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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- J. Hempel, -manifolds, Ann. of Math. Studies, No. 86, Princeton Univ. Press, Princeton, N. J., 1976. MR 0415619 (54:3702)
- A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992 (50:2485)
- B. Maskit, On Poincaré's theorem for fundamental polygons, Adv. in Math. 7 (1971), 219-230. MR 0297997 (45:7049)
- W. Thurston, The geometry and topology of -manifolds, Lecture Notes, Princeton Univ., 1978-79.
- N. Wielenberg, Hyperbolic -manifolds which share a fundamental polyhedron, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Princeton Univ. Press, Princeton, N. J., 1981, pp. 505-513. MR 624835 (82i:57012)
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