Higher rank hyperbolicity

Bruce Kleiner, Urs Lang

Research output: Contribution to journalArticlepeer-review


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≥ 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 rn 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 CAT (0) spaces of asymptotic rank n extends to a class of (n- 1) -cycles in the Tits boundaries.

Original languageEnglish (US)
Pages (from-to)597-664
Number of pages68
JournalInventiones Mathematicae
Issue number2
StatePublished - Aug 1 2020

ASJC Scopus subject areas

  • General Mathematics


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