An extension of the Maskit slice for $4$-dimensional Kleinian groups

Abstract:

Let $\Gamma$ be a $3$-dimensional Kleinian punctured torus group with accidental parabolic transformations. The deformation space of $\Gamma$ in the group of Möbius transformations on the $2$-sphere is well known as the Maskit slice ${\mathcal {M}}_{1,1}$ of punctured torus groups. In this paper, we study deformations $\Gamma ’$ of $\Gamma$ in the group of Möbius transformations on the $3$-sphere such that $\Gamma ’$ does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of $3$-space $\mathbb {R}^3$, which contains the Maskit slice ${\mathcal {M}}_{1,1}$ as a slice through a plane. Furthermore, we will show that the space also contains the Maskit slice $\mathcal {M}_{0,4}$ of fourth-punctured sphere groups as a slice through another plane. Some of the other slices of the space will be also studied.