Hyperbolic quasi-geodesics in CAT(0) spaces

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 $γ$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $γ$ is Morse, (ii) $γ$ is (b,c)--contracting, (iii), $γ$ is strongly contracting, and (iv) in every asymptotic cone $X_ω,$ any two distinct points in the ultralimit $γ_ω$ 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.