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arXiv · 1707.05773

On the hyperbolic distance of $n$-times punctured spheres

Toshiyuki Sugawa·Matti Vuorinen·Tanran Zhang
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Abstract

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 $Q(A).$

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Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang. 2017-07-18. On the hyperbolic distance of $n$-times punctured spheres. https://arxiv.org/abs/1707.05773

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