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Conformal Geometry and Dynamics

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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
MathSciNet review: 2904566
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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

Yasushi Yamashita
Affiliation: Department of Information and Computer Sciences, Nara Women’s University, 630-8506 Nara, Japan
MR Author ID: 310816

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.