## The fundamental groups of almost nonnegatively curved manifolds.(English)Zbl 0770.53028

A group is called almost nilpotent if it contains a nilpotent subgroup of finite index. The authors prove in their main result that there exists a positive number $$\varepsilon=\varepsilon(n)$$ such that the fundamental group of a compact Riemannian $$n$$-manifold $$M$$ is almost nilpotent if the sectional curvature $$K_ M$$ and the diameter $$d_ M$$ satisfy $K_ Md^ 2_ M>-\varepsilon(n).$ This was conjectured by Gromov and follows from his almost flat manifold theorem under the stronger assumption $$| K_ M| d^ 2_ M<\varepsilon(n)$$. The proof uses a collapsing technique.
Reviewer: W.Ballmann (Bonn)

### MSC:

 53C20 Global Riemannian geometry, including pinching
Full Text: