Discrete groups of intermediate rank

这个项目的主题是最近发现的一族无限有限呈现的离散群，称为中等秩群。这些群处于两类众所周知的和被积极研究的离散群的前沿：格罗莫夫意义下的双曲群，以及作用于非紧型对称空间的李群中的格子，或欧几里德的Tits建筑。在这种情况下，人们经常观察到等级1和更高等级空间之间的明显区别；中间等级的组在这两个世界之间插入。 该建议围绕三个主要主题：遍历群论，它通过群在标准空间上的概率保持度量作用来研究群；几何群论，它更多地依赖于几何性质，如非正曲率；以及算子代数，它使用无限维分析技术。这些学科之间存在着持续和持久的相互作用，最突出的例子之一可能是莫斯托-齐默尔刚性理论。与算子代数的联系始于40年代von Neumann关于动力系统和算子代数的工作，以及对酉表示和(群)C*-代数的研究。例如，Haagerup地产在中级群体的发展中发挥了主导作用。自从大约10年前在这些群体上首次发现以来，一些最重要的工具来自这些领域之间的联系。 首席研究员与他的合著者和学生将进一步加强这些联系；秩插值法首先是一个需要探索和绘制的新领域。提案中详细说明了明确的开放问题和新的方向。此外，PI将针对高级学生进行介绍性讲座和撰写课堂讲稿。他还将致力于促进长期国际关系的发展，加强现有的联系。

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.