Dicks, W. and Schick, T.: The spectral measure of certain elements of the complex group
ring of a wreath
product
Abstract:
We use elementary methods to compute the $L^2$-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 $L^2$-Betti numbers.
We use our results to construct manifolds with certain $L^2$-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.
Thomas Schick