Periodic Reporting for period 5 - IGOC (Interactions between Groups, Orbits, and Cartans)
Reporting period: 2024-07-01 to 2025-02-28
The project’s overall objective is to develop interactions between several areas in mathematics (C*-algebras, topological dynamics and group theory), based on several key notions such as Cartan subalgebras or continuous orbit equivalence which have not been systematically studied before or which have been introduced only recently. A better understanding of these notions and their interplay leads to a transfer of knowledge between mathematical areas and helps to make progress on outstanding open problems in these areas.
The mathematical areas have connections to and applications in areas going beyond mathematics. For instance, operator algebras provide the mathematical foundation for quantum physics, dynamical systems lead to mathematical models for physical systems, and group theory provides the abstract framework to understand symmetry, which is a key concept in a variety of sciences.
At a more technical level, topological groupoids are the central structure which this project focuses on. All in all, we have achieved a better understanding of topological groupoids, their structure, invariants and algebras.
The mathematical areas have connections to and applications in areas going beyond mathematics. For instance, operator algebras provide the mathematical foundation for quantum physics, dynamical systems lead to mathematical models for physical systems, and group theory provides the abstract framework to understand symmetry, which is a key concept in a variety of sciences.
At a more technical level, topological groupoids are the central structure which this project focuses on. All in all, we have achieved a better understanding of topological groupoids, their structure, invariants and algebras.
The first part of the project focused on developing a better understanding for key concepts and important invariants of these. One main result was a description of spectra of Cartan subalgebras in classifiable C*-algebras. This was the first such result in this direction. For instance, it led to progress on the question of uniqueness and classification of Cartan subalgebras. As another important result, we established a K-theory formula for C*-algebras attached to combinatorial data, generalizing previous work. This is significant as K-theory is one of the most important invariants for C*-algebras. In addition, we also obtained other structural results about this class of C*-algebras, with a particular focus of distinguished quotients of them.
The second part of the project focused on developing machinery for understanding and computing important homological invariants of topological groupoids and topological full groups. The growing interest in these results and developments is evidenced by several seminar series (in Glasgow, Kyoto and Oslo) dedicated to them and the invited lecture series and talks the PI has given on the topic (for example in Oberwolfach, at ICMS, at AIM, at the Global Noncommutative Geometry Seminar, and in Binghamton, Copenhagen, Muenster, Purdue, Shanghai).
The second part of the project focused on developing machinery for understanding and computing important homological invariants of topological groupoids and topological full groups. The growing interest in these results and developments is evidenced by several seminar series (in Glasgow, Kyoto and Oslo) dedicated to them and the invited lecture series and talks the PI has given on the topic (for example in Oberwolfach, at ICMS, at AIM, at the Global Noncommutative Geometry Seminar, and in Binghamton, Copenhagen, Muenster, Purdue, Shanghai).
In addition to the main results, one new insight of our work was that Garside structures help to study C*-algebras constructed from left regular representations. These Garside structures first arose in the context of Braid groups and more general Artin-Tits groups, but it turns out that they naturally come up in the theory of C*-algebras as well, for instance in the context of graph algebras or higher rank graph C*-algebras.
Another major breakthrough was a complete characterization of C*-simplicity for etale groupoids. Previous results in the groupoid case often assumed some sort of amenability, but we succeeded in giving a completely general characterization. Interestingly, the key notions that appear are closely related to dynamical properties of the groupoids in question.
Moreover, our results on homological invariants of topological groupoids and topological full groups have made progress on our understanding of these important invariants, going beyond the state of the art. In particular, our results verified Matui’s AH conjecture for a large class of groupoids. We also expect that our work in this direction will be influential in the future when discussing other, related invariants of groupoids and their algebras, for instance K-theory.
Another major breakthrough was a complete characterization of C*-simplicity for etale groupoids. Previous results in the groupoid case often assumed some sort of amenability, but we succeeded in giving a completely general characterization. Interestingly, the key notions that appear are closely related to dynamical properties of the groupoids in question.
Moreover, our results on homological invariants of topological groupoids and topological full groups have made progress on our understanding of these important invariants, going beyond the state of the art. In particular, our results verified Matui’s AH conjecture for a large class of groupoids. We also expect that our work in this direction will be influential in the future when discussing other, related invariants of groupoids and their algebras, for instance K-theory.