Mathematical Sciences: Two-Generator Discrete Groups: Algebra and Geometry

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The principal investigator will complete the study of two- generator subgroups of PSL(2,R). The investigation will result in a complete and unified account of the geometric and algebraic approaches to this classification problem, as well as a complete description of the relationship between this algebra and geometry. The investigator will also study an extension of these methods to the complex numbers and to several real variables. The work to be completed by the principal investigator has as one of its ramifications the complete understanding of tilings of the plane by tiles invariant under some sort of discrete group action. These have been of interest to mathematicians for more than a century and their study was recently revived by W. Thurston. Connections between this investigation and the topology of three dimensional hypersurfaces as well as to dynamical systems, abound.