Explicit equivalences between CAT(0) hyperbolic type geodesics

Abstract

We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and $\gamma$ a quasi-geodesic, the following four statements are equivalent and moreover the quantifiers in the equivalences are explicit: (i) $\gamma$ is S-Slim, (ii) $\gamma$ is M-Morse, (iii) $\gamma$ is (b,c)-contracting, and (iv) $\gamma$ is C-strongly contracting. In particular, this explicit equivalence proves that for $f$ a (K,L)-quasi-isometry between CAT(0) spaces, and $\gamma$ a C-strongly contracting (K',L')-quasi-geodesic, then $f(\gamma)$ is a C'(C,K,L,K',L')-strongly contracting quasi-geodesic. This result is necessary for a key technical point with regard to Charney's contracting boundary for CAT(0) spaces.