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