The spectral measure of certain elements of the complex group ring of a wreath product

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-Zuk. The latter result was used by Grigorchuk-Linnell-Schick-Zuk 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.

14 pages, AMS-LaTeX

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