## 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 ${\mathcal L}_c$.

## References

- Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada, and Yasushi Yamashita,
*Punctured torus groups and 2-bridge knot groups. I*, Lecture Notes in Mathematics, vol. 1909, Springer, Berlin, 2007. MR**2330319** - R. D. Canary, D. B. A. Epstein, and P. L. Green,
*Notes on notes of Thurston [MR0903850]*, Fundamentals of hyperbolic geometry: selected expositions, London Math. Soc. Lecture Note Ser., vol. 328, Cambridge Univ. Press, Cambridge, 2006, pp. 1–115. With a new foreword by Canary. MR**2235710** - Linda Keen and Caroline Series,
*Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori*, Topology**32**(1993), no. 4, 719–749. MR**1241870**, DOI 10.1016/0040-9383(93)90048-Z - Linda Keen and Caroline Series,
*How to bend pairs of punctured tori*, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359–387. MR**1476997**, DOI 10.1090/conm/211/02830 - Linda Keen and Caroline Series,
*Pleating invariants for punctured torus groups*, Topology**43**(2004), no. 2, 447–491. MR**2052972**, DOI 10.1016/S0040-9383(03)00052-1 - Yohei Komori,
*On the boundary of the Earle slice for punctured torus groups*, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001) London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 293–304. MR**2044555**, DOI 10.1017/CBO9780511542817.014 - Yohei Komori and Jouni Parkkonen,
*On the shape of Bers-Maskit slices*, Ann. Acad. Sci. Fenn. Math.**32**(2007), no. 1, 179–198. MR**2297885** - Yohei Komori and Caroline Series,
*Pleating coordinates for the Earle embedding*, Ann. Fac. Sci. Toulouse Math. (6)**10**(2001), no. 1, 69–105 (English, with English and French summaries). MR**1928990**, DOI 10.5802/afst.985 - Christos Kourouniotis,
*Complex length coordinates for quasi-Fuchsian groups*, Mathematika**41**(1994), no. 1, 173–188. MR**1288062**, DOI 10.1112/S0025579300007270 - Katsuhiko Matsuzaki and Masahiko Taniguchi,
*Hyperbolic manifolds and Kleinian groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR**1638795** - Curtis T. McMullen,
*Complex earthquakes and Teichmüller theory*, J. Amer. Math. Soc.**11**(1998), no. 2, 283–320. MR**1478844**, DOI 10.1090/S0894-0347-98-00259-8 - Robert Meyerhoff,
*A lower bound for the volume of hyperbolic $3$-manifolds*, Canad. J. Math.**39**(1987), no. 5, 1038–1056. MR**918586**, DOI 10.4153/CJM-1987-053-6 - Yair N. Minsky,
*The classification of punctured-torus groups*, Ann. of Math. (2)**149**(1999), no. 2, 559–626. MR**1689341**, DOI 10.2307/120976 - Jean-Pierre Otal,
*Sur le coeur convexe d’une variété hyperbolique de dimension 3*, Preprint. - John R. Parker and Jouni Parkkonen,
*Coordinates for quasi-Fuchsian punctured torus spaces*, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 451–478. MR**1668328**, DOI 10.2140/gtm.1998.1.451 - Ser Peow Tan,
*Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures*, Internat. J. Math.**5**(1994), no. 2, 239–251. MR**1266284**, DOI 10.1142/S0129167X94000140 - William P. Thurston,
*The geometry and topology of three-manifolds*, Princeton lecture notes, 1980.

## 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
- © 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

Dedicated: This paper is dedicated to Professor Caroline Series on the occasion of her 60th birthday.