arXiv: math.GT/9811180

Abstract

In recently published work Maskit constructs a fundamental domain ${\mathcal{D}}_g$ for the Teichmüller modular group of a closed surface $\mathcal{S}$ of genus $g\ge 2$. Maskit’s technique is to demand that a certain set of $2g$ non-dividing geodesics ${\mathcal{C}}_{2g}$ on $\mathcal{S}$ satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in ${\mathcal{C}}_{2g}$ 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 $\mathcal{S}\setminus {\mathcal{C}}_4$ for $\mathcal{S}$; and is realised (as an equality) on one or other of two special surfaces.

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