Graded Riemann surfaces

There has been a lot of activity directed at describing super Riemann surfaces and the super Teichmuller spaces that classify them. Most descriptions use a subcategory of G∞-supermanifolds in which the coordinate charts have a particularly simple form (''de Witt'' supermanifolds). This paper considers the more restrictive case of Riemann surfaces in the category of graded manifolds. The gain in doing this is the evident role of the double cover Sl(2, C) of the Lorentz group in the classification of graded Riemann surfaces. The results are as follows. 1. The group Sl(2, C) plays the role of the Mobius group as the automorphism group of the algebra of ''graded rational functions''. 2. All graded Riemann surfaces occur as quotients of ''simply connected'' graded Riemann surfaces by discrete subgroups of Sl(2, C). (orig.)