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Conformal Geometry and Dynamics

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


Authors: Yoshiaki Araki and Kentaro Ito
Journal: Conform. Geom. Dyn. 12 (2008), 199-226
DOI: https://doi.org/10.1090/S1088-4173-08-00187-2
Published electronically: December 15, 2008
MathSciNet review: 2466017
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Abstract | References | Additional Information

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.


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Additional Information

Yoshiaki Araki
Affiliation: Synclore Corporation, Hakuyo Building, 3-10 Nibancho Chiyoda-ku, Tokyo 102-0084, Japan
Email: yoshiaki.araki@synclore.com

Kentaro Ito
Affiliation: Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Email: itoken@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/S1088-4173-08-00187-2
Received by editor(s): April 1, 2008
Published electronically: December 15, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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