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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Thrice-punctured spheres in hyperbolic $ 3$-manifolds

Author: Colin C. Adams
Journal: Trans. Amer. Math. Soc. 287 (1985), 645-656
MSC: Primary 57N10; Secondary 57M25
MathSciNet review: 768730
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Abstract: The work of $ {\text{W}}$. Thurston has stimulated much interest in the volumes of hyperbolic $ 3$-manifolds. In this paper, it is demonstrated that a $ 3$-manifold $ M\prime$ obtained by cutting open an oriented finite volume hyperbolic $ 3$-manifold $ M$ along an incompressible thrice-punctured sphere $ S$ and then reidentifying the two copies of $ S$ by any orientation-preserving homeomorphism of $ S$ will also be a hyperbolic $ 3$-manifold with the same hyperbolic volume as $ M$. It follows that an oriented finite volume hyperbolic $ 3$-manifold containing an incompressible thrice-punctured sphere shares its volume with a nonhomeomorphic hyperbolic $ 3$-manifold. In addition, it is shown that two orientable finite volume hyperbolic $ 3$-manifolds $ {M_1}$ and $ {M_2}$ containing incompressible thrice-punctured spheres $ {S_1}$ and $ {S_2}$, respectively, can be cut open along $ {S_1}$ and $ {S_2}$ and then glued together along copies of $ {S_1}$ and $ {S_2}$ to yield a $ 3$-manifold which is hyperbolic with volume equal to the sum of the volumes of $ {M_1}$ and $ {M_2}$. Applications to link complements in $ {S^3}$ are included.

References [Enhancements On Off] (What's this?)

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Keywords: Hyperbolic $ 3$-manifold, thrice-punctured sphere, link complement, hyperbolic volume
Article copyright: © Copyright 1985 American Mathematical Society