Hyperbolicity via Geodesic Stability

TL;DR

This paper establishes that a proper geodesic space where all geodesics are Morse must be hyperbolic, providing a converse to the Morse lemma and characterizing hyperbolicity via geodesic stability.

Contribution

It proves that the Morse property for all geodesics characterizes hyperbolic spaces among homogeneous proper geodesic spaces.

Findings

Spaces with all geodesics Morse are hyperbolic.

The Morse property characterizes hyperbolicity in homogeneous proper geodesic spaces.

Application to infinite groups with all geodesics Morse.

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.