Hyperbolic models for \(\mathrm{CAT}(0)\) spaces (Q6562851)

Let \(X\) be a \(\mathrm{CAT}(0)\) space. In this paper, the authors introduce two new tools for studying \(X\): curtain and curtain model. A curtain is \(\pi^{-1}_{\alpha}(P)\), where \(\alpha\) is a geodesic of \(X\), \(\pi_{\alpha}\) is the closest-point projection and \(P\) is a length-one subinterval of the interior of \(\alpha\).\N\NGiven \(X\), they use curtains to define a new metric \(\mathrm{D}\) on \(X\) that can be seen the space \(X_{\mathsf{D}}=(X,\mathrm{D})\), which they call the curtain model of \(X\) (a very close analogue of the curve graph). Moreover, the full isometry group \(\operatorname{Isom} X\) acts on \(X_{\mathrm{D}}\), providing a new way to study isometry groups of \(\mathrm{CAT}(0)\) spaces and not only groups with a geometric action on a \(\mathrm{CAT}(0)\) space.\N\NThe first result is Theorem B: There exists some \(\delta\) such that for every \(\mathrm{CAT}(0)\) space \(X\), the curtain model \(X_{\mathrm{D}}\) is \(\delta\)-hyperbolic and \(\operatorname{Isom} X \leq \operatorname{Isom} X_{\mathrm{D}}\). Furthermore, there is some \(k\) such that each \(\mathrm{CAT}(0)\) geodesic is an unparametrised \((1,k)\)-quasigeodesic in \(X_{\mathrm{D}}\).\N\NThe paper under review contains ten other theorems of great interest. These results allow the authors 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 \(\mathrm{CAT}(0)\) space and characterise rank-one isometries both in terms of their action on the curtain model and in terms of curtains. Furthermore, they show that the curtain is universal for WPD (weak proper discontinuity) actions over all groups acting properly on the \(\mathrm{CAT}(0)\) space.