Topics at the Intersection of Geometry, Topology and Group Theory

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ABSTRACT FOR NSF PROPOSAL 0604633There are three main components to this project. First, Farb will continue to investigate mapping class groups and the moduli space of Riemann surfaces. This topic lies at the intersection of many areas of mathematics, from algebraic geometry to low-dimensional topology to string theory to geometric group theory. Farb will continue to apply methods from discrete subgroups of Lie groups in order to understand these objects, especially the "Torelli group", which is a part of the oldest but least understood part of the theory. Symmetry is a core idea in mathematics. Farb will continue his work with S. Weinberger on the broad program of classifying all spaces (that is Riemannian manifolds) with symmetry. The ideas used so far in this work have included the theories of harmonic maps, large-scale geometry, and transformation groups. In a third project, Farb will continue his work with C. Hruska on bringing together techniques and ideas from geometric group theory with those from discrete subgroups of Lie groups in order to build the theory of lattices in automorphism groups of 2-complexes. This is a 2-dimensional extension of Bass-Lubotzky's theory of tree lattices, where wild new phenomena can occur. Throughout each of the projects just described, Farb will continue to work with and mentor many young students and researchers.