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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=531015
• 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群体和几何空间，这些空间可以被认为是平面和负曲线片的(部分)复杂混合物，沿着规定的局部奇点组装在一起。这个项目是探索这些群和空间的性质，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。该提案包含一系列公开问题和实现这些目标的新想法，包括例如关于“中阶”空间、随机空间、随机群或Bruhat-Tits建筑物及其自同构群的问题。统一的主题是无限群论，从几何的观点出发，遵循格罗莫夫的观点，特别是涉及非正曲率的几何。这通常会自然而然地引入相邻的数学领域，如动力系统或算子代数，其中的几个联系也会被研究。例如，这类群在概率空间上的保测量作用所引起的许多关于动力系统的公开问题，特别是关于它们的轨道结构、它们的不变量(代价、等参常数、SOFIC维)或它们的刚性性质。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 Betti数或算子K-理论)本质上是算子代数。其他密切相关的领域包括图论和计算机科学。后者既有算法的理论部分，又有软件编程的部分。特别是，一个非常具体的输出将是关于非正曲率无限群的计算机程序，它将有助于对它们进行分类并了解它们的性质。该提案包括一系列难度不同的开放问题，从令人兴奋并提供指导的经典问题，到(想必)更包含的开放问题，这些问题可以由不同水平的学生处理。教育活动构成了这项提案的一个重要方面。致力于这些问题的学生将能够为该领域的进步做出贡献，并打开他们自己的研究视角。