A1 Refereed original research article in a scientific journal
On the hyperbolic distance of n-times punctured spheres
Authors: Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang
Publisher: HEBREW UNIV MAGNES PRESS
Publication year: 2020
Journal: Journal d'Analyse Mathématique
Journal name in source: JOURNAL D ANALYSE MATHEMATIQUE
Journal acronym: J ANAL MATH
Number of pages: 25
ISSN: 0021-7670
eISSN: 1565-8538
DOI: https://doi.org/10.1007/s11854-020-0112-9
Self-archived copy’s web address: https://arxiv.org/abs/1707.05773
Abstract
The shortest closed geodesic in a hyperbolic surface X is called a systole of X. When X is an n-times punctured sphere C∖^A" role="presentation">C∖ˆA where A⊂C^" role="presentation">A⊂Cˆis a finite set of cardinality n >= 4, we define a quantity Q(A) in terms of cross ratios of quadruples in A so that Q(A) is quantitatively comparable with the systole length of X. We next propose a method to construct a distance function dX on a punctured sphere X which is Lipschitz equivalent to the hyperbolic distance hX on X. In particular, when the construction is based on a modified quasihyperbolic metric, dX is Lipschitz equivalent to hX with a Lipschitz constant depending only on Q(A).
The shortest closed geodesic in a hyperbolic surface X is called a systole of X. When X is an n-times punctured sphere C∖^A" role="presentation">C∖ˆA where A⊂C^" role="presentation">A⊂Cˆis a finite set of cardinality n >= 4, we define a quantity Q(A) in terms of cross ratios of quadruples in A so that Q(A) is quantitatively comparable with the systole length of X. We next propose a method to construct a distance function dX on a punctured sphere X which is Lipschitz equivalent to the hyperbolic distance hX on X. In particular, when the construction is based on a modified quasihyperbolic metric, dX is Lipschitz equivalent to hX with a Lipschitz constant depending only on Q(A).
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