An exotic deformation of the hyperbolic space

TL;DR

This paper introduces a novel family of non-isometric CAT(-1) spaces with minimal and cocompact isometry group actions, providing new models for simple Lie groups and classifying related infinite-dimensional hyperbolic actions.

Contribution

It constructs the first non-standard CAT(0) model spaces for simple Lie groups and classifies all continuous non-elementary actions of hyperbolic isometry groups on infinite-dimensional hyperbolic spaces.

Findings

Constructed a continuous family of exotic CAT(-1) spaces.

Provided the first examples of non-standard CAT(0) model spaces.

Classified all continuous non-elementary actions of hyperbolic isometry groups.

Abstract

On the one hand, we construct a continuous family of non-isometric proper CAT(-1) spaces on which the isometry group ${\rm Isom}(\mathbf{H}^{n})$ 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.