Deformation spaces of isometric group actions

Geometric group theory studies infinite groups via their isometric actions on metric spaces. By fixing particular classes of actions/spaces, we can recover many classical deformation spaces, such as Teichmuller space (actions on the hyperbolic plane), quasi-Fuchsian space (actions on 3 hyperbolic spaces), Outer space (actions on trees), etc.
In this talk, I will explain the construction of a "universal" deformation space, which for a given finitely generated group parameterizes all its isometric actions that preserve the large-scale geometry of the group. I will mention some properties of this space in the case of surface groups, as well as its connection with other known deformation spaces. Part of this is joint work with Stephen Cantrell.