Abstract
Let X be a compact Riemann surface of genus g>1 and let G be a group of biholomorphic mappings on X onto itself. Consider all pairs (X,G). We say that (X,G) is topologically equivalent to (X′,G′) if there exists an o.p. (orientation preserving) homeomorphism h of X onto X′ such that G′h=hG. In this paper, we shall classify the (X,G)'s up to topological equivalence in the case g=4.
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