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 infinitedimensional. 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 higherdimensional graphs. Hence tools including groupoid homology and the Ktheory 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.
