Higher rank hyperbolicity

Autor: Urs Lang, Bruce Kleiner
Rok vydání: 2020
Předmět:
Zdroj: Inventiones mathematicae. 221:597-664
ISSN: 1432-1297
0020-9910
DOI: 10.1007/s00222-020-00955-w
Popis: 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 CAT(0) spaces of asymptotic rank $n$ extends to a class of $(n-1)$-cycles in the Tits boundaries.
Comment: 59 pages. Visual metrics added, minor improvements
Databáze: OpenAIRE
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