Let ${M}$ be a compact Riemannian manifold of (strictly) negative curvature, so that ${M}$ is a ${K(\pi_1 M, 1)}$. In the previous post, we saw that the group ${\pi_1 M}$ was significantly restricted: for example, every solvable subgroup of ${\pi_1 M}$ had to be infinite cyclic. The goal of this post and the next is to understand the result of Milnor that the group ${\pi_1 M}$ is of exponential growth in an arithmetic sense.

Milnor’s (wonderful) idea is to translate this into a problem in geometry: that is, to relate the growth of the group ${\pi_1 M}$ to the volume growth of expanding balls in the universal cover ${\widetilde{M}}$. As I understand, this idea has proved enormously influential on future work on the fundamental groups of Riemannian manifolds with restricted curvature. Note that Milnor’s result also highlights the difference between positive and negative curvature: in positive curvature, the fundamental group of every compact manifold is finite. Most of this material is from Chavel’s Riemannian geometry: a modern introduction.

1. Volume growth

To begin with, let’s say something about volume growth. Let ${M}$ be a complete, simply connected Riemannian manifold whose sectional curvatures are ${\leq c < 0}$. If we choose ${p \in M}$, we know that the exponential map

$\displaystyle \exp_p : T_p M \rightarrow M$

is a diffeomorphism. Note that ${\exp_p}$ sends the euclidean ball of radius ${r}$ diffeomorphically onto the (metric) ball of radius ${r}$ in ${M}$.

Our goal is to prove:

Theorem 13 The function ${r \mapsto \mathrm{vol} ( B_M(p, r))}$ which sends ${r}$ to the volume of the ball in ${M}$ of radius ${r}$ centered at ${p}$ grows exponentially.

This theorem also highlights the sense in which negative curvature corresponds to the “spreading” of geodesics: the geodesics spread so much that the volumes of linearly expanding balls actually grow exponentially. (more…)