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 { M i } {\displaystyle \{M_{i}\}} in H converges to M in H if there are a sequence of positive real numbers ϵ i {\displaystyle \epsilon _{i}} converging to 0, and a sequence of ( 1 + ϵ i ) {\displaystyle (1+\epsilon _{i})} -bi-Lipschitz diffeomorphisms ϕ i : M i , [ ϵ i , ∞ ) → M [ ϵ i , ∞ ) , {\displaystyle \phi _{i}:M_{i,[\epsilon _{i},\infty )}\rightarrow M_{[\epsilon _{i},\infty )},}
where the domains and ranges of the maps are the ϵ i {\displaystyle \epsilon _{i}} -thick parts of either the M i {\displaystyle 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.
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.
