The PI proposes to investigate the large scale geometry of negatively curved spaces by using quasiconformal analysis on metric spaces. The proposed research includes large scale geometry of solvable groups, metric structure on the boundary of relatively hyperbolic groups, and quasiisometric rigidity of Hadamard manifolds. The proposed research on solvable groups involves the study of quasiconformal analysis on nilpotent Lie groups. Although the metrics on the nilpotent Lie groups are not Riemannian, they are left invariant and admit a one-parameter family of dilations. They include Carnot metrics and also many other metrics. Stated in the language of quasiconformal analysis, some of the goals are: to classify these nilpotent Lie groups up to quasiconformal equivalence; to show that (in most cases) every quasiconformal map is biLipschitz. In terms of large scale geometry, the goal is to show that quasiisometries between (most) negatively curved solvable Lie groups preserve distance up to an additive constant and sometimes are even at a finite distance from isometries. If successful, this research will lead to progress on the large scale geometry of finitely generated solvable groups. The proposed research on relatively hyperbolic groups will try to determine if there is a canonical quasiconformal structure on the boundary of relatively hyperbolic groups. % and the limit sets of geometrically finite groups. It would have applications to rigidity questions about these groups. The proposed research about Hadamard manifolds concerns the question whether every quasiisometry between Hadamard manifolds is at a finite distance from a biLipschitz homeomorphism. The proposed projects on negatively curved solvable Lie groups and Hadamard manifolds are continuation of the PI's previous work on these topics. A common theme of these proposed projects is the interplay between geometry of negatively curved spaces and analysis on the ideal boundary of these spaces.
The proposed research lies in geometric group theory and geometric analysis. The questions in geometric group theory often concern the large scale properties of the spaces or maps, while in analysis the main focus is often the local properties. Surprisingly, these are related for spaces with negative curvature: a negatively curved space has an ideal boundary, and the large scale properties of the spaces are encoded in the local structure of the ideal boundary. The PI proposes to study the large scale properties of negatively curved spaces by investigating the ideal boundary using geometric analysis.