TY - JOUR
T1 - Higher rank hyperbolicity
AU - Kleiner, Bruce
AU - Lang, Urs
N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00222-020-00955-w
DO - 10.1007/s00222-020-00955-w
M3 - Article
AN - SCOPUS:85079758900
SN - 0020-9910
VL - 221
SP - 597
EP - 664
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 2
ER -