Studying generalised Thompson's group with tools from geometric group theory and operator algebra

使用几何群论和算子代数的工具研究广义汤普森群

• 批准号: EP/W007371/1
• 负责人: Brita Nucinkis
• 金额: $ 10.14万
• 依托单位: Royal Holloway University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW007371%2F1
• 关键词: Studying; generalised; Thompson; group; tools

Many mathematical concepts first arose in descriptions of physical systems, and later took on a life of their own after they proved to be deep and relevant for other areas of mathematics. A remarkable insight of Heisenberg in 1925 suggested that the observables of a quantum system could be realised as infinite matrices satisfying certain commutation relations. This did not really make sense at the the time, but mathematicians quickly developed the necessary tools, and now we know that he was really talking about operators, which are linear transformations on a vector space, and his insight was that this vector space had to be infinite-dimensional. Thus mathematicians were led to the study of operator algebras, which is now a vast area of mathematics that influences many other areas of mathematics, such as group theory, ergodic theory, dynamics, geometric topology, differential topology, noncommutative geometry, logic and set theory, and number theory.We shall study this connection with group theory from a group theorist's point of view: a group is a mathematician's tool to capture the notion of symmetry in the abstract. The study of symmetry provides a powerful guiding principle in a wide varietyof research problems not only in operator algebra, but in many areas of mathematics and the sciences. For that reason applications of groups abound in these fields.The study of examples is essential to the general understanding of the theory. One class of examples in particular are R. Thompson's groups F,T and V and their generalisations, which exhibit some very surprising properties, and, for the past 50 years, have been studied extensively in a wide variety of mathematical subjects: homotopy theory, dynamical systems, infinite simple groups, the word problem, group cohomology, logic and analysis. For instance, Thompson's groups provided the first known examples of infinite, finitely presented simple groups. Since then, Thompson's groups and their various generalisations have generated a large body of research trying to understand their properties, some of which are not completely settled.One powerful approach to generalised Thompson's groups is their description as automorphism groups of certain Cantor algebras; under some mild conditions one can use this viewpoint to apply discrete Morse theory to determine cohomological finiteness properties of these groups. On the other hand, many of the generalised Thompson's groups can be viewed as topological full groups of a Cuntz algebra. This was recently generalised to include groups that are obtained from higher-dimensional graphs. Hence tools including groupoid homology and the K-theory of the groupoid C*-algebra have become available.The purpose of this project is to develop a comprehensive dictionary between the two approaches to be able to answer open questions arising in both fields. For example, we expect to apply Morse theoretic methods to the groups arising from higher rank graphs to determine their cohomological finiteness conditions. On the other hand, tools like groupoid homology promise to be helpful when distinguishing isomorphism types of automorphism groups of Cantor algebras. This project is a feasibility study designed to not only answer questions such as these but also to to develop a far reaching programme for tackling other involved problems from either area.

许多数学概念最初出现在对物理系统的描述中，后来在它们被证明对数学的其他领域具有深刻意义和相关性之后，它们才有了自己的生命。海森伯在1925年提出了一个惊人的见解，即量子系统的可观测量可以实现为满足某些对易关系的无限矩阵。这在当时并没有什么意义，但数学家们很快就开发出了必要的工具，现在我们知道他真正谈论的是算子，这是向量空间上的线性变换，他的见解是这个向量空间必须是无限维的。因此，数学家们导致研究算子代数，这是一个巨大的数学领域，影响了许多其他领域的数学，如群论，遍历理论，动力学，几何拓扑，微分拓扑，非交换几何，逻辑和集合论，数论。我们将研究这种联系与群论从一组理论家的观点：群是数学家用来抽象地描述对称性的工具。对称性的研究为各种各样的研究问题提供了强有力的指导原则，不仅在算子代数中，而且在数学和科学的许多领域。由于这个原因，群在这些领域中有大量的应用，研究例子对于一般理解理论是必不可少的。其中一类例子是R。汤普森的群F，T和V和他们的概括，其中表现出一些非常令人惊讶的性质，并在过去的50年里，已被广泛研究各种各样的数学科目：同伦理论，动力系统，无限简单的群体，字的问题，组上同调，逻辑和分析。例如，汤普森的群提供了已知的第一个无限的例子，并提出了简单的群。从那时起，汤普森的群体和他们的各种推广产生了大量的研究试图了解他们的性质，其中一些是不完全解决。一个强大的方法广义汤普森的群体是他们的描述为自同构群的某些康托代数;在一些温和的条件下，人们可以使用这种观点适用于离散莫尔斯理论，以确定上同调有限性质的这些团体。另一方面，许多广义汤普森群可以被看作是一个Cuntz代数的拓扑全群。这是最近推广到包括从高维图获得的群体。因此，工具，包括广群同源和K-理论的广群C*-代数已成为可用。本项目的目的是开发一个全面的字典之间的两种方法，能够回答开放的问题，在这两个领域。例如，我们期望将莫尔斯理论方法应用于由高秩图产生的群，以确定它们的上同调有限性条件。另一方面，广群同调等工具在区分康托代数的自同构群的同构类型时是有帮助的。该项目是一项可行性研究，不仅旨在回答这些问题，而且还旨在制定一项影响深远的方案，以解决这两个领域的其他相关问题。