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 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 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M. F.: Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32-33 (1976), 43-72.
Baumslag, G.: A finitely presented metabelian group with a free abelian derived group of infinite rank, Proc. Amer. Math. Soc. 35 (1972), 61-62.
Grigorchuk, R., Linnell, P., Schick, T. and Zuk, A.: On a question of Atiyah, C. R. Acad. Sci. Ser I Math. 331 (2000), 663-668, earlier version available via http://arxiv.org/abs/math.GT/0009175
Grigorchuk, R. and Zuk, A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata 87 (2001), 209-244.
Hardy, G. H. and Wright, E. M.: An Introduction to the Theory of Numbers, 5th edn, Oxford Univ. Press, 1979.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dicks, W., Schick, T. The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product. Geometriae Dedicata 93, 121–137 (2002). https://doi.org/10.1023/A:1020381532489
Issue Date:
DOI: https://doi.org/10.1023/A:1020381532489