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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Linear slices of the quasi-Fuchsian space of punctured tori
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by Yohei Komori and Yasushi Yamashita PDF
Conform. Geom. Dyn. 16 (2012), 89-102 Request permission

Abstract:

After fixing a marking $(V, W)$ of a quasi-Fuchsian punctured torus group $G$, the complex length $\lambda _V$ and the complex twist $\tau _{V,W}$ parameters define a holomorphic embedding of the quasi-Fuchsian space ${\mathcal Q}{\mathcal F}$ of punctured tori into $\mathbf {C}^2$. It is called the complex Fenchel-Nielsen coordinates of $\mathcal {QF}$. For $c \in \mathbf {C}$, let $\mathcal {Q}_{\gamma , c}$ be the affine subspace of $\mathbf {C}^2$ defined by the linear equation $\lambda _V=c$. Then we can consider the linear slice ${\mathcal L}_c$ of $\mathcal {QF}$ by $\mathcal {QF} \cap \mathcal {Q}_{\gamma , c}$ which is a holomorphic slice of $\mathcal {QF}$. For any positive real value $c$, ${\mathcal L}_c$ always contains the so-called Bers-Maskit slice ${\mathcal BM}_{\gamma , c}$ defined in [Topology 43 (2004), no. 2, 447–491]. In this paper we show that if $c$ is sufficiently small, then ${\mathcal L}_c$ coincides with ${\mathcal BM}_{\gamma , c}$ whereas ${\mathcal L}_c$ has other components besides ${\mathcal BM}_{\gamma , c}$ when $c$ is sufficiently large. We also observe the scaling property of ${\mathcal L}_c$.
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Additional Information
  • Yohei Komori
  • Affiliation: Advanced Mathematical Institute and Department of Mathematics, Osaka City University, 558-8585, Osaka, Japan
  • Email: komori@sci.osaka-cu.ac.jp
  • Yasushi Yamashita
  • Affiliation: Department of Information and Computer Sciences, Nara Women’s University, 630-8506 Nara, Japan
  • MR Author ID: 310816
  • Email: yamasita@ics.nara-wu.ac.jp
  • Received by editor(s): November 7, 2011
  • Published electronically: April 4, 2012
  • Additional Notes: The first author was partially supported by Grant-in-Aid for Scientific Research (C) (19540194), Ministry of Education, Science and Culture of Japan

  • Dedicated: This paper is dedicated to Professor Caroline Series on the occasion of her 60th birthday.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 16 (2012), 89-102
  • MSC (2010): Primary 30F40; Secondary 20H10, 57M50
  • DOI: https://doi.org/10.1090/S1088-4173-2012-00237-8
  • MathSciNet review: 2904566