Convex-cocompact representations of finitely generated groups into the isometry group of the infinite-dimensional hyperbolic space.
- David Xu (Université Paris Saclay, France)
Abstract
The Anosov representations of a hyperbolic group into a rank one Lie group are exactly those that are convex-cocompact. They have some important properties, for example, a representation of a finitely generated group into PO(1,n) is convex-cocompact if and only if its orbit maps are quasi-isometric embeddings into the associated symmetric space, the n-dimensional hyperbolic space. Moreover, the set of convex-cocompact representations into PO(1,n) is open in the space of all the representations in PO(1,n) (this is called the stability of convex-cocompact representations). In this talk, we will be interested in some infinite-dimensional representations, namely representations into PO(1,infinity), where the same properties for convex-cocompact representations remain true. In particular, the stability property allows the use of deformation techniques to produce other convex-cocompact representations of a given group.