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.