As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well‐defined since quasi‐isometric CAT(0) spaces can have non‐homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up of rays satisfying one of five hyperbolic‐like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi‐isometry invariant. We use this invariant to distinguish the quasi‐isometry classes of certain right‐angled Coxeter groups.

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 X a CAT(0) space and γ ⊂ X a quasi-geodesic, 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 Xω, any two distinct points in the ultralimit γω 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, originally proven in Behrstock-Druţu [BD], that in CAT(0) spaces Morse quasi-geodesics have at least quadratic divergence.

ABSTRACT. We study the Asymptotic Cone of Teichmüller space equipped with the WeilPetersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as a similar characterization for right angled Artin groups in [3] and for mapping class groups in [7]. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type, thereby answering questions of Behrstock-Druţu [4], Behrstock-Druţu-Mosher [5], and Brock-Masur [15]. In particular, we prove that Teichmüller space of the genus two surface with one boundary component (or puncture) can be uniquely characterized in the following two senses: it is thick of order two, and it has superquadratic yet at most cubic divergence. In addition, we characterize strongly contracting quasi-geodesics in Teichmüller space, generalizing results of BrockMasur-Minsky [17]. As a tool, we develop a complex of separating multicurves, which may be of independent interest.

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