Geometriae Dedicata

, Volume 93, Issue 1, pp 121–137

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

  • Warren Dicks
  • Thomas Schick
Article

DOI: 10.1023/A:1020381532489

Cite this article as:
Dicks, W. & Schick, T. Geometriae Dedicata (2002) 93: 121. doi:10.1023/A:1020381532489

Abstract

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L2-Betti numbers. We use our results to construct manifolds with certain L2-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.

Markov operator lamplighter group spectral measure Atiyah conjecture wreath product 
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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Warren Dicks
    • 1
  • Thomas Schick
    • 2
  1. 1.Departament de MatemátiquesUniversitat Autónoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.FB MathematikUniversitat GöttingenGöttingenGermany

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