Hyperbolic quasi-geodesics in CAT(0) spaces

TL;DR

This paper characterizes Morse quasi-geodesics in CAT(0) spaces as those that are contracting, establishing equivalences among several properties and providing a converse to the Morse stability lemma.

Contribution

It proves the equivalence of Morse and contracting properties for quasi-geodesics in CAT(0) spaces and offers a new proof of quadratic divergence for Morse quasi-geodesics.

Findings

Morse quasi-geodesics are equivalent to contracting quasi-geodesics in CAT(0) spaces.

A converse to the Morse stability lemma is established in the CAT(0) setting.

Morse quasi-geodesics exhibit at least quadratic divergence.

Abstract

We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $\gamma$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $\gamma$ is Morse, (ii) $\gamma$ is (b,c)--contracting, (iii), $\gamma$ is strongly contracting, and (iv) in every asymptotic cone $X_{\omega},$ any two distinct points in the ultralimit $\gamma_{\omega}$ are separated by a cutpoint. As a corollary, we provide a converse to the usual Morse stability lemma in the CAT(0) setting. In addition, as a warm up we include an alternative proof of the fact that in CAT(0) spaces Morse quasi-geodesics have at least quadratic divergence, originally proven by Behrstock-Drutu.