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Characters, $L^2$-Betti numbers and an equivariant approximation theorem

Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results are approximation theorems for $L^2$-multiplicities which extend the approximation theorems for $L^2$-Betti numbers of Lück, Farber and Elek-Szabó respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.