Large Scale Geometry, Topology, and Rigidity in Geometric Group Theory

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9704640 Farb Farb's project is in the area of geometric group theory. A new geometric/topological approach to rigidity questions for lattices in semisimple Lie groups, generalizing Mostow rigidity, has been developed by several authors, including Farb. He plans to continue his work with L. Mosher in bringing these new ideas and themes to bear on understanding quasi-isometries of broader classes of groups arising in combinatorial group theory. He will also work on characterizing a broad class of finitely generated groups by certain dynamics they exhibit. Finally, Farb will study the geometry of deformation spaces of discrete subgroups of rank-one Lie groups over local fields. Geometric group theory is a mixture of geometry, topology, algebra, and algorithms. Farb's project in this area is an attempt to understand the relationship between the symmetries exhibited by a space and its geometric structure on very large scales. ***