Fundamental groups of closed positively curved manifolds with almost discrete Abelian group actions

Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1×ℤpk-action with p prime. The main result of the article is that if k=1 for n=3 or for n ≥ 5, then there exists a positive constant p(n), depending only on n, such that π1(M) is cyclic if p≥ p(n).