Questions in and around Geometric Group Theory

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Central to this project is the question, first proposed by Gromov, of classifying finitely generated groups by their (quasi-isometric) geometry. The PI's proposal follows two principle lines of study. The first direction is to shed light on the question of quasi-isometrically classifying various families of groups. The second direction of study, involves applying knowledge gained from analysis of the quasi-isometric structure of a family of groups to illuminate questions of a different nature: for instance to embeddings of 3-dimensional manifold groups or to functional analytic properties of mapping class groups.Quasi-isometric geometry is a way of considering geometric properties of a space which are unchanged by a "small" change of the distance function. In this proposal the PI considers the geometry of various natural families of groups and studies question such as: to what extent do geometric properties determine algebraic ones. One family of groups which the PI continues to study with Walter Neumann (Barnard College) are those associated to 3-dimensional manifolds---these spaces are of crucial importance throughout low-dimensional topology and in physics as well.