Rigidity in negative curvature and quasiconformal analysis

本文提出利用度量空间的拟共形分析来研究负弯曲空间的大尺度几何。提出的研究内容包括可解群的大尺度几何、相对双曲群边界上的度量结构以及Hadamard流形的拟等距刚性。可解群的研究涉及幂零李群的拟共形分析。虽然幂零李群上的度量不是黎曼的，但它们是左不变的，并且承认一个单参数的扩张族。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，一些目标是：将这些幂零李群分类到拟共形等价；来证明（在大多数情况下）每个拟共形映射都是biLipschitz。就大尺度几何而言，目标是证明（大多数）负弯曲可解李群之间的准等距保持到一个加性常数，有时甚至与等距保持有限距离。如果成功，这项研究将导致有限生成可解群的大规模几何的进展。关于相对双曲群的研究将试图确定在相对双曲群的边界上是否存在正则拟共形结构。%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。关于阿达玛流形的研究涉及到阿达玛流形之间的每一个拟等距是否与biLipschitz同纯存在有限距离的问题。关于负弯曲可解李群和Hadamard流形的项目是PI先前在这些主题上的工作的延续。这些拟议项目的一个共同主题是负弯曲空间的几何形状和对这些空间的理想边界的分析之间的相互作用。本文主要研究几何群论和几何分析。几何群论中的问题往往涉及空间或映射的大尺度性质，而在几何群论分析中，主要关注的往往是局部性质。令人惊讶的是，这些与负曲率的空间有关：负弯曲的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。该方法提出利用几何分析方法研究负弯曲空间的理想边界，从而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.