Geometric topology (object)

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 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.

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.