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

Fukaya, Kenji; Yamaguchi, Takao

Ann. Math. (2) 136, No. 2, 253-333 (1992).

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

Keywords:

almost nonnegative curvature; nilpotent subgroup; fundamental group; collapsing technique