Key points are not available for this paper at this time.
We show that a bijection f: H²² 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 H². 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 H². This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles.
Lo et al. (Mon,) studied this question.