Rigidity on horocycles and hypercycles

Abstract

We show that a bijection $f:{\mathbb{H}}^2\to {\mathbb{H}}^2$ of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in ${\mathbb{H}}^2$. We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of ${\mathbb{H}}^2$. This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles.