On the hyperbolic distance of \(n\)-times punctured spheres

The length of the shortest closed geodesic in a hyperbolic surface \(X\) is called the systole of \(X.\) When \(X\) is an \(n\)-times punctured sphere \(\hat{ \mathbb{C}} \setminus A\) where \(A \subset \hat{\mathbb{C}}\) is a finite set of cardinality \(n\ge4,\) we define a quantity \(Q(A)\) in ter...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-07
Main Authors: Sugawa, Toshiyuki, Vuorinen, Matti, Zhang, Tanran
Format: Article
Language:English
Subjects:
Equivalence
Systole
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The length of the shortest closed geodesic in a hyperbolic surface \(X\) is called the systole of \(X.\) When \(X\) is an \(n\)-times punctured sphere \(\hat{ \mathbb{C}} \setminus A\) where \(A \subset \hat{\mathbb{C}}\) is a finite set of cardinality \(n\ge4,\) 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 of \(X.\) We next propose a method to construct a distance function \(d_X\) on a punctured sphere \(X\) which is Lipschitz equivalent to the hyperbolic distance \(h_X\) on \(X.\) In particular, when the construction is based on a modified quasihyperbolic metric, \(d_X\) is Lipschitz equivalent to \(h_X\) with Lipschitz constant depending only on \(Q(A).\)
ISSN:2331-8422