Geometric topology (object)

Short description: Topological Object

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

## Contents

* 1 Use
* 2 Definition
* 3 Alternate definition
* 3.1 On framed manifolds
* 4 See also
* 5 References

## 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 {Mi} in H converges to M in H if there are

* a sequence of positive real numbers ϵi converging to 0, and
* a sequence of (1+ϵi)-bi-Lipschitz diffeomorphisms ϕi:Mi,[ϵi,∞)->M[ϵi,∞),

where the domains and ranges of the maps are the ϵi-thick parts of either the Mi'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.

## See also

* Algebraic topology (object)

## References

* William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981).
* Canary, R. D.; Epstein, D. B. A.; Green, P., Notes on notes of Thurston. Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), 3--92, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987.

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