Mathematics > Metric Geometry
[Submitted on 28 Jul 2022 (v1), last revised 1 Jul 2024 (this version, v4)]
Title:Hyperbolic models for CAT(0) spaces
View PDF HTML (experimental)Abstract:We introduce two new tools for studying CAT(0) spaces: \emph{curtains}, versions of cubical hyperplanes; and the \emph{curtain model}, a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a dichotomy of a rank-rigidity flavour, establish Ivanov-style rigidity theorems for isometries of the curtain model, find isometry-invariant copies of its Gromov boundary in the visual boundary of the underlying CAT(0) space, and characterise rank-one isometries both in terms of their action on the curtain model and in terms of curtains. Finally, we show that the curtain model is universal for WPD actions over all groups acting properly on the CAT(0) space.
Submission history
From: Harry Petyt [view email][v1] Thu, 28 Jul 2022 14:48:18 UTC (298 KB)
[v2] Thu, 8 Dec 2022 13:27:23 UTC (425 KB)
[v3] Thu, 6 Jul 2023 12:59:03 UTC (559 KB)
[v4] Mon, 1 Jul 2024 10:12:22 UTC (454 KB)
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