# Project B7: Measurable group theory and L^{2}-invariants

## Principal investigator

## Participating scientist

## Summary

Measurable group theory and L^{2}-invariants have been rapidly evolving fields in recent years. L

^{2}-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 L

^{2}-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 L

^{2}-invariants of foliations suggested by number theory.

Originally, L

^{2}-invariants were defined for a compact Riemannian manifold in terms of the heat kernel of its universal covering; examples are L

^{2}-Betti numbers and L

^{2}-torsion, which are topological invariants. The concept of L

^{2}-Betti numbers was continually extended leading to an algebraic definition of L

^{2}-Betti numbers for arbitrary group actions on topological spaces. L

^{2}-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 L

^{2}-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 L

^{2}-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 L

^{2}-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 L

^{2}-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 L
^{2}-invariants - Spectral density, random walks, and isoperimetric profile
- Measure equivalence rigidity
- Dynamical systems on foliated spaces
- Miscellaneous