Mathematische Annalen

, Volume 371, Issue 1–2, pp 405–444 | Cite as

Characters, \(L^2\)-Betti numbers and an equivariant approximation theorem

  • Steffen KionkeEmail author


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.

Mathematics Subject Classification

Primary 55N10 Secondary 20J06 57S30 11F75 



The research described in this article was conducted at the Hausdorff Research Institute for Mathematics in Bonn during the trimester program “Topology” in 2016. I would like to thank the institute for the hospitality and support. Moreover, I would like to thank the members of the research group \(L^2\)-Invariants, namely Gerrit Hermann, Holger Kammeyer, Aditi Kar and Jean Raimbault for the stimulating discussions during the program.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany

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