An exotic deformation of the hyperbolic space

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