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

David Griffiths

Geometry & Topology Monographs 1 (1998) 167–180

DOI: 10.2140/gtm.1998.1.167

arXiv: math.GT/9811180

Abstract

In recently published work Maskit constructs a fundamental domain Dg for the Teichm\"uller modular group of a closed surface S of genus g≥2. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C2g on S satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in C2g 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\C4 for S; and is realised (as an equality) on one or other of two special surfaces.

Keywords

fundamental domain, non-dividing geodesic, Teichmüller modular group, hyperelliptic involution, Weierstrass point

Mathematical Subject Classification

Secondary: 14H55, 30F60

References
Publication

Received: 18 November 1997
Published: 9 November 1998

Authors
David Griffiths
Laboratoire de Mathematiques Pures de Bordeaux
Universite Bordeaux 1, 351 cours de la liberation
Talence 33405, Cedex, France