5CAT(0) spaces and groups

IV Topics in Geometric Group Theory

5.5 Cartan–Hadamard theorem

Theorem

(Cartan–Hadamard theorem)

.

If

X

is a complete, connected length

space of non-positive curvature, then the universal cover

˜

X, equipped with the

induced length metric, is CAT(0).

This was proved by Cartan and Hadamard in the differential geometric

setting.

Corollary.

A (torsion free) group Γ is CAT(0) iff it is the

π

1

of a complete,

connected space X of non-positive curvature.

We’ll indicate some steps in the proof of the theorem.

Lemma.

If

X

is proper, non-positively curved and uniquely geodesic, then

X

is CAT(0).

Proof idea.

The idea is that given a triangle, we cut it up into a lot of small

triangles, and since

X

is locally CAT(0), we can use Alexandrov’s lemma to

conclude that the large triangle is CAT(0).

Recall that geodesics vary continuously with their endpoints. Consider a

triangle ∆ = ∆(

x, y, z

)

⊆

¯

B ⊆ X

, where

¯

B

is a compact ball. By compactness,

there is an ε such that for every x ∈

¯

B, the ball B

x

(¯ε) is CAT(0).

We let

β

t

be the geodesic from

x

to

α

(

t

). Using continuity, we can choose

0 < t

1

< · · · < t

N

= 1 such that

d(β

t

i

(s), β

t

i+1

(s)) < ε

for all s ∈ [0, 1].

Now divide ∆ up into a “patchwork” of triangles, each contained in an

ε

ball,

so each satisfies the CAT(0) condition, and apply induction and Alexandrov’s

lemma to conclude.

Now to prove the Cartan–Hadamard theorem, we only have to show that the

universal cover is uniquely geodesic. Here we must use the simply-connectedness

condition.

Theorem.

Let

X

be a proper length space of non-positive curvature, and

p, q ∈ X

. Then each homotopy class of paths from

p

to

q

contains a unique

(local) geodesic representative.