Rigidity in negative curvature and quasiconformal analysis

PI建议通过对度量空间的拟共形分析来研究负弯曲空间的大尺度几何。 拟研究的内容包括可解群的大规模几何、相对双曲群边界上的度量结构和Hadamard流形的拟等距刚性。 可解群的研究涉及到拟共形分析在幂零李群上的研究。尽管幂零李群上的度量不是黎曼度量，但它们是左不变的，并且允许单参数伸缩族。 它们包括卡诺度量和许多其他度量。 用拟共形分析的语言表述，其中一些目标是：将这些幂零李群分类到拟共形等价;证明（在大多数情况下）每个拟共形映射都是双Lipschitz。 在大规模几何方面，目标是证明（大多数）负弯曲可解李群之间的拟等距保持距离直到一个加性常数，有时是 即使在距离等距线有限的地方。 如果成功，这项研究将导致大规模几何学的进展， 可解群的生成。拟对相对双曲群进行的研究将试图确定在相对双曲群的边界上是否存在典范拟共形结构。 %和 几何有限群的极限集它可以应用于这些群体的刚性问题。 关于Hadamard流形的拟议研究涉及的问题是，Hadamard流形之间的每个拟等距是否都离biLipschitz同胚有有限距离。关于负弯曲可解李群和阿达玛流形的建议项目是PI以前在这些主题上工作的延续。这些建议的共同主题是 项目是负弯曲空间的几何学和这些空间的理想边界的分析之间的相互作用。拟议的研究在于几何群论和几何分析。 几何群论中的问题通常涉及空间或映射的大尺度性质，而在分析中，主要焦点通常是局部性质。令人惊讶的是，这些都与负曲率空间有关：负曲率空间有一个理想边界，空间的大尺度性质被编码在理想边界的局部结构中。PI建议通过使用几何分析研究理想边界来研究负弯曲空间的大尺度性质。

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.