichard Schwartz plans to continue his research ingeometry and discrete groups.One of his main goals is to explore the connectionsbetween three dimensional real hyperbolic geometryand four dimensional complex hyperbolic geometry.Schwartz observed that certain complex hyperbolicdeformations of the reflection triangle groupslead to complex hyperbolic four-manifolds whoseideal boundaries are real hyperbolic three-manifolds.As the angles in the triangle group change, theideal boundary appears to undergo Dehn surgery.Schwartz also plans to study quotients of the circle,based on patterns of geodesics in the hyperbolicplane which are invariant under the action of asurface group. The idea is to make topological models for the limit sets which couldarise in connection with deformations of surfacegroups into Lie groups and then use the modelsto study actual deformations of surface groups intoLie groups.Broadly speaking, Schwartz' research deals withthe geometry of infinite repeating patterns.A crystal lattice is an example of an infiniterepeating pattern in Euclidean space. Analogouspatterns exist in curved spaces, and oftenthe curvature of the space allows for theexistence of more exotic and geometrically rich patterns.Schwartz is interested in the studying theseexotic patterns when they are generated by a simplemechanism which has a finite description.For example, one can place several mirrors ina curved space and look at the pattern generatedby the images of an object which is reflectedendlessly in the mirrors.The central question is: When does the finitemechanism of generation lead to an infinitediscrete pattern?