This research proposal concerns Gromov's program on the large scale geometry of finitely generated groups. The main theme present in the PI's research is how analysis on a suitably defined boundary at infinity can be used to answer questions about the large scale geometry of certain groups. While the PI plans to study traditional boundaries of hyperbolic groups the main focus of this proposal is on boundaries of solvable groups. One of the main tool used by the PI is recent work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable groups. By studying boundaries of these groups the PI has contributed to and hopes to expand the understanding of quasi-isometries between solvable groups as well as lattice envelopes of these groups. More importantly the PI has been able to contribute to basic questions raised by the Gromov program. In particular, the PI has providing counter examples to demonstrate the distinction between quasi-isometric and bilipschitz equivalence. This is a little understood distinction that the PI plans to explore.
Quasi-isometries provide a notion of similarity between mathematical spaces when viewed from a far away distance. A useful analogy is to imagine that two forests composed of different trees will look like indistinguishable green regions when viewed from a satellite. Alternatively, one can visualize the `large scale structure' of a space by placing the viewer inside the (infinite) space and viewing the 'boundary' at infinity. This is much like a viewer gazing out to the horizon. It is these two perspectives and their interplay that are at the core of this proposal. Both notions have been fruitful recently in the study of problems in many areas of mathematics but particularly when the mathematical spaces arise from algebraic objects (finitely generated groups). The PI's research focuses on how restricting to specific quasi-isometries can change the notion of similarity between finitely generated groups as well as how comparing boundaries of spaces can be used to show two groups are similar.
