By a rigidity theorem of Karpelevich (1953) and Mostow (1955), given two finite integers n<m, it is a classical fact that representations of the isometry group of n-dimensional hyperbolic space to that of m-dimensional hyperbolic space must preserves the image of a totally geodesic embedding. But this does not hold in infinite dimensions. An evidence is the existence of embedding homothety classes of convex bodies into infinite dimensional hyperbolic spaces. In this talk, I will talk about the stochastic construction of this embedding and how it is related to exotic representations of the isometry group of hyperbolic plane in the isometry of the infinite dimensional hyperbolic space. Some of this work is joint with François Fillastre and David Xu.
