In this paper, we construct an explicit fundamental domain tor the action of the Teichmtillcr modular group on the Teichmiiller space of closed surfaces of genus o 2: 2. Fundamental domains for the modular groups of surfaces of low signature have been obtained by Keen [K], Rosenbcrger [R] and others, including this author [Ml, M2]. Fundamental domains for modular groups of hyperbolic surfuccs of finite area with punctures have been obtained by Penuer |P], His technique makes essential use of the punctures, while our technique seems to work only for closed surfaces; There is also a related fundamental domain for closed surfaces of genus 2 obtained by Semmler [S]; his method uses dividing geodesies, while we use non-dividing geodesies, (Throughout this paper, the word "geodesic" refers to a closed geodesic, which we will usually regard as being unoricnted.
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