Approximating L2‐invariants and the Atiyah conjecture

By J. Dodziuk, P. Linnell, Varghese Mathai, Thomas Schick, Stuart Yates. 2001

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Let G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion‐free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2‐Betti numbers over ℚG, 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 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 𝒢. © 2003 Wiley Periodicals, Inc.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Integrality of $L^2$-Betti numbers (2000, background): “ could not be ﬁxed. Consequently, in this new version everything based on this result had to be removed. ∗e-mail: schick@math-math.gwdg.de www:http://www.uni-math.gwdg.de/schickt/ 1 2 ThomasSchick In [1] Atiyah introduced L2-Betti numbers of closed manifolds in terms of the kernel of the Laplacian o”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, background): “1, we note by passing a slightly more general result from [22].”
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “A conjecture due to Dodziuk, Linnell, Mathai, Schick, and Yates known as algebraic eigenvalues conjectures (see [33]) asserts that elements x of ZΓ regarded as linear operators on l2Γ have algebraic integers as eigenvalues.”
Finite group extensions and the Atiyah conjecture (2004, background): “e for CG if G is a trivial group. 3 More positive results about the Atiyah conjecture In this section, we recall a few more known results about the Atiyah conjecture, proved in particular in [48] and [12]. 3.1 Deﬁnition. Let Dbe the smallest non-empty class of groups such that: Finite group extensio”
Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants (2010, background): “Because of Linnell’s work on the strong Atiyah conjecture [45] (see also [18, 69]), we know that the zero divisor conjecture holds for all torsion-free groups in the smallest class of groups containing all free groups and being closed under extensions with elementary amenable quotients and under dir”