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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. 925957. ISSN 0003486X
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Official URL: http://www.jstor.org/stable/3597321
Abstract
Given a nonempty compact connected subset X subset of S2 with complement a simplyconnected open subset Omega subset of S2, let Dome (Omega) be the boundary of the hyperbolic convex hull in H3 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 2quasiconformal homeomorphism Omega > Dome (Omega) which extends to the identity map on their common boundary in S2. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take S2 = 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 torsionfree 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:  0003486X  
Official Date:  March 2005  
Dates: 


Volume:  Vol.161  
Number:  No.2  
Number of Pages:  33  
Page Range:  pp. 925957  
Status:  Peer Reviewed  
Publication Status:  Published  
URI:  http://wrap.warwick.ac.uk/id/eprint/6805 
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