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

## Contents

## 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*H*converges to

*M*in

*H*if there are

*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.