# $L^2$-Betti numbers and Riemannian volume

## Sabine Braun

created by collari on 09 Jun 2017

15 jun 2017 -- 15:00

Sala Riunioni, Dipartimento di Matematica, Pisa

Seminari dei Baby-Geometri

Abstract.

Gromov raised the question whether there is a universal bound for the $L^2$-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 of $L^2$-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 for $L^2$-Betti numbers.

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