Thrice-punctured spheres in hyperbolic -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 . 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.

**[1]**C. Adams,*Hyperbolic structures on link complements*, Ph.D. Thesis, University of Wisconsin, Madison, August 1983.**[2]**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****[3]**Albert Marden,*The geometry of finitely generated kleinian groups*, Ann. of Math. (2)**99**(1974), 383–462. MR**0349992****[4]**Bernard Maskit,*On Poincaré’s theorem for fundamental polygons*, Advances in Math.**7**(1971), 219–230. MR**0297997****[5]**W. Thurston,*The geometry and topology of*-*manifolds*, Lecture Notes, Princeton Univ., 1978-79.**[6]**Norbert J. Wielenberg,*Hyperbolic 3-manifolds which share a fundamental polyhedron*, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 505–513. MR**624835**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1985-0768730-6

Keywords:
Hyperbolic -manifold,
thrice-punctured sphere,
link complement,
hyperbolic volume

Article copyright:
© Copyright 1985
American Mathematical Society