The almost cyclicity of the fundamental groups of positively curved manifolds

Recall that a pure F-structure is a kind of generalized torus action. The main result asserts that if a compact positively curved manifold Mⁿ admits an invariant pure F-structure such that each orbit has positive dimension, then the fundamental group has a finite cyclic subgroup with index less than w_n, a constant depending only on n. As an application, we conclude that for all 0 < δ ≦ 1, the fundamental group of a δ-pinched n-manifold either has a cyclic subgroup with index less than wn or has order less than w(n,δ), a constant depending only on n and δ. In particular, this substantially improves the main result in [Ro1].