Invariants at infinity for asymptotically hyperbolic manifolds

Asymptotically hyperbolic manifolds arise in general relativity to model some isolated gravitational systems.
Although they are described using particular charts, called "charts at infinity", in which the metric tensor decays toward that of the hyperbolic space, one can attach to them a number, called the mass, which is invariant under changes between such charts, therefore having a geometric meaning.
In a joint work with Mattias Dahl and Romain Gicquaud, we aim at classifying all objects which have a similar invariance property at infinity. I will present our method based on the study of the action of the group O+(n,1) of hyperbolic isometries, and the results obtained in the 3-dimensional case.