Abstract: In recent years several interesting results in group theory and topology were obtained using so-called L²-invariants. In the theory of L²-invariants, methods from functional analysis are used to generalize classical homological invariants of finite simplicial complexes to infinite simplicial complexes with a cocompact proper action of a group. For example, the Betti numbers of a finite simplicial complex are the vector space dimensions of the rational homology groups of the complex. In a similar vein, L²-Betti numbers are defined as von Neumann dimensions of the so-called L²-homology groups of the complex. We will introduce the underlying tools and discuss the definition of L²-Betti numbers. In fact, we will see how a modification yields a notion of dimension (and Betti numbers) for every character of the underlying group. Some of these new Betti numbers, the L²-multiplicities, turn out to be themselves natural generalizations of classical finite dimensional invariants. In the end, we discuss an approximation theorem which relates the L²-multiplicities to their finite dimensional analogues and mention some applications.
Betti numbers, L²-multiplicities and an equivariant approximation theorem