On closed $\delta$-pinched manifolds with discrete abelian group actions

Let $M^{n}$ be a closed odd $n$-manifold with sectional curvature $\delta <\sec _{M}\le 1$, and let $M$ admit an effective isometric $\mathbb{Z}^{k}_{p}$-action with $p$ prime. The main results in the paper are: (1) if $\delta >0$ and $n\ge 5$, then there exists a constant $p(n,\delta )$, depending only on $n$ and $\delta$, such that $p\ge p(n,\delta )$ implies that (i) $k\le \frac{n+1}{2}$, (ii) the universal covering space of $M$ is homeomorphic to $S^{n}$ if $k>\frac{3}{8}n+1$, (iii) the fundamental group $\pi _{1}(M)$ is cyclic if $k>\frac{n+1}{4}+1$; (2) if $\delta =0$ and $n=3$, then $k\le 4$ for $p=2$ and $k\le 2$ for $p\ge 3$, and $\pi _{1}(M)$ is cyclic if $p\ge 5$ and $k=2$.