Higher rank hyperbolicity

• 成果类型: Article
• 署名作者: Kleiner, Bruce; Lang, Urs
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00955-w
• 发表日期: 2020
• 页码: 597-664
• 关键词: quasi-isometries; symmetric-spaces; morse lemma; geodesics; MANIFOLDS; RIGIDITY; BOUNDARIES; CURVATURE; geometry; currents

摘要

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 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.