##### Department of Mathematics,

University of California San Diego

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### Analysis Seminar

## Stefaan Vaes

#### K.U. Leuven

## $L^2$-Betti numbers of locally compact groups and their cross section equivalence relations.

##### Abstract:

$L^2$-Betti numbers were defined by Atiyah for compact manifolds X, by Cheeger and Gromov for countable groups $\Gamma$ and by Gaboriau for countable probability measure preserving equivalence relations. Henrik D. Petersen proposed a definition of $L^2$-Betti numbers for locally compact groups. I will present this definition and a recent joint work with Kyed and Petersen in which we prove that the $L^2$-Betti numbers of a locally compact group coincide with those of any associated cross section equivalence relation. As a consequence, we obtain several vanishing theorems for the reduced $L^2$-cohomology of locally compact groups.

Adrian Ioana

### March 6, 2013

### 12:00 PM

### AP&M 5829

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