5CAT(0) spaces and groups

IV Topics in Geometric Group Theory

5.3 Length metrics

In differential geometry, if we have a covering space

˜

X → X

, and we have a

Riemannian metric on

X

, then we can lift this to a Riemannian metric to

˜

X

.

This is possible since the Riemannian metric is a purely local notion, and hence

we can lift it locally. Can we do the same for metric spaces?

Recall that if γ : [a, b] → X is a path, then

`(γ) = sup

a=t

0

<t

1

<···<t

n

=b

n

X

i=1

d(γ(t

i−1

), γ(t

i

)).

We say γ is rectifiable if `(γ) < ∞.

Definition

(Length space)

.

A metric space

X

is called a length space if for all

x, y ∈ X, we have

d(x, y) = inf

γ:x→y

`(γ).

Given any metric space (

X, d

), we can construct a length pseudometric

ˆ

d : X × X → [0, ∞] given by

ˆ

d(x, y) = inf

γ:x→y

`(γ).

Now given a covering space

p

:

˜

X → X

and (

X, d

) a metric space, we can define

a length pseudometric on

˜

X by, for any path ˜γ : [a, b] →

˜

X,

`(˜γ) = `(p ◦ ˜γ).

This induces an induced pseudometric on

˜

X.

Exercise. If X is a length space, then so is

˜

X.

Note that if

A, B

are length spaces, and

X

=

A ∪ B

(such that the metrics

agree on the intersection), then

X

has a natural induced length metric. Recall we

previously stated the Hopf–Rinow theorem in the context of differential geometry.

In fact, it is actually just a statement about length spaces.

Theorem

(Hopf–Rinow theorem)

.

If a length space

X

is complete and locally

compact, then X is proper and geodesic.

This is another application of the Arzel´a–Ascoli theorem.