Geometric topology (object)

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For the mathematical subject area, see geometric topology.

In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume. Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.

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.

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.

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


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