A note on the fundamental group of a manifold of negative curvature

Abstract

Let Y be a compact connected C∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π1(Y). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G, a ≠ e, such that for all b belonging to a set of generators for G we have ambn = bqap for some integers m, n, p, q with either m = p or n = q.