Rigidity for Quasi-Mobius group actions
  Suppose G is a hyperbolic group whose boundary has topological dimension k. If the boundary is quasisymmetrically homeomorphic to an Ahlfors k-regular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group of hyperbolic (k+1)-space. 
Mario Bonk, Bruce Kleiner