Hyperbolicity via Geodesic Stability

Abstract

A geodesic $g$ is Morse, for every $L \geq 1, A \geq 0$ there exists a $C=C_g(L,A)$ such that any $(L,A)$-quasi-geodesic connecting two points on $g$ stays $C$-close to $g$. The Morse lemma implies that in a hyperbolic space every geodesic is Morse. Here we prove the converse: If a homogeneous proper geodesic space is such that for every geodesic $g$ and every $L\geq 1, A \geq 0$ there exists a constant $C=C_g(L,A)$ such that any $(L,A)$-quasi-geodesic between any two points on $g$ stays $C$-close, then the space is hyperbolic. This applies in particular to infinite groups in which all geodesics are Morse.