L2-Betti numbers, the Atiyah conjecture and related conjectures

In 1976 M. Atiyah, in connection with his work with Singer, introduced a certain kind of cohomology in order to extend his famous Atiyah-Singer Index Theorem to the noncompact setting. This cohomology is nowadays commonly known as l2-cohomology. He also defined the generalized notion of l2-Betti numbers as von Neumann dimensions of the resulting l2-cohomology groups. In his paper Atiyah observed that in many cases these are in fact rational numbers, and indeed he asked if it is possible to obtain irrational ones. That was the beginning of what is now called the Atiyah conjecture.

In this talk I will introduce the basic notions of von Neumann dimension and l2-Betti numbers in a purely algebraic setting. I will also present some modern variants of the Atiyah conjecture, as well as known results concerning them.

Lastly, I will talk about other famous conjectures on group rings, and their relations with the previously stated Atiyah conjecture.