Quasi-Isometries and Boundaries in Rigidity

准等距和刚度边界

• 批准号: 1207296
• 负责人: Tullia Dymarz
• 金额: $ 15.7万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207296&HistoricalAwards=false
• 关键词: Quasi; Isometries; Boundaries; Rigidity

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.

本研究计划涉及Gromov的计划，在大规模几何学的非线性生成的群体。PI研究的主要主题是如何在无穷远处适当定义的边界上进行分析，以回答某些群体的大规模几何问题。虽然PI计划研究双曲群的传统边界，但本提案的主要重点是可解群的边界。PI使用的主要工具之一是Eskin-Fisher-Whyte和Peng最近关于某些可解群的准等距结构的工作。通过研究这些群体的边界，PI有助于并希望扩大可解群体之间的准等距的理解，以及这些群体的格包络。更重要的是，PI已经能够为Gromov计划提出的基本问题做出贡献。特别是，PI提供了反例来证明准等距和bilipschitz等价之间的区别。这是PI计划探索的一个不太容易理解的区别。准等距提供了一个从远处观察时数学空间之间相似性的概念。一个有用的类比是想象两个由不同树木组成的森林在从卫星上看时看起来像是无法区分的绿色区域。或者，人们可以通过将观察者置于（无限）空间内并观察无限远处的“边界”来可视化空间的“大尺度结构”。这很像一个观众凝视着地平线。这两个观点及其相互作用是本建议的核心。这两个概念已经卓有成效最近在研究问题的许多领域的数学，但特别是当数学空间出现的代数对象（代数生成群）。PI的研究重点是如何限制到特定的准等距可以改变的概念之间的相似性生成的群体，以及如何比较空间的边界可以用来显示两个群体是相似的。