## 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

### Keywords:

K-flat groups; geodesic flow
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