Milnor's Theorem

If a Compact Manifold $M$ has Nonnegative Ricci Curvature, then its Fundamental Group has at most Polynomial growth. On the other hand, if $M$ has Negative curvature, then its Fundamental Group has exponential growth in the sense that $n(\lambda)$ grows exponentially, where $n(\lambda)$ is (essentially) the number of different ``words'' of length $\lambda$ which can be made in the Fundamental Group.

References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.