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K-theory and dynamics. I. (English) Zbl 0653.58035

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.
Reviewer: V.B.Marenich

MSC:

37C10 Dynamics induced by flows and semiflows
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
58J47 Propagation of singularities; initial value problems on manifolds
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