Discrete groups of intermediate rank

中级离散组

• 批准号: RGPIN-2019-05172
• 负责人: Pichot, Mikael
• 金额: $ 1.24万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675823
• 关键词: Discrete; groups; intermediate; rank

***The theme of this project is a recently discovered family of infinite finitely presented discrete groups called the groups of intermediate rank. These groups are at the frontier of two well---known and actively studied classes of discrete groups: hyperbolic groups, in the sense of Gromov, and lattices in Lie groups, which act on symmetric spaces of non compact type, or Euclidian Tits buildings. In this context, one often observes a sharp distinction between rank 1 and higher rank spaces; groups of intermediate rank interpolate between these two worlds.******The proposal is centered around three main themes: ergodic group theory, which studies groups through their probability measure preserving actions on standard spaces, geometric group theory, which relies more on geometric properties, such as nonpositive curvature, and operator algebras, which uses the techniques of infinite dimensional analysis. There has been a continuous and long--lasting interplay between these subjects, one of most prominent example being perhaps the Mostow----Margulis----Zimmer rigidity theory. The connections with operator algebras started with the work of von Neumann on dynamical systems and operator algebras in the 40s, and the study of unitary representations and (group) C*-algebras. The Haagerup property, for example, has played a leading role in the developments of groups of intermediate rank. Some of the most important tools, since the first discoveries on these groups around 10 years ago, have come from connections between these fields. ******The principal investigator, together with his co-authors and students, will further these connections; rank interpolation is, first and foremost, a new domain that needs to be explored and charted. Explicit open problems and new directions are detailed in the proposal. Additionally, the PI will give introductory lectures and write lecture notes aimed at advanced students. He will also work to foster the development of long term international relationships strengthening the existing ties.*****

***这个项目的主题是最近发现的无限有限呈现的离散群族，称为中间秩群。这些群处于两类众所周知且被积极研究的离散群的前沿：格罗莫夫意义上的双曲群和李群中的格，它们作用于非紧型对称空间，或欧几里得Tits建筑。在这种情况下，人们经常会注意到第1级和更高级别空间之间的明显区别；中等等级的群体在这两个世界之间插入。******该提案围绕三个主要主题：遍历群论，通过它们在标准空间上的概率度量保持作用来研究群，几何群论，更多地依赖于几何性质，如非正曲率，以及算子代数，使用无限维分析技术。这些学科之间一直存在着持续而持久的相互作用，最突出的例子之一可能是Mostow----马古利斯----齐默刚性理论。与算子代数的联系始于20世纪40年代冯·诺伊曼对动力系统和算子代数的研究，以及对酉表示和（群）C*-代数的研究。例如，哈格鲁普性质在中等等级群体的发展中起了主导作用。自从大约10年前在这些群体中首次发现以来，一些最重要的工具来自于这些领域之间的联系。******首席研究员将与他的共同作者和学生一起进一步建立这些联系；排名插值首先是一个需要探索和绘制图表的新领域。明确开放的问题和新的方向在建议中详细说明。此外，PI还将针对高级学生进行介绍讲座和撰写课堂笔记。他还将努力促进长期国际关系的发展，加强现有关系。*****