Abstract
We prove the equivalence of various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, in our main theorem we prove that for \(X\) a CAT(0) space and \(\gamma \subset X\) a quasi-geodesic, the following four statements are equivalent: (1) \(\gamma \) is Morse, (2) \(\gamma \) is (b,c)—contracting, (3) \(\gamma \) is strongly contracting, and (4) in every asymptotic cone \(X_{\omega },\) any two distinct points in the ultralimit \(\gamma _{\omega }\) are separated by a cut-point. In particular, this characterization provides a converse to a generalized Morse stability lemma in the CAT(0) setting. In addition, under mild conditions we prove the equivalence of wideness and unconstrictedness in the CAT(0) setting.
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Acknowledgments
I want to express my gratitude to my advisors Jason Behrstock and Walter Neumann for their extremely helpful advice and insights throughout my research, and specifically regarding this paper. I would also like to acknowledge Igor Belegradek, Ruth Charney, and Amey Kaloti for useful conversations and insights regarding arguments and ideas in this paper.
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Sultan, H. Hyperbolic quasi-geodesics in CAT(0) spaces. Geom Dedicata 169, 209–224 (2014). https://doi.org/10.1007/s10711-013-9851-4
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DOI: https://doi.org/10.1007/s10711-013-9851-4