The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product
By Warren Dicks, Thomas Schick. 2001
75 citations recorded by the source corpus.
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.
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1d4bd9c271f8b3d15a66e3760eb0893a8beda649
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
- R. Exel
- P. Ara
- Laurent Bartholdi
- Wolfgang Woess
- Markus Neuhauser
- Ben Steinberg
- Mark V. Lawson
- Enrique Pardo
Citations in the corpus
- Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, background): “This group provided the first counter-example to the Strong Atiyah’s Conjecture, see [19], [31].”
- Horocyclic products of trees (2006, result): “This extends previous results regarding the lamplighter group and the basic Diestel-Leader graphs DL(q1, q2), see Grigorchuk and Żuk [20], Dicks and Schick [15] and Bartholdi and Woess [3].”
- Finite group extensions and the Atiyah conjecture (2004, unknown): “…. . . . . . 20 4.4 Generalized Galois cohomology of profinite groups . . . . . . . . 21 Filtration by skeleta . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.6 Spectral sequences . . . . . . . . . . . . . . . .…”
- Dead end words in lamplighter groups and other wreath products (2003, background): “We then describe an interesting phenomenon which occurs in this Cayley graph, namely the existence of dead end elements.”