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

许多数学概念最初出现在对物理系统的描述中，后来在它们被证明是深刻的并与数学的其他领域相关后，又有了自己的生命。1925年海森堡的一项杰出的洞察表明，量子系统的可观测性可以实现为满足某些对易关系的无限矩阵。这在当时并不是很有意义，但数学家们很快开发出了必要的工具，现在我们知道他真正谈论的是运算符，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，数学家们开始研究算符代数，它现在是一个广泛的数学领域，影响了许多其他数学领域，如群论、遍历论、动力学、几何拓扑学、微分拓扑学、非交换几何、逻辑和集合论以及数论。我们将从群论家的角度研究这种联系与群论：群是数学家从抽象的角度捕捉对称性概念的工具。对称性的研究为各种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中。由于这个原因，群在这些领域中的应用很多。对例子的研究对于理论的一般理解是必不可少的。一类特别的例子是R.Thompson的群F，T和V及其推广，它们展示了一些非常令人惊讶的性质，并且在过去的50年里，在各种各样的数学学科中得到了广泛的研究：同伦理论，动力系统，无限单群，字问题，群上同调，逻辑和分析。例如，Thompson的群提供了第一个已知的无限、有限表示的单群的例子。从那时起，Thompson群和它们的各种推广产生了大量的研究，试图理解它们的性质，其中一些还没有完全确定。推广Thompson群的一个强有力的方法是将它们描述为某些Cantor代数的自同构群；在一些温和的条件下，人们可以使用这个观点来应用离散Morse理论来确定这些群的上同调有限性质。另一方面，许多推广的Thompson群可以看作是Cuntz代数的拓扑全群。这最近被推广到包括从更高维图获得的群。因此，包括群胚同调和群胚C*-代数的K理论在内的工具成为可能。本项目的目的是在这两种方法之间开发一本全面的词典，以便能够回答这两个领域中出现的公开问题。例如，我们期望将Morse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条件。另一方面，像群同调这样的工具在区分Cantor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，其目的不仅是为了回答这些问题，而且是为了制定一项影响深远的方案，以解决这两个领域中涉及的其他问题。

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.