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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Linear slices of the quasi-Fuchsian space of punctured tori


Authors: Yohei Komori and Yasushi Yamashita
Journal: Conform. Geom. Dyn. 16 (2012), 89-102
MSC (2010): Primary 30F40; Secondary 20H10, 57M50
Published electronically: April 4, 2012
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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
Email: yamasita@ics.nara-wu.ac.jp

DOI: http://dx.doi.org/10.1090/S1088-4173-2012-00237-8
PII: S 1088-4173(2012)00237-8
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.