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

TL;DR

This paper extends the classical Maskit slice for 3D Kleinian groups to 4D, exploring the deformation space of punctured torus groups in 3-sphere Möbius transformations and revealing its geometric structure.

Contribution

It introduces a higher-dimensional deformation space that includes known Maskit slices as slices, providing new insights into Kleinian group deformations in 4D.

Findings

Deformation space realized as a 3D domain containing known slices

Maskit slice of punctured torus groups appears as a slice in the space

Maskit slice of four-punctured sphere groups also appears as a slice

Abstract

Let $\Gamma$ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of $\Gamma$ in the group of M\"{o}bius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations $\Gamma'$ of $\Gamma$ in the group of M\"{o}bius 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 of punctured torus groups as a slice through a plane. Furthermore, we will show that the space also contains the Maskit slice of fourth-punctured sphere groups as a slice through another plane. Some of another slices of the space will be also studied.