A Morse geodesic is defined by the property that quasi-geodesics with endpoints on the geodesic remain a bounded distance from the geodesic. Contracting geodesics on the other hand are defined by the property that metric balls disjoint from the geodesic have uniformly bounded nearest point projections onto the geodesic. In hyperbolic space all geodesics are both Morse and contracting, while in Euclidean space there are no Morse or contracting geodesics. More generally, throughout the literature on geometric group theory, one is frequently interested in understanding families of (quasi-)geodesics which admit hyperbolic behavior, such as being Morse or contracting. In this talk, I prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. In particular, as a corollary this resolves an open question of Drutu-Mozes-Sapir regarding a converse to their characterization of Morse quasi-geodesics in the asymptotic cone.
←Back to the seminar home page