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

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

2007-07-17

arxiv.org/abs/0707.2427v3

Conform. Geom. Dyn. 12 (2008), 199-226

Mathematics

Geometric Topology

34 pages, 11 figures. v3: The title is changed and some typo are fixed. To appear in Conform. Geom. dyn. The paper including m

Scientific paper

10.1090/S1088-4173-08-00187-2

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.

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