Project B7: Measurable group theory and L2-invariants
SummaryMeasurable group theory and L2-invariants have been rapidly evolving fields in recent years. L2-Betti numbers of groups are useful invariants in measurable group theory; they only depend on the orbit structure of a free measure-preserving action of the group on a probability space. We seek to exploit this fact to address geometric conjectures about L2-Betti numbers of aspherical manifolds. Further, rigidity questions in measurable group theory will be investigated. Finally, we aim to achieve progress on the Atiyah conjecture and on L2-invariants of foliations suggested by number theory.
Originally, L2-invariants were defined for a compact Riemannian manifold in terms of the heat kernel of its universal covering; examples are L2-Betti numbers and L2-torsion, which are topological invariants. The concept of L2-Betti numbers was continually extended leading to an algebraic definition of L2-Betti numbers for arbitrary group actions on topological spaces. L2-Invariants have connections and applications to problems arising in topology, K-theory, group theory, ergodic theory, von Neumann algebras, and the spectral theory of the Laplace operator. Outstanding conjectures in the field are the Atiyah Conjecture, the Singer Conjecture, and the conjecture relating the simplicial volume to L2-invariants. A better understanding of the meaning of these conjectures and further progress towards their proofs in certain cases is one of the main goals of this project.
A new direction in L2-invariants that gained prominence in recent years is their application to questions in measurable group theory and II1-factors. Conversely, techniques from measurable group theory surprisingly allowed to prove deep results about L2-invariants. Another promising approach specifically to the Atiyah Conjecture is provided by the recent progress on the K-theoretic isomorphism conjecture.
In another direction, within a dictionary between arithmetic geometry and foliated dynamical systems new problems on R-actions and L2-invariants for foliations suggested by number theory should be investigated.
The project is subdivided into the following subprojects:
- Atiyah Conjecture
- Singer Conjecture
- Asymptotic invariants and L2-invariants
- Spectral density, random walks, and isoperimetric profile
- Measure equivalence rigidity
- Dynamical systems on foliated spaces