An exotic deformation of the hyperbolic space

On the one hand, we construct a continuous family of non-isometric proper CAT(-1) spaces on which the isometry group $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathbf{H}}^n)$ of the real hyperbolic $n$-space acts minimally and cocompactly. This provides the first examples of non-standard CAT(0) model spaces for simple Lie groups. On the other hand, we classify all continuous non-elementary actions of ${\rm Isom}($\mathbf{H}$^{n})$ on the infinite-dimensional real hyperbolic space. It turns out that they are in correspondence with the exotic model spaces that we construct.