Almost homogeneous manifolds with boundary
  Let $\rho_0$ be an action of a Lie group on a manifold with boundary that is transitive on the interior. We study the set of actions that are topologically conjugate to $\rho_0$, up to smooth or analytic change of coordinates. We show that in many cases, including the compactifications of negatively curved symmetric spaces, this set is infinite. 
Benoit Kloeckner (IF)