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| {{for|the mathematical subject area|geometric topology}}
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| In [[mathematics]], the '''geometric topology''' is a [[topological space|topology]] one can put on the set ''H'' of [[hyperbolic 3-manifold]]s 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.
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| The following is a definition due to [[Troels Jorgensen]]: | |
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| :A sequence <math>\{M_i\}</math> in ''H'' converges to ''M'' in ''H'' if there are
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| :* a sequence of positive real numbers <math>\epsilon_i</math> converging to 0, and
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| :* a sequence of <math>(1+\epsilon_i)</math>-bi-Lipschitz [[diffeomorphism]]s <math>\phi_i: M_{i, [\epsilon_i, \infty)} \rightarrow M_{[\epsilon_i, \infty)},</math>
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| :where the domains and ranges of the maps are the <math>\epsilon_i</math>-thick parts of either the <math>M_i</math>'s or ''M''. | |
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| There is an alternate definition due to [[Mikhail Gromov (mathematician)|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 [[homeomorphism]]s 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.
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| 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 group]]s with the [[Chabauty topology]].
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| ==See also==
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| *[[Algebraic topology (object)]]
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| ==References==
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| * [[William Thurston]], [http://www.msri.org/publications/books/gt3m/ ''The geometry and topology of 3-manifolds''], Princeton lecture notes (1978-1981).
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| * Canary, R. D.; [[David B. A. Epstein|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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| [[Category:3-manifolds]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Topological spaces]]
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| {{topology-stub}}
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The writer is known as Wilber Pegues. The preferred hobby for him and his children is to perform lacross and he would never give it up. Distributing production has been his profession for some time. Her family members lives in Alaska but her husband desires them to transfer.
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