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Higher rank hyperbolicity

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Abstract

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank \(n \ge 2\) in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) n-cycles of \(r^n\) volume growth; prime examples include n-cycles associated with n-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper \({\text {CAT}}(0)\) spaces of asymptotic rank n extends to a class of \((n-1)\)-cycles in the Tits boundaries.

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  1. Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY, 10012, USA

    Bruce Kleiner

  2. Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland

    Urs Lang

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  1. Bruce Kleiner
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BK was supported by NSF Grant DMS-1711556, and a Simons Collaboration grant.

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Kleiner, B., Lang, U. Higher rank hyperbolicity. Invent. math. 221, 597–664 (2020). https://doi.org/10.1007/s00222-020-00955-w

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