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.
This was a highly successful project in which a number of papers were written and important questions in the field answered. As well, this project had significant broader impact through mentoring (including a number of underrepresented minorities), conference organizing, and other activities. Two highlights which indicated the intellectual merit of the work completed during the period of this project by the PI are the following. The PI successfully gave the quasi-isometric classification of almost all fundamental groups of 3-dimensional manifolds, in joint work with Walter Neumann. The PI successfully resolved a long-standing question formuated by Gromov and Gersten concerning the divergence of non-positively curved spaces, this was done in joint work with Cornelia Drutu. This broader impact of this proposal includes the following. During this period the PI mentored a large number of undergraduate, graduate and postdoctoral students from his home institution as well as from other institutions. Among the students the PI mentored there were a number of underrepresentated minorities from the PI's home institution. Also, during the duration of this project the PI supervised a number of PhD students, two of whom have now graduated; one is currently a postdoc and the other works for the National Security Agency. The PI organized a number of conferences.
