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 $$n \ge 2$$ 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 $$r^n$$ 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 $${\text {CAT}}(0)$$ CAT ( 0 ) spaces of asymptotic rank n extends to a class of $$(n-1)$$ ( n - 1 ) -cycles in the Tits boundaries.

中文翻译：

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高阶双曲线

双曲度量空间的大规模几何表现出许多独特的特征，例如准测地线（莫尔斯引理）的稳定性、可见性属性以及由准等距引起的视觉边界之间的同胚。我们在渐近意义上证明了秩为 $$n \ge 2$$ n ≥ 2 的空间的许多非常相似的结果，在一些让人想起非正曲率的弱假设下。为此，我们用 $$r^n$$rn 体积增长的准最小化（局部有限）n 周期替换准测地线；主要例子包括与 n -quasiflats 相关的 n -cycles。解决渐近高原问题并为此类循环在无穷远处产生独特的切锥，