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

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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
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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