Abstract: For a closed Riemannian manifold M with universal covering M, Atiyah introduced the L^2-Betti numbers b^(2)_p(M) which measure the size of the space of harmonic square-integrable p-forms on M. Let G denote the fundamental group of M. Atiyah asked whether the L^2 Betti are always rational, and many others have asked whether they are integers in the case G is torsion free. These questions can be phrased in a more algebraic/analytic way in terms of the left regular representation of G on L^2(G), without any reference to M. For simplicity we will concentrate on the case G is torsion free, so we want to know whether the L^2 Betti numbers are alway integers. This has been verified for free groups and elementary amenable groups (that includes solvable groups). Recent work of Dicks and Schick has under mild conditions, verified it for fundamental groups of graphs of group.
In this talk I will describe how it is proved for Braid groups. This is joint work with Thomas Schick. The proof depends on showing that Braid groups have nontrivial torsion free virtually nilpotent quotients.