5CAT(0) spaces and groups

IV Topics in Geometric Group Theory

5.1 Some basic motivations

Given a discrete group Γ, there are two basic problems you might want to solve.

Question. Can we solve the word problem in Γ?

Question. Can we compute the (co)homology of Γ?

Definition

(Group (co)homology)

.

The (co)homology of a group Γ is the

(co)homology of K(Γ, 1).

We can define this in terms of the group itself, but would require knowing

some extra homological algebra. A very closely related question is

Question.

Can we find an explicit

X

such that Γ =

π

1

X

and

˜

X

is contractible?

We know that these problems are not solvable in general:

Theorem

(Novikov–Boone theorem)

.

There exists a finitely-presented group

with an unsolvable word problem.

Theorem

(Gordon)

.

There exists a sequence of finitely generated groups Γ

n

such that H

2

(Γ

n

) is not computable.

As before, we might expect that we can solve these problems if our groups

come with some nice geometry. In the previous chapter, we talked about

hyperbolic groups, which are negatively curved. In this section, we shall work

with slightly more general spaces, namely those that are non-positively curved.

Let

M

be a compact manifold of non-positive sectional curvature. It is a

classical fact that such a manifold satisfies a quadratic isoperimetric inequality.

This is not too surprising, since the “worst case” we can get is a space with

constant zero curvature, which implies

˜

M

∼

=

R

n

.

If we know this, then by the Filling theorem, we know the Dehn function of

the fundamental group is at worst quadratic, and in particular it is computable.

This solves the first question.

What about the second question?

Theorem

(Cartan–Hadamard theorem)

.

Let

M

be a non-positively curved com-

pact manifold. Then

˜

M

is diffeomorphic to

R

n

. In particular, it is contractible.

Thus, M = K(π

1

M, 1).

For example, this applies to the torus, which is not hyperbolic.

So non-positively curved manifolds are good. However, there aren’t enough

of them. Why? In general, the homology of a group can be very complicated,

and in particular can be infinite dimensional. However, manifolds always have

finite-dimensional homology groups. Moreover, they satisfy Poincar´e duality.

Theorem

(Poincar´e duality)

.

Let

M

be an orientable compact

n

-manifold.

Then

H

k

(M; R)

∼

=

H

n−k

(M; R).

This is a very big constraint, and comes very close to characterizing manifolds.

In general, it is difficult to write down a group whose homology satisfies Poincar´e

duality, unless we started off with a manifold whose universal cover is contractible,

and then took its fundamental group.

Thus, we cannot hope to realize lots of groups as the

π

1

of a non-positively

curved manifold. The idea of CAT(0) spaces is to mimic the properties of

non-positively curved manifolds in a much more general setting.