Research in Geometry and Topology

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Most of the proposal is dedicated to the large scale geometry of the outer automorphism group of a free group of finite rank and of the associated Outer space. Mapping class group theory developed in the last 15 years serves as a guide and much of this approach has already been implemented in the last 2-3 years, but the hard questions remain, mostly due to the fact that there aren't "enough" projections analogous to subsurface projections. Understanding this could lead to quasi-isometric rigidity of automorphism groups of free groups. The proposal concludes with several smaller sections involving specific questions about the second homology of Torelli groups, homology growth of pseudo-Anosov homeomorphisms in finite covers, and the geometry of quadratic forms as seen from the well rounded retract.The subject of the proposal is the study of topological and geometric properties of several objects that naturally appear in mathematics. For example, consider the collection of all possible finite graphs with a fixed number of cycles. This collection forms a space, called Outer space. There is a natural geometry associated with it, whose intrinsic feature is that it is not symmetric (not unlike in a city with one-way streets, where it might be quicker to get from A to B than from B to A). Nevertheless, this geometry displays hyperbolic features as well. In hyperbolic geometry, if one considers shortest paths joining each pair of three points A,B,C there is always a "point of congestion" T with all three of these paths coming close to T. A part of the proposal is to explore this phenomenon further.