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

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

1998-11-09

arxiv.org/abs/math/9811180v1

Geom. Topol. Monogr. 1 (1998), 167-180

Mathematics

Geometric Topology

14 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper9.abs.html

Scientific paper

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.

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