Quasi-Isometries and Boundaries in Rigidity

这一研究方案涉及Gromov关于有限生成群的大规模几何的计划。PI的研究中提出的主要主题是，如何使用对无穷大处适当定义的边界的分析来回答有关某些群体的大规模几何图形的问题。虽然PI计划研究双曲群的传统边界，但这项建议的主要重点是可解群的边界。PI使用的主要工具之一是Ekin-Fisher-Whyte和Peng最近关于某些可解群的拟等距结构的工作。通过研究这些群的边界，PI贡献并希望扩大对可解群之间的准等距以及这些群的格包络的理解。更重要的是，国际和平研究所能够对格罗莫夫计划提出的基本问题作出贡献。特别是，PI提供了反例来证明准等距等价和bilipschitz等价之间的区别。这是PI计划探索的一个鲜为人知的区别。准等距提供了一种从远处观察数学空间之间的相似性的概念。一个有用的类比是，想象一下，从卫星上看，由不同树木组成的两片森林看起来像是难以区分的绿色区域。或者，人们可以通过将观察者放置在(无限的)空间内并查看无穷远处的“边界”来可视化空间的“大尺度结构”。这就像一个观察者凝视着地平线。正是这两种观点及其相互作用是这项提议的核心。这两个概念最近在许多数学领域的问题研究中都取得了丰硕的成果，特别是当数学空间由代数对象(有限生成群)产生时。PI的研究重点是限制到特定的准等距线如何改变有限生成群之间的相似性概念，以及如何使用空间的比较边界来表明两个群是相似的。

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.