Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space

Rigidity and Stability for Isometry Groups in Hyperbolic 4-Space by Youngju Kim Advisor: Professor Ara Basmajian 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. These isometries are topologically conjugate to strictly parabolic isometries. However, we show that screw parabolic isometries are not quasiconformally conjugate to strictly parabolic isometries. In addition, we show that two screw parabolic isometries are generically not quasiconformally conjugate to each other. We also give some geometric properties of a hyperbolic 4-manifold related to screw parabolic isometries. A Fuchsian thrice-punctured sphere group has a trivial deformation space in hyperbolic 3-space. Thus, it is quasiconformally rigid. We prove that the Fuchsian thrice-punctured sphere group has a large deformation space in hyperbolic 4-space which is in contrast to lower dimensions. In particular, we prove that there is a 2-dimensional parameter space in the deformation space of the Fuchsian thrice-punctured sphere group for which the deformations are all geometrically finite and generically quasiconformally distinct. In contrast, the thrice-punctured sphere group is still quasiconformally rigid in hyperbolic 4-space.