4Hyperbolic groups
IV Topics in Geometric Group Theory
4.1 Definitions and examples
We now want to define a negatively-curved space in great generality. Let
X
be a geodesic metric space. Given
x, y ∈ X
, we will write [
x, y
] for a choice of
geodesic between x and y.
Definition
(Geodesic triangle)
.
A geodesic triangle ∆ is a choice of three points
x, y, z and geodesics [x, y], [y, z], [z, x].
Geodesic triangles look like this:
Note that in general, the geodesics may intersect.
Definition
(
δ
-slim triangle)
.
We say ∆ is
δ
-slim if every side of ∆ is contained
in the union of the δ-neighbourhoods of the other two sides.
Definition
(Hyperbolic space)
.
A metric space is (Gromov) hyperbolic if there
exists
δ ≥
0 such that every geodesic triangle in
X
is
δ
-slim. In this case, we say
it is δ-hyperbolic.
Example. R
2
is not Gromov hyperbolic.
Example.
If
X
is a tree, then
X
is 0-hyperbolic! Indeed, each triangle looks
like
We call this a tripod.
Unfortunately, none of these examples really justify why we call these things
hyperbolic. Let’s look at the actual motivating example.
Example.
Let
X
=
H
2
, the hyperbolic plane. Let ∆
⊆ H
2
be a triangle. Then
∆ is
δ
-slim, where
δ
is the maximum radius of an inscribed semi-circle
D
in ∆
with the center on one of the edges.
But we know that the radius of
D
is bounded by some increasing function
of the area of
D
, and the area of
D
is bounded above by the area of ∆. On
the other hand, by hyperbolic geometry, we know the area of any triangle is
bounded by π. So H
2
is δ-hyperbolic for some δ.
If we worked a bit harder, then we can figure out the best value of
δ
. However,
for the arguments we are going to do, we don’t really care about what δ is.
Example.
Let
X
be any bounded metric space, e.g.
S
2
. Then
X
is Gromov
hyperbolic, since we can just take δ to be the diameter of the metric space.
This is rather silly, but it makes sense if we take the “coarse point of view”,
and we have to ignore bounded things.
What we would like to do is to say is that a group Γ if for every finite
generating set
S
, the Cayley graph
Cay
S
(Γ) equipped with the word metric is
δ-hyperbolic for some δ.
However, this is not very helpful, since we have to check it for all finite
generating sets. So we want to say that being hyperbolic is quasi-isometry
invariant, in some sense.
This is slightly difficult, because we lose control of how the geodesic behaves
if we only look at things up to isometry. To do so, we have to talk about
quasi-geodesics.