L²-Betti Numbers of Locally Compact Groups

We introduce a notion of $L^2$-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of $L^2$-Betti numbers of countable discrete groups for lattices. In this way, several new computations are obtained for countable groups, including lattices in algebraic groups over local fields, and Kac-Moody lattices. We also extend the vanishing of reduced $L^2$-cohomology for countable amenable groups, a well known theorem due to Cheeger and Gromov, to cover all amenable, second countable, unimodular locally compact groups.