Rigidity on horocycles and hypercycles

TL;DR

The paper proves that bijections of the hyperbolic plane preserving horocycles or hypercycles are isometries, extending previous results and linking automorphisms of geometric graphs to earthquake maps and isometries.

Contribution

It extends Jeffers' geodesic rigidity result to all constant curvature curves and characterizes automorphisms of horocycle and hypercycle graphs as induced by earthquake maps and isometries.

Findings

Bijections sending horocycles to horocycles are isometries.

Automorphisms of horocycle and hypercycle graphs are induced by earthquake maps and isometries.

Extension of rigidity results from geodesics to all constant curvature curves.

Abstract

We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ 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.