Explicit equivalences between CAT(0) hyperbolic type geodesics

TL;DR

This paper establishes explicit equivalences between different hyperbolic-like properties of quasi-geodesics in CAT(0) spaces, aiding the understanding of their geometric structure and boundary behavior.

Contribution

It provides explicit quantifier bounds for the equivalence of slimness, Morse, and contracting properties of quasi-geodesics in CAT(0) spaces, and applies this to quasi-isometries.

Findings

Equivalence of four hyperbolic properties with explicit quantifiers.

Quasi-isometries preserve strong contracting properties of geodesics.

Supports technical development of Charney's contracting boundary.

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.