Fundamental groups of closed manifolds with positive curvature and torus actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T ^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n - 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n ≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T ^k -fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.