Kernels of Möbius type

Given a Möbius structure or cross-ratio defined on a set, we will be provide necessary and sufficient conditions for this cross-ratio to be induced by an embedding in the visual boundary of a real or complex hyperbolic space.

This is not the first effort in this direction. The idea of studying hyperbolic representations through kernels (of hyperbolic type) is largely based on the work of Pierre Py and Nicolas Monod.The kernels of hyperbolic type "model" the maps of a set into a real or complex hyperbolic space. In their equivariant version those kernels can be used to study the hyperbolic representations through the orbit maps of points contained in the hyperbolic space. With the kernels of Möbius type, introduced in this talk, and their equivariant version, the study of hyperbolic representations is done by looking at the induced action by Möbius transformations on the visual boundary of the hyperbolic spaces. This presents some advantages due to the dynamic properties of such actions.

With the language of the kernels of Möbius type at hand, one can give necessary and sufficient conditions for two irreducible representations of a group on a real or complex hyperbolic space to be equivalent. Kim, developing on techniques by Otal, showed that in the locally compact case, the marked length of an irreducible representation is a complete invariant.Using the kernels of Möbius type is straightforward to generalize that result for any hyperbolic representation, possibly on an infinite-dimensional hyperbolic space.