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 space parameterizing its geometric actions on Gromov hyperbolic spaces, simultaneously encoding 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. As an application, for fundamental groups of negatively curved manifolds we produce a dictionary between this space of geometric actions and the space of reparameterizations of the geodesic flow. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.