Sala Riunioni (Dip. Matematica).
Gromov raised the question whether there is a universal bound for theL^2-Betti numbers of an aspherical manifold by its simplicial volume. A positive answer would yield, in combination with Gromov’s main inequality, an upper bound ofL^2-Betti numbers of an aspherical manifold by its Riemannian volume provided a lower Ricci curvature bound. While the above conjecture remains open, the implication was shown by Sauer using so-called randomization techniques. After a short introduction to L^2-Betti numbers, I will address the randomization techniques and new developments around curvature-free versions of the main inequality forL^2-Betti numbers.