I will present the "space of metrics of a group'', a metric space parameterizing the geometric actions of
an arbitrary hyperbolic group on Gromov hyperbolic spaces. Even for the surface group case, this space is much larger than
the classical Teichmüller space, encompassing negatively curved Riemannian metrics, geodesic currents,
random walks, and more. I will discuss how Green metrics—those associated with admissible random walks on the group—are dense in
the space of metrics. This is joint work in progress with Stephen Cantrell and Eduardo Reyes.