  The solution space of three-dimensional asymptotically anti-de Sitter or flat
Einstein gravity is given by the coadjoint representation of two copies of the
Virasoro group in the former and the centrally extended BMS$_3$ group in the
latter case. Dynamical actions that control these solution spaces are usually
constructed by starting from the Chern-Simons formulation and imposing all
boundary conditions. In this note, an alternative route is followed. We study
in detail how to derive these actions from a group-theoretical viewpoint by
constructing geometric actions for each of the coadjoint orbits, including the
appropriate Hamiltonians. We briefly sketch relevant generalizations and
potential applications beyond three-dimensional gravity.
