In mathematics, the **geometric topology** is a topology one can put on the set *H* of hyperbolic 3-manifolds of finite volume.

Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.

The following is a definition due to Troels Jorgensen:

A sequence

{
M
i
}
in

*H* converges to

*M* in

*H* if there are

a sequence of positive real numbers
ϵ
i
converging to 0, and
a sequence of
(
1
+
ϵ
i
)
-bi-Lipschitz diffeomorphisms
ϕ
i
:
M
i
,
[
ϵ
i
,
∞
)
→
M
[
ϵ
i
,
∞
)
,
where the domains and ranges of the maps are the

ϵ
i
-thick parts of either the

M
i
's or

*M*.

There is an alternate definition due to Mikhail Gromov. Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on *pointed* hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphisms on larger and larger balls. This results in the same notion of convergence as above as the thick part is always connected; thus, a large ball will eventually encompass all of the thick part.

As a further refinement, Gromov's metric can also be defined on *framed* hyperbolic 3-manifolds. This gives nothing new but this space can be explicitly identified with torsion-free Kleinian groups with the Chabauty topology.