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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
Article
  • 185 Downloads

Abstract

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J., Samet, I.: On the growth of Betti numbers of locally symmetric spaces. C. R. Math. Acad. Sci. Paris 349(15–16), 831–835 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bergeron, N., Gaboriau, D.: Asymptotique des nombres de Betti, invariants \(l^2\) et laminations. Comment. Math. Helv. 79, 362–395 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borel, A.: The \(L^2\)-cohomology of negatively curved riemannian symmetric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 95–105 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourbaki, N.: Algebra 1. Chapters 1–3, reprint of the: English translation, p. 1998. Springer, Berlin (1989)Google Scholar
  5. 5.
    Capraro, V., Valerio, Lupini, M.: Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, with an appendix by Vladimir Pestov. Lecture Notes Math., vol. 2136. Springer, Cham (2015)Google Scholar
  6. 6.
    Dixmier, J.: Von Neumann Algebras. North-Holland Publishing Company, Amsterdam (1981)zbMATHGoogle Scholar
  7. 7.
    Dudko, A., Medynets, K.: Finite factor representations of the Higman-Thompson groups. Groups Geom. Dyn. 8, 375–389 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elek, G., Szabó, E.: On sofic groups. J. Group Theory 9, 161–171 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elek, G., Szabó, E.: Hyperlinearity, essentially free actions and \(L^2\)-invariants. The sofic property. Math. Ann. 332, 421–441 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Elstrodt, J.: Maß- und Integrationstheorie, 4th edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  11. 11.
    Farber, M.: Geometry of growth: approximation theorems for \(L^2\) invariants. Math. Ann. 311, 335–375 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Folland, G.B.: A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  13. 13.
    Gaboriau, D.: Invariants \(\ell ^2\) de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95, 93–150 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Geoghegan, R.: Topological Methods in Group Theory, Graduate Texts in Math., vol. 243. Springer, New York (2008)Google Scholar
  15. 15.
    de George, D.L., Wallach, N.R.: Limit formulas for multiplicities in \(L^2(\Gamma \backslash G)\). Ann. Math. (2) 107(1), 133–150 (1978)MathSciNetGoogle Scholar
  16. 16.
    Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1(2), 109–197 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lück, W.: Approximating \(L^2\)-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4(4), 455–481 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lück, W.: \(L^2\)-Invariants: Theory and Applications to Geometry and \(K\)-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge, vol. 44. Springer, Berlin (2002)Google Scholar
  19. 19.
    Lück, W.: Approximating \(L^2\)-invariants by their classical counterparts. EMS Surv. Math. Sci. 3(2), 269–344 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Petersen, H.D., Sauer, R., Thom, A.: \(L^2\)-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices, preprint (2016). arXiv:1612.04559
  21. 21.
    Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of charges. A Study of Finitely Additive Measures. Academic, Inc., New York (1983)Google Scholar
  22. 22.
    Rohlfs, J., Speh, B.: On limit multiplicities of representations with cohomology in the cuspidal spectrum. Duke Math. J. 55(1), 199–211 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Savin, G.: Limit multiplicities of cusp forms. Invent. Math. 95(1), 149–159 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schick, T.: \(L^2\)-determinant class and approximation of \(L^2\)-Betti numbers. Trans. Am. Math. Soc. 353, 3247–3265 (2001)CrossRefzbMATHGoogle Scholar
  25. 25.
    Thoma, E.: Über unitäre Darstellungen abzählbarer, diskreter Gruppen. Math. Ann. 153, 111–138 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes, Springer Series in Statistics. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wehrfritz, B.A.F.: Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 76. Springer, New York (1973)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany

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