-Betti Numbers of Locally Compact Groups

We introduce a notion of L-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L-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 elds, and Kac-Moody lattices. We also extend the vanishing of reduced L-cohomology for countable amenable groups, a well known theorem due to Cheeger and Gromov, to cover all amenable, second countable, unimodular locally compact groups. Resumé Vi introducerer L-Betti tal for lokalkompakte, anden tællelige, unimodulære grupper. Disses relation med L-Betti tal for gitre i lokalkompakte grupper undersøges. Således fås ere nye udregninger for tællelige grupper, for eksempel for gitre i algebraiske grupper over lokale legemer, og Kac-Moody gitre. Vi viser også at den reducerede L-kohomologi forsvinder for alle amenable anden tællelige, unimodulære lokalkompakte grupper. Dette udvider et velkendt resultat af Cheeger og Gromov for tællelige grupper.