On the hyperbolic distance of n-times punctured spheres

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 where A⊂ ℂ^ is 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 d_X onapunctured 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 a Lipschitz constant depending only on Q(A).