L2-Betti numbers, Isomorphism Conjectures and Noncommutative Localization

Abstract

Abstract In this paper we want to discuss how the question about the rationality of L 2 -Betti numbers is related to the Isomorphism Conjecture in algebraic K -theory and why in this context noncommutative localization appears as an important tool. L 2 - Betti numbers are invariants of spaces which are defined analogously to the ordinary Betti-numbers but they take information about the fundamental group into account and are a priori real valued. The Isomorphism Conjecture in algebraic K-theory predicts that K 0 (ℂΓ), the Grothendieck group of finitely generated projective ℂΓ-modules, should be computable from the K -theory of the complex group rings of finite subgroups of Γ. Given a commutative ring one can always invert the set of all non-zerodivisors. Elements in the resulting ring have a nice description in terms of fractions. For noncommutative rings like group rings this may no longer be the case and other concepts for a noncommutative localization can be more suitable for specific problems. The question whether L 2 -Betti numbers are always rational numbers was asked by Atiyah in. The question turns out to be a question about modules over the group ring of the fundamental group Γ. In Linnell was able to answer the question affirmatively if Γ belongs to a certain class of groups which contains free groups and is stable under extensions by elementary amenable groups (one also needs a bound on the orders of finite subgroups).