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In recently published work Maskit constructs a fundamental domain D_g forthe Teichmueller modular group of a closed surface S of genus g>1. Maskit'stechnique 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 setof length inequalities that the geodesics in C_{2g} must satisfy. Maskit showsthat this set of inequalities is finite and that for genus g=2 there are atmost 45. In this paper we improve this number to 27. Each of theseinequalities: compares distances between Weierstrass points in the fundamentaldomain S-C_4 for S; and is realised (as an equality) on one or other of twospecial surfaces.

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