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### The logarithmic spiral : a counterexample to the K=2 conjecture

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Epstein, D. B. A. and Markovic, V. (Vladimir).
(2005)
*The logarithmic spiral : a counterexample to the K=2 conjecture.*
Annals of Mathematics, Vol.161
(No.2).
pp. 925-957.
ISSN 0003-486X

**Full text not available from this repository.**

Official URL: http://www.jstor.org/stable/3597321

## Abstract

Given a nonempty compact connected subset X subset of S-2 with complement a simply-connected open subset Omega subset of S-2, let Dome (Omega) be the boundary of the hyperbolic convex hull in H-3 of X. We show that if X is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism Omega -> Dome (Omega) which extends to the identity map on their common boundary in S-2. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take S-2 = C boolean OR {infinity}). We also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion-free quasifuchsian group with compact quotient. Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately .98 pi/2, which is substantially larger than that of any previously known example.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Journal or Publication Title: | Annals of Mathematics |

Publisher: | Mathematical Sciences Publishers |

ISSN: | 0003-486X |

Official Date: | March 2005 |

Volume: | Vol.161 |

Number: | No.2 |

Number of Pages: | 33 |

Page Range: | pp. 925-957 |

Status: | Peer Reviewed |

Publication Status: | Published |

URI: | http://wrap.warwick.ac.uk/id/eprint/6805 |

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