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

We use elementary methods to compute the L-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 [4]. The latter result was used by Grigorchuk-Linnell-Schick-Zuk [3] to produce a counterexample to a strong version of the Atiyah conjecture about the range of L-Betti numbers. We use our results to construct manifolds with certain L-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. 1 Notation and statement of main result In this section we introduce notation that will be fixed throughout and will be used in the statement of the main result. Let U denote a discrete group with torsion. Let e be a nontrivial projection (so e = e = e, e 6= 0, 1) in C[U ]. For example, U could be finite and nontrivial, and e could be the ‘average’ of the elements of U , avg(U) := 1 |U | ∑ u∈U u. e-mail: [email protected] www: http://mat.uab.es/dicks/ Fax: ++34 -93/581 2790 Research funded by the DGI (Spain) through Grant BFM2000-0354. e-mail: [email protected] www: http://wwwmath.uni-muenster.de/u/lueck/ Fax: ++49 -251/83 38370