Dodziuk, J., Linnell, P., Mathai, V., Schick, T. and Yates, S.: Approximating L2-invariants, and the Atiyah conjecture
Abstract:
Let G be a torsion free discrete group and let
\overline{Q} denote the
field of algebraic numbers in C. We prove that
\overline{Q}[G] fulfills the Atiyah conjecture if G lies in
a certain class of groups D, which contains in
particular all groups which are
residually
torsion free elementary amenable or which are residually free.
This result implies that there are no non-trivial zero-divisors
in C[G].
The statement relies on new approximation results for L2-Betti numbers over
\overline{Q}[G], which are the core of the work done in this
paper.
Another set of results in the paper is concerned with certain number
theoretic properties of eigenvalues for the combinatorial Laplacian
on L2-cochains on any normal covering space of a finite CW complex.
We establish the absence of eigenvalues that are transcendental numbers,
whenever the covering transformation group is either amenable or in the
Linnell class \mathcal{C}. We also establish the absence of eigenvalues
that are Liouville transcendental numbers whenever the covering transformation
group is either residually finite or more generally in
a certain large bootstrap class \mathcal{G}.
MSC: 55N25 (homology with local coefficients), 16S34 (group rings,
Laurent rings), 46L50
(non-commutative measure theory)
Thomas Schick