Milnor's Theorem

If a compact manifold M has nonnegative Ricci curvature tensor, 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.

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

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Milnor's Theorem

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Weisstein, Eric W. "Milnor's Theorem." From MathWorld--A Wolfram Web Resource.

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