Premium
Approximating L 2 ‐invariants and the Atiyah conjecture
Author(s) -
Dodziuk Józef,
Linnell Peter,
Mathai Varghese,
Schick Thomas,
Yates Stuart
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10076
Subject(s) - mathematics , conjecture , betti number , transcendental number , torsion (gastropod) , algebraic number , combinatorics , eigenvalues and eigenvectors , discrete group , group (periodic table) , pure mathematics , discrete mathematics , mathematical analysis , medicine , chemistry , physics , surgery , organic chemistry , quantum mechanics
Let G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚ G 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 ℂ G . The statement relies on new approximation results for L 2 ‐Betti numbers over 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 L 2 ‐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 . 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 . © 2003 Wiley Periodicals, Inc.