A compact hyperbolic manifold and fundmental group

Question

I just start to learn differential geometry and have a problem. Let $M$ be a compact hyperbolic manifold, how to prove Z $\oplus$ Z is not a subgroup of $\pi_1(M)$ .

Answer 1

One way to prove this is to use Preissman's Theorem, which says that if $M$ is a compact negatively curved Riemannian manifold, then every non-trivial abelian subgroup of $\pi_1(M)$ must be isomorphic to $\mathbb{Z}$. In particular, if $M$ is a compact hyperbolic manifold, then $\mathbb{Z}\oplus\mathbb{Z}$, which is abelian and is not isomorphic to $\mathbb{Z}$, cannot be a subgroup of $\pi_1(M)$.

But using Preissman's Theorem may be overkill: if $M$ is a hyperbolic manifold, i.e. has constant negative sectional curvature, there may be some simpler way to prove it.