IGOC · Interactions between Groups, Orbits, and Cartans

Recently, we discovered that the notion of Cartan subalgebras builds bridges between C*-algebras, topological dynamics, and geometric group theory. The goal of this research project is to develop our understanding of this concept in order to attack the following major open questions:I. The UCT questionII. The Baum-Connes conjectureIII. The conjugacy problem for topological shiftsIV. Quasi-isometry rigidity for polycyclic groupsUCT stands for Universal Coefficient Theorem and is a crucial ingredient in classification. I want to make progress on the open question whether sufficiently regular C*-algebras satisfy the UCT, taking my joint work with Barlak as a starting point.The Baum-Connes conjecture predicts a K-theory formula for group C*-algebras which has far-reaching applications in geometry and algebra as it implies open conjectures of Novikov and Kaplansky. My new approach to II will be based on Cartan subalgebras and the notion of independent resolutions due to Norling and myself.Problem III asks for algorithms deciding which shifts are topologically conjugate. It has driven a lot of research in symbolic dynamics.Conjecture IV asserts that every group quasi-isometric to a polycyclic group must already be virtually polycyclic. A solution would be a milestone in our understanding of solvable Lie groups.To attack III and IV, I want to develop the new notion of continuous orbit equivalence which (as I recently showed) is closely related to Cartan subalgebras.Problems I to IV address important challenges, so that any progress will result in a major breakthrough. On top of that, my project will initiate new interactions between several mathematical areas. It is exactly the right time to develop the proposed research programme as it takes up recent breakthroughs in classification of C*-algebras, orbit equivalence for Cantor minimal systems, and measured group theory, where measure-theoretic analogues of our key concepts have been highly successful.