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

In recently published work Maskit constructs a fundamental domain Dg for the Teichmüller 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. AMS Classification 57M50; 14H55, 30F60