A closed Riemannian manifold of nonpositive sectional curvature with finite $\pi_1 (M)$

Question

Let $ \left( M,g \right)$ be a closed Riemannian manifold of nonpositive sectional curvature $K \leq 0$. Is there then a variety with a finite fundamental group $\pi_1 \left(M \right)$. I know that if our manifold is compact and $K < 0$ then by the Preissman's theorem its fundamental group doesn't need to contain any abelian subgroup.

Answer 1

By the Cartan-Hadamard theorem a manifold with a complete metric of non-positive sectional curvature has universal cover diffeomorphic to $\mathbb{R}^{n}$. If $M$ is compact this implies that that $\pi_{1}(M)$ is infinite, since it acts on $\mathbb{R}^{n}$ with a compact quotient space.