Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space

It is known that a geometrically finite Kleinian group is quasiconformally stable. We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of screw parabolic isometries in dimension 4. We also prove that a Fuchsian thrice-punctured sphere group has a large deformation space in hyperbolic 4-space which is in contrast to lower imensions where the Fuchsian thrice-punctured sphere group has a trivial deformation space. However, the thrice-punctured
sphere group is still quasiconformally rigid.