Abstract
ABSTRACr. Let SI and S2 be compact Riemann surfaces of genus g > 1. Let T: SI S2 be a continuous map. The map T induces a group homomorphism from the group of divisors on S, into the group of divisors on S2* This group homomorphism will be denoted by the same letter T throughout this paper. If D = I mjp is a divisor on SI, then T(D)= XnI miT(p). If T is a holomorphic or an anti-holomorphic homeomorphism, then T(D) is a principal divisor on S2 if D is a principal divisor on SI. To what extent is the converse of this statement true? The answer to this question is provided by Theorem I of this paper: If T(D) is a principal divisor on S2 whenever D is a principal divisor on SI, then T is either a holomorphic or an anti-holomorphic homeomorphism.
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