#### Geometry & Topology Monographs 1 (1998),
The Epstein Birthday Schrift,
paper no. 9, pages 167-180.

## At most 27 length inequalities define Maskit's fundamental domain for the modular group in genus 2

### David Griffiths

**Abstract**.
In recently published work Maskit constructs a fundamental domain D_g for the Teichmueller modular group of a closed surface S of genus g>1. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C_{2g} on S satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in C_{2g} must satisfy. Maskit shows that this set of inequalities is finite and that for genus g=2 there are at most 45. In this paper we improve this number to 27. Each of these inequalities: compares distances between Weierstrass points in the fundamental domain S-C_4 for S; and is realised (as an equality) on one or other of two special surfaces.
**Keywords**.
Fundamental domain, non-dividing geodesic, Teichmueller modular group, hyperelliptic involution, Weierstrass point

**AMS subject classification**.
Primary: 57M50. Secondary: 14H55, 30F60.

**E-print:** `arXiv:math.GT/9811180`

Submitted: 18 November 1997.
Published: 9 November 1998.

Notes on file formats
David Griffiths

Laboratoire de Mathematiques Pures de Bordeaux

Universite Bordeaux 1, 351 cours de la liberation

Talence 33405, Cedex, France

Email: griffith@math.u-bordeaux.fr

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