Approximating L2 invariants and the Atiyah conjecture

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>G</jats:italic> be a torsion‐free discrete group, and let <jats:styled-content>ℚ</jats:styled-content> denote the field of algebraic numbers in ℂ. We prove that <jats:styled-content>ℚ</jats:styled-content><jats:italic>G</jats:italic> fulfills the Atiyah conjecture if <jats:italic>G</jats:italic> lies in a certain class of groups <jats:italic>D</jats:italic>, 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 ℂ<jats:italic>G</jats:italic>. The statement relies on new approximation results for <jats:italic>L</jats:italic><jats:sup>2</jats:sup>‐Betti numbers over <jats:styled-content>ℚ</jats:styled-content><jats:italic>G</jats:italic>, 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 <jats:italic>L</jats:italic><jats:sup>2</jats:sup>‐cochains on any normal covering space of a finite <jats:italic>CW</jats:italic> 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.</jats:p>