 Equality in hyperbolic Schwarz-Pick contraction

A holomorphic function between hyperbolic Riemann surfaces lifts to a disk map. If the lift is not an automorphism, strict Schwarz-Pick lemma derivative inequality makes every nonconstant geodesic shorter. Equality of distance for one distinct pair therefore forces the map to be a covering map and a local isometry. Covering maps attain equality for sufficiently close pairs in a single lift of a small convex metric ball; they need not preserve all global distances. A punctured disk covers a thrice-punctured sphere by including a cyclic parabolic subgroup into $\Gamma (2)$, so equality is possible in that example.

// Destination: analysis.bigb