5CAT(0) spaces and groups

IV Topics in Geometric Group Theory 5.2 CAT(κ) spaces
Let
κ
=
1
,
0 or 1, and let
M
κ
be the unique, simply connected, complete
2-dimensional Riemannian manifold of curvature κ. Thus,
M
1
= S
2
, M
0
= R
2
, M
1
= H
2
.
M
κ
for other
κ
, but we can just obtain those by scaling
M
±1
.
Instead of working with Riemannian manifolds, we shall just think of these
as complete geodesic metric spaces. We shall now try to write down a “CAT(
κ
)”
condition, that says the curvature is bounded by κ in some space.
Definition (Triangle). A triangle with vertices {p, q, r} X is a choice
∆(p, q, r) = [p, q] [q, r] [r, p].
If we want to talk about triangles on a sphere, then we have to be a bit more
careful since the sides cannot be too long. Let
D
κ
=
diam M
κ
, i.e.
D
κ
=
for
κ = 0, 1 and D
κ
= π for κ = +1.
Suppose ∆ = ∆(
x
1
, x
2
, x
3
) is a triangle of perimeter
2
D
κ
in some complete
geodesic metric space (
X, d
). Then there is, up to isometry, a unique comparison
triangle
¯
∆ = ∆(¯x
1
, ¯x
2
, ¯x
3
) M
κ
such that
d
M
k
(¯x
i
, ¯x
j
) = d(x
i
, x
j
).
This is just the fact we know from high school that a triangle is determined
by the lengths of its side. The natural map
¯
is called the comparison
triangle.
Similarly, given a point
p
[
x
i
, x
j
], there is a comparison point
¯p
[
¯x
i
, ¯y
j
].
Note that
p
might be on multiple edges, so it could have multiple comparison
points. However, the comparison point is well-defined as long as we specify the
edge as well.
Definition
(CAT(
κ
) space)
.
We say a space (
X, d
) is CAT(
κ
) if for any geodesic
triangle
X
of diameter
2
D
κ
, any
p, q
and any comparison points
¯p, ¯q
¯
∆,
d(p, q) d
M
κ
(¯p, ¯q).
If X is locally CAT(κ), then K is said to be of curvature at most κ.
In particular, a locally CAT(0) space is called a non-positively curved space.
We are mostly interested in CAT(0) spaces. At some point, we will talk
Example. The following are CAT(0):
(i) Any Hilbert space.
(ii) Any simply-connected manifold of non-positive sectional curvature.
(iii) Symmetric spaces.
(iv) Any tree.
(v) If X, Y are CAT(0), then X × Y with the `
2
metric is CAT(0).
(vi) In particular, a product of trees is CAT(0).
Lemma
(Convexity of the metric)
.
Let
X
be a CAT(0) space, and
γ, δ
: [0
,
1]
X be geodesics (reparameterized). Then for all t [0, 1], we have
d(γ(t), δ(t)) (1 t)d(γ(0), δ(0)) + td(γ(1), δ(1)).
Note that we have strict equality if this is in Euclidean space. So it makes
sense that in CAT(0) spaces, we have an inequality.
Proof. Consider the rectangle
α
γ(0) γ(1)
δ(0) δ(1)
γ
δ
Let
α
: [0
,
1]
X
be a geodesic from
γ
(0) to
δ
(1). Applying the CAT(0) estimate
to ∆(γ(0), γ(1), δ(1)), we get
d(γ(t), α(t)) d(γ(t), α(t)) = td(γ(1), α(1)) = td(γ(1), α(1)) = td(γ(1), δ(1)),
using what we know in plane Euclidean geometry. The same argument shows
that
d(δ(t), α(t)) (1 t)d(δ(0), γ(0)).
So we know that
d(γ(t), δ(t)) d(γ(t), α(t)) + d(α(t), δ(t)) (1 t)d(γ(0), δ(0)) + td(γ(1), δ(1)).
Lemma.
If
X
is CAT(0), then
X
is uniquely geodesic, i.e. each pair of points is
joined by a unique geodesic.
Proof.
Suppose
x
0
, x
1
X
and
γ
(0) =
δ
(0) =
x
0
and
γ
(1) =
δ
(1) =
x
1
. Then
by the convexity of the metric, we have
d
(
γ
(
t
)
, δ
(
t
))
0. So
γ
(
t
) =
δ
(
t
) for all
t.
This is not surprising, because this is true in the Euclidean plane and the
hyperbolic plane, but not in the sphere. Of course, one can also prove this result
directly.
Lemma.
Let
X
be a proper, uniquely geodesic metric space. Then geodesics in
X
vary continuously with their end points in the compact-open topology (which
is the same as the uniform convergence topology).
This is actually true without the word “proper”, but the proof is harder.
Proof. This is an easy application of the Arzel´a–Ascoli theorem.
Proposition. Any proper CAT(0) space X is contractible.
Proof.
Pick a point
x
0
X
. Then the map
X Maps
([0
,
1]
, X
) sending
x
to
the unique geodesic from
x
0
to
x
X ×
[0
,
1]
X
is then a homotopy from the constant map at x
0
to the identity map.
Definition
(CAT(0) group)
.
A group is CAT(0) if it acts properly discontinu-
ously and cocompactly by isometries on a proper CAT(0) space.
Usually, for us, the action will also be free. This is the case, for example,
when a fundamental group acts on the covering space.
Note that a space being CAT(0) is a very fine-grained property, so this is
not the same as saying the Cayley graph of the group is CAT(0).
Example. Z
n
for any n is CAT(0), since it acts on R
n
.
Example.
More generally,
π
1
M
for any closed manifold
M
of non-positive
curvature is CAT(0).
Example.
Uniform lattices in semi-simple Lie groups. Examples include
SL
n
O
K
for certain number fields K.
Example. Any free group, or direct product of free groups is CAT(0).
We remark, but will not prove
Proposition.
Any CAT(0) group Γ satisfies a quadratic isoperimetric inequality,
that is δ
Γ
' n or n
2
.
Note that if Γ is in fact CAT(-1), then Γ is hyperbolic, which is not terribly
difficult to see, since
H
2
is hyperbolic. But if a group is hyperbolic, is it necessarily
CAT(-1)? Or even just CAT(0)? This is an open question. The difficulty in
answering this question is that hyperbolicity is a “coarse condition”, but being
CAT(0) is a fine condition. For example, Cayley graphs are not CAT(0) spaces
unless they are trees.