Let be a compact Riemannian manifold of (strictly) negative curvature, so that
is a
. In the previous post, we saw that the group
was significantly restricted: for example, every solvable subgroup of
had to be infinite cyclic. The goal of this post and the next is to understand the result of Milnor that the group
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 to the volume growth of expanding balls in the universal cover
. 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 be a complete, simply connected Riemannian manifold whose sectional curvatures are
. If we choose
, we know that the exponential map
is a diffeomorphism. Note that sends the euclidean ball of radius
diffeomorphically onto the (metric) ball of radius
in
.
Our goal is to prove:
Theorem 13 The function
which sends
to the volume of the ball in
of radius
centered at
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.
In order to prove this result, we need to understand the pullback of the metric on
to
via
. Without loss of generality, we can ignore zero, and we work in polar coordinates. We have a diffeomorphism
where is the unit sphere in
. We thus get “coordinates”
on
where
is the radius and
. Our goal is to express the metric
on
as
where the (abusive) notation refers to a positive definite quadratic form
on
(under the polar identification of the tangent space at
with
). The existence of such a decomposition of the metric is a consequence of the Gauss lemma.
Now, our goal is to understand . In other words, given
perpendicular to
, we would like to understand
If we understand these values, then we understand the pulled back metric . However, we can understand these values in terms of geodesic deviation or Jacobi fields. We recall that the value
is exactly the value (at ) of a Jacobi field
along the geodesic
with
,
.
The claim is that, using the assumed upper bounds for curvature, we can bound below the quadratic form , and thus the volume. This will require a special case of the Rauch comparison theorem.
2. The case of constant curvature
Let’s see how the computation of plays out when we are working with a space of constant curvature
. In this case, if we keep the same notation as above, we need to understand the Jacobi field
along the geodesic
with
. This satisfies the Jacobi equation
Now, started out orthogonal to
(as
), so this is always true by the following lemma.
Lemma 14 Let
be a Jacobi field along a geodesic
with
perpendicular to
. Then
is always perpendicular to
.
Proof: Indeed, we have
by the Jacobi equation and the antisymmetry of the curvature tensor. So if and its derivative vanish at
, it must vanish identically.
Anyway, this means that
If is a parallel orthonormal basis along
of
, then the basis of solutions of this second-order equation are
Taking , it follows that the quadratic form
on
is exactly
— or rather, that times the given (euclidean) metric on
. In other words, we find that
in polar coordinates is
where is the metric on
. This is the polar expression of hyperbolic space.
Now, we can get from here the exponential growth of the volume. The volume form associated to is given by
where is the volume form on
. Then
is an exponentially growing function on . (If
, we would see polynomial growth instead.)
3. The general case
The goal is to generalize the above analysis in the case of a space of constant negative curvature to allow the curvature to vary. The principle is that since the curvature is less than , geodesics should deviate (i.e., Jacobi fields should grow) more than they do in the space of constant curvature
— which we have seen is already exponential. The Rauch comparison theorem provides a formalization of this principle, but we will only need a fairly limited analysis.
Let be as before, except we now assume its sectional curvatures are
. Consider the geodesic
as before and let
be a unit vector, and let
be the Jacobi field along
with
. We note that
is always perpendicular to
, as before, and we have that
It follows that if , then
By Cauchy-Schwarz applied to the last two terms, we find that this is at least
So we get the differential inequality:
rather than the differential equation that we got before with solutions the function. Moreover, by
and L’Hopital’s rule, we get as
is a unit vector.
The idea is that a solution of the differential inequality will have to be at least the solution of the corresponding differential equation, and this is the heart of the comparison with constant sectional curvature. Let’s state this as follows:
Lemma 15 Let
. Let
be a nonnegative
function with
satisfying the differential inequality (4). Then
.
Proof: Consider ; we have that
. We’d like to show that
which will imply the result. Write . Then (5) follows from
if we can show that . To see this, we note that
takes the value
at
, so we will be done if we can show that its derivative is nonnegative. But we have
It follows that the pull-back metric has the property that the quadratic form
on
is greater than
times the euclidean metric on
. That is, if we interpret
as a linear transformation
, then
. It follows that
so the volume form we get is bigger than the one we got before in constant curvature.
We have thus proved the following:
Theorem 16 Let
be a complete, simply connected Riemannian manifold with sectional curvatures
. Let
. Let
be the ball of radius
in
centered at
. Then
grows exponentially. In fact, it grows as fast as the analogous volumes of balls in the constant curvature space of curvature
.
In the next post, we’ll see what this has to do with the growth rate of fundamental groups.
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