Discrete groups of intermediate rank

这个项目的主题是最近发现的一个家庭的无限的离散群的提出称为集团的中间排名。这些群体是在前沿的两个众所周知的和积极研究类离散群体：双曲群，在意义上的格罗莫夫，和格在李群，它的作用对称空间的非紧凑型，或欧几里德山雀建筑物。在这种情况下，人们经常观察到秩1和更高秩空间之间的明显区别;中间秩的群在这两个世界之间插值。该提案围绕三个主题：遍历群论，它通过标准空间上的概率测度保持作用来研究群，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，它使用无限维分析的技术。这些主题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-Margulis-Zimmer刚性理论。与算子代数的联系始于冯·诺依曼在40年代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，Haagerup财产在中等级别集团的发展中发挥了主导作用。自大约10年前首次发现这些群体以来，一些最重要的工具来自这些领域之间的联系。 首席研究员，连同他的合著者和学生，将进一步这些连接;排名插值是，首先，一个新的领域，需要探索和绘制。明确的开放问题和新的方向在提案中详细说明。此外，PI将提供介绍性讲座，并针对先进的学生写讲义。他还将致力于促进长期国际关系的发展，加强现有的关系。

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.