Infinite groups of nonpositive curvature

无限非正曲率群

• 批准号: 418144-2012
• 负责人: Pichot, Mikael
• 金额: $ 1.46万
• 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: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508727
• 关键词: Infinite; groups; nonpositive; curvature

The research program described in the proposal concerns groups and geometric spaces that can be thought of as intricate mixtures of (portions of) flat planes and negatively curved pieces, assembled together along prescribed local singularities. The project is to explore the properties of these groups and spaces, classify them, provide new constructions, and study invariants associated with them. The proposal contains a list of open problems and new ideas to achieve these goals, including for example on spaces of "intermediate rank", random spaces, random groups, or on Bruhat-Tits buildings and their automorphism groups. The unifying theme is the theory of infinite groups, scrutinized from a geometric standpoint, following Gromov, and especially involving geometry of nonpositive curvature. This often naturally brings in adjacent domains of mathematics, like dynamical systems or operator algebras, and several of these connections are investigated too. For example, many open problems on dynamical systems arising from measure preserving actions of these groups on probability spaces are described in the proposal, especially concerning their orbit structures, their invariants (cost, isomperimetric constant, sofic dimension), or their rigidity properties. The theory of operator algebras will play a important role. We will be studying invariants of infinite groups and dynamical systems, and many of them (e.g. L2 Betti numbers or operator K-theory) are operator algebraic in nature. Other closely related areas include graph theory and computer science. The latter has both a theoretic component in algorithmic and a software programming component. In particular, a very tangible output will be a computer program on infinite groups of nonpositive curvature that will help classifying them and understanding their properties. The proposal includes a range of open problems with various degrees of difficulty, from classical problems that are exciting and provide guidance to (presumably) more contained open problems that can be approached by students at various level. Educational activities form an essential aspect of this proposal. Students working on these questions will be able to contribute to progress in the field and open their own research perspectives.

提案中描述的研究计划涉及的群体和几何空间可以被认为是平面（的一部分）和负弯曲件的复杂混合物，沿着沿着规定的局部奇点组装在一起。该项目旨在探索这些群体和空间的属性，对它们进行分类，提供新的构造，并研究与它们相关的不变量。该提案包含了一系列的开放问题和新的想法，以实现这些目标，包括例如空间的“中间等级”，随机空间，随机组，或对布鲁哈特-山茨建筑物及其自同构群。统一的主题是理论的无限群，审查从几何的立场，以下格罗莫夫，特别是涉及几何的非正曲率。这通常会自然地引入相邻的数学领域，如动力系统或算子代数，并且也研究了其中的几个连接。例如，许多开放的问题所产生的动力系统的措施保持行动，这些群体的概率空间中的建议，特别是有关他们的轨道结构，他们的不变量（成本，isomperimetric常数，sofic维数），或他们的刚性属性。算子代数理论将发挥重要作用。我们将学习无限群和动力系统的不变量，其中许多（例如L2贝蒂数或算子K理论）本质上是算子代数的。 其他密切相关的领域包括图论和计算机科学。后者既有理论部分的算法和软件编程组件。特别是，一个非常有形的输出将是一个关于非正曲率的无限群的计算机程序，这将有助于对它们进行分类并理解它们的性质。该提案包括一系列具有不同难度的开放性问题，从令人兴奋并提供指导的经典问题到（可能）更多包含的开放性问题，这些问题可以由不同级别的学生解决。教育活动是这项建议的一个重要方面。研究这些问题的学生将能够为该领域的进步做出贡献，并打开自己的研究视角。