Mathematical Sciences: The Large-Scale Geometry of Groups, and Spaces with Nonpositive Curvature

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9626911 Kleiner The proposed research lies in the interface amongst group theory, differential geometry, dynamics and topology, stemming from the study of negatively curved surfaces. More specifically, the investigator is interested in relations between a discrete group and the large scale structure of its Cayley graphs, and applications of these to geometric and dynamical problems involving negatively curved surfaces. In mathematics as well as in physics and chemistry, groups are often used to describe various symmetry properties. Discrete groups are finite, hence, objects that can be written down rather explicitly. The fundamental group of a curved space is a group carrying information about large scale symmetries of the space; this group turns out to be discrete in many cases.