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:


 * A sequence $$\{M_i\}$$ in H converges to M in H if there are


 * a sequence of positive real numbers $$\epsilon_i$$ converging to 0, and
 * a sequence of $$(1+\epsilon_i)$$-bi-Lipschitz diffeomorphisms $$\phi_i: M_{i, [\epsilon_i, \infty)} \rightarrow M_{[\epsilon_i, \infty)},$$


 * where the domains and ranges of the maps are the $$\epsilon_i$$-thick parts of either the $$M_i$$'s or M.

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.