Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [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 (54 #3702)
  • [3] Albert Marden, The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383–462. MR 0349992 (50 #2485)
  • [4] Bernard Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971), 219–230. MR 0297997 (45 #7049)
  • [5] W. Thurston, The geometry and topology of $ 3$-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 (82i:57012)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57M25

Retrieve articles in all journals with MSC: 57N10, 57M25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0768730-6
PII: S 0002-9947(1985)0768730-6
Keywords: Hyperbolic $ 3$-manifold, thrice-punctured sphere, link complement, hyperbolic volume
Article copyright: © Copyright 1985 American Mathematical Society