K-theory and dynamics. I
type: Item
instance of: scholarly article
title: K-theory and dynamics. I
zbMATH Open document ID: 0653.58035
DOI: 10.2307/2007092
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: We say that the group \(\Gamma\) is K-flat if for all free abelian groups \(C^ n\) of rank n \(Wh(\Gamma \times C^ n)=0\), where Wh(G) denotes the Whitehead group of a group G. The authors prove that the class of K-flat groups contains admissible groups with the following definition: We say that connected complete riemannian manifold M is admissible, if M has finite volume and negative pinched curvature: \(-b^ 2<k_{\sigma}<-a^ 2<0\). A smooth fibre bundle \(F\to E\to^{p}M\) is called admissible if M is an admissible manifold, F is a closed connected manifold and for each poly-\({\mathbb{Z}}\) by finite subgroup S of \(\pi_ 1(M)\) the group \(P_{\#}^{-1}(S)\) is K-flat, where \(P_{\#}: \pi_ 1(E)\to \pi_ 1(M)\) is a homeomorphism induced by p. (A group S is called poly- \({\mathbb{Z}}\) by finite if it contains a finite index subgroup \(\Gamma\) which is poly-\({\mathbb{Z}}\); i.e.: \(\Gamma\) has filtration: \(1=\Gamma_ 0\triangleleft \Gamma_ 1\triangleleft \Gamma_ 2\triangleleft...\triangleleft \Gamma_ n=\Gamma,\) where \(\Gamma_{i+1}/\Gamma_ i\) are \(\infty\)-cyclic.) The group \(\Gamma\) is called admissible if it is isomorphic to the fundamental group of the total space E of an admissible fibre bundle. The proof of the K-flatness of all admissible groups uses Anosov's result for the geodesic flow on the admissible manifold and foliated version of Ferry's metric h- cobordism theorem obtained by the authors.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 58J47
zbMATH DE Number: 4065998
zbMATH Keywords: K-flat groups
zbMATH Keywords: geodesic flow
author: F. Thomas Farrell
author: Lowell Jones
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2007092
OpenAlex ID: W2319262301
description: scientific article; zbMATH DE number 4065998