Geometric topology (object)

In mathematics, the geometric topology is a topology one can put on the set "H" of hyperbolic 3-manifolds of finite volume. 
Use.
Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds. 
Definition.
The following is a definition due to Troels Jorgensen: 
Alternate definition.
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. 
On framed manifolds.
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.