Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



An explicit counterexample to the equivariant $ K=2$ conjecture

Authors: Yohei Komori and Charles A. Matthews
Journal: Conform. Geom. Dyn. 10 (2006), 184-196
MSC (2000): Primary 30F40, 30F60, 32G15, 57M50
Published electronically: August 24, 2006
MathSciNet review: 2261047
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct an explicit example of a geometrically finite Kleinian group $ G$ with invariant component $ \Omega$ in the Riemann sphere $ {\bf\hat{C}}$ such that any quasiconformal map from $ \Omega$ to the boundary of the convex hull of $ {\bf\hat{C}} - \Omega$ in $ {\bf H^3}$ which extends to the identity map on their common boundary in $ {\bf\hat{C}}$, and which is equivariant under the group of Möbius transformations preserving $ \Omega$, must have maximal dilatation $ K > 2.002$.

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

  • [Bis02] Christopher J. Bishop, Quasiconformal Lipschitz maps, Sullivan's convex hull theorem, and Brennan's conjecture, Ark. Mat. 40 (2002), no. 1, 1-26. MR 1948883 (2003i:30063)
  • [Bis04] -, An explicit constant for Sullivan's convex hull theorem, In the Tradition of Ahlfors and Bers, III, Contemp. Math., 355, Amer. Math. Soc., Providence, RI, 2004, pp. 41-69. MR 2145055 (2006c:30022)
  • [EM87] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113-253. MR 0903852 (89c:52014)
  • [EM05] D. B. A. Epstein and V. Markovic, The logarithmic spiral: a counterexample to the $ {K}=2$ conjecture, Ann. of Math. (2) 161 (2005), no. 2, 925-957. MR 2153403 (2006f:30048)
  • [EMM04] D. B. A. Epstein, A. Marden, and V. Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. (2) 159 (2004), no. 1, 305-336. MR 2052356 (2005d:30067)
  • [KRV79] L. Keen, H. E. Rauch, and A. T. Vasquez, Moduli of punctured tori and the accessory parameter of Lamé's equation, Trans. Amer. Math. Soc. 255 (1979), 201-230. MR 0542877 (81j:30074)
  • [KS93] Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719-749. MR 1241870 (95g:32030)
  • [KS97] -, How to bend pairs of punctured tori, Lipa's Legacy, Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359-388. MR 1476997 (98m:30063)
  • [KS04] -, Pleating invariants for punctured torus groups, Topology 43 (2004), no. 2, 447-491. MR 2052972 (2005f:30077)
  • [Mat01] Charles A. Matthews, Approximation of a map between one-dimensional Teichmüller spaces, Experiment. Math. 10 (2001), no. 2, 247-265. MR 1837674 (2002g:30037)
  • [PP98] John Parker and Jouni Parkkonen, Coordinates for quasi-Fuchsian punctured torus space, The Epstein Birthday Schrift, Geometry and Topology Monographs, Geom. Topol. Publ., Coventry, 1998, pp. 451-478. MR 1668328 (2000d:30065)
  • [Sul81] Dennis Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $ 3$ fibrées sur $ {S}\sp{1}$, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981, pp. 196-214. MR 0636524 (83h:58079)
  • [Thu98] William P. Thurston, Hyperbolic structures on 3-manifolds, II: surface groups and 3-manifolds which fiber over the circle, eprint available at as paper number math.GT/9801045, 1998.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F40, 30F60, 32G15, 57M50

Retrieve articles in all journals with MSC (2000): 30F40, 30F60, 32G15, 57M50

Additional Information

Yohei Komori
Affiliation: Department of Mathematics, Osaka City University, Osaka 558-8585, Japan

Charles A. Matthews
Affiliation: Department of Mathematics, Southeastern Oklahoma State University, Durant, Oklahoma 74701

Received by editor(s): April 20, 2006
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society