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

TL;DR

This paper introduces a new quantity related to cross ratios for n-times punctured spheres and constructs a Lipschitz equivalent distance function to the hyperbolic metric, facilitating easier analysis of hyperbolic geometry.

Contribution

It defines a new invariant $Q(A)$ for punctured spheres and proposes a method to construct a Lipschitz equivalent distance function to the hyperbolic metric.

Findings

$Q(A)$ is comparable to the systole of the surface.

Constructed distance $d_X$ is Lipschitz equivalent to hyperbolic distance $h_X$.

Lipschitz constant depends only on $Q(A)$.

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 $Q(A).$