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 [math]\displaystyle{ \{M_i\} }[/math] in H converges to M in H if there are
- a sequence of positive real numbers [math]\displaystyle{ \epsilon_i }[/math] converging to 0, and
- a sequence of [math]\displaystyle{ (1+\epsilon_i) }[/math]-bi-Lipschitz diffeomorphisms [math]\displaystyle{ \phi_i: M_{i, [\epsilon_i, \infty)} \rightarrow M_{[\epsilon_i, \infty)}, }[/math]
- where the domains and ranges of the maps are the [math]\displaystyle{ \epsilon_i }[/math]-thick parts of either the [math]\displaystyle{ M_i }[/math]'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
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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