Hyperbolic models for CAT(0) spaces

TL;DR

This paper introduces new geometric tools called curtains and the curtain model for CAT(0) spaces, providing insights into their structure, isometry groups, and boundary behaviors, with applications to rigidity and universality of group actions.

Contribution

It develops curtains and the curtain model as novel tools for analyzing CAT(0) spaces, leading to new rigidity theorems and a universal framework for group actions.

Findings

Proves a rank-rigidity type dichotomy for CAT(0) spaces.

Establishes Ivanov-style rigidity theorems for the curtain model.

Shows the curtain model is universal for WPD actions on CAT(0) spaces.

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.