The Space of 4-dimensional Kleinian Punctured Torus Groups without Screw Parabolic Transformations

Let Γ be a 3-dimensional Kleinian punctured torus group with accidental parabolic transformations. The deformation space of Γ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice M1,1 of punctured torus groups. In this paper, we study deformations Γ of Γ in the group of Möbius transformations on the 3-sphere such that Γ does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of 3-space R, which contains the Maskit slice M1,1 as a slice through a plane. Furthermore, we will show that the space also contains the Maskit slice M0,4 of fourth-punctured sphere groups as a slice through another plane. Some of another slices of the space will be also studied.