Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 32, 2007, 179-198

# ON THE SHAPE OF BERS-MASKIT SLICES

## Yohei Komori and Jouni Parkkonen

Osaka City University, Department of Mathematics

3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585 Japan;
komori 'at' sci.osaka-cu.ac.jp

University of Jyväskylä, Department of Mathematics and Statistics

P.O. Box 35, 40014 University of Jyväskylä, Finland;
parkkone 'at' maths.jyu.fi

**Abstract.**
We consider complex
one-dimensional Bers-Maskit slices through the deformation space of
quasifuchsian groups which uniformize a pair of punctured tori.
In these slices, the pleating locus on one of the components
of the convex hull boundary of the quotient three-manifold has
constant rational pleating and constant hyperbolic length.
We show that the boundary of such a slice is a
Jordan curve which is cusped at a countable dense set of points.
We will also show that the slices are not vertically convex, proving
the phenomenon observed numerically by Epstein, Marden and Markovic.

**2000 Mathematics Subject Classification:**
Primary 30F40, 30F60, 57M50.

**Key words:**
Kleinian groups, punctured torus groups,
Teichmüller space, pleating coordinates, end invariants.

**Reference to this article:** Y. Komori and J. Parkkonen:
On the shape of Bers-Maskit slices.
Ann. Acad. Sci. Fenn. Math. 32 (2007), 179-198.

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Copyright © 2007 by Academia Scientiarum Fennica