Variety of Actions on a Hyperbolic Space

In this talk, we will consider all isometric actions of a group on a hyperbolic space $H^n$ where dimension n is arbitrary and possibly infinite. By grouping equivalent representations if they are "homothetic" (a notion generalizing conjugation), we will see that for a finitely generated group, this variety of actions is compact. We will also see rigid examples where the variety of non-elementary actions is reduced to a single point, for example, for the group of automorphisms of a tree or the group $PGL_2(K)$ where $K$ is a valued field.