An exotic deformation of the hyperbolic space
Let \(H^n\) be the real hyperbolic space of dimension \(n\) (in this paper \(n \geq 2\)). A \(\mathrm{CAT}(k)\)-space (\(k \in \mathbb R\)) is a metric geodesic space in which every geodesic triangle is thinner than the corresponding `model triangles' in a standard space of constant curvature \(k\). For \(k=-1\) the model space is \(H^n\).NEWLINENEWLINEIt is proved in this paper that there is a continuous family \(\{ C_t\}_{0<t \leq 1}\), of proper \(\mathrm{CAT}(-1)\) spaces equipped with a continuous family of cocompact minimal isometric \(\mathrm{Isom}(H^n) = PO(1,n)\)-actions, corresponding to \(C_1 = H ^n\). The spaces \(C_t\) are mutually non-isometric (even after rescaling) and \(\mathrm{Isom}(C_t) \simeq \mathrm{Isom}(H^n)\). So these \(C_t\) may be considered as exotic hyperbolic spaces and give the first examples of nonstandard \(\mathrm{CAT}(0)\) model spaces for simple Lie groups. It contrasts with the fact that if \(X\) is a non-compact \(\mathrm{CAT}(0)\) space on which some group \(G\) acts continuously and cocompactly by isometries, and if X is geodesically complete, then X is isometric to the model space (possibly rescaled).NEWLINENEWLINEThe spaces \(C_t\) are constructed as canonical minimal invariant convex subsets in the infinite-dimensional hyperbolic space \(H^\infty\) for a family of representations of \(\mathrm{Isom}(H^n)\) on \(H^\infty \). These representations are deformations of the standard embedding of \(\mathrm{Isom}(H^n)\) into \(\mathrm{Isom}(H^\infty )\) and so it is possible to consider \(C_t\) as a deformation of the standard image \(C_1 = H^n \subset H^\infty \) arising as the limit of the composed standard embeddings \(H^n \subseteq H^{n+1} \subseteq H^{n+2} \dots\). A large part of this article is devoted to the study of representations of \(\mathrm{Isom}(H^n)\) \(X^0\) in \(\mathrm{Isom}(H^\infty )\).
scientific article; zbMATH DE number 6365640