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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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An extension of the Maskit slice for $4$-dimensional Kleinian groups
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by Yoshiaki Araki and Kentaro Ito
Conform. Geom. Dyn. 12 (2008), 199-226
Published electronically: December 15, 2008


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.
Bibliographic Information
  • Yoshiaki Araki
  • Affiliation: Synclore Corporation, Hakuyo Building, 3-10 Nibancho Chiyoda-ku, Tokyo 102-0084, Japan
  • Email:
  • Kentaro Ito
  • Affiliation: Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
  • Email:
  • Received by editor(s): April 1, 2008
  • Published electronically: December 15, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 12 (2008), 199-226
  • DOI:
  • MathSciNet review: 2466017