Teichmüller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary hyperbolic group we consider a metric space that parameterizes its geometric actions on Gromov hyperbolic spaces. Even in the surface group case, this space turns out to be much larger than Teichmüller space, and we can find points induced by negatively curved Riemannian metrics, Anosov representations, random walks, geometric cubulations, etc. In particular, I will discuss how Green metrics (those encoding admissible random walks on the group) are dense in this space. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.