On homeomorphisms preserving principal divisors

Abstract:

Let ${S_1}$ and ${S_2}$ be compact Riemann surfaces of genus $g > 1$. Let $\tau :{S_1} \to {S_2}$ be a continuous map. The map $\tau$ induces a group homomorphism from the group of divisors on ${S_1}$ into the group of divisors on ${S_2}$. This group homomorphism will be denoted by the same letter $\tau$ throughout this paper. If $D = \sum _{i = 1}^n{m_i}{p_i}$ is a divisor on ${S_1}$, then $\tau (D) = \sum _{i = 1}^n{m_i}\tau ({p_i})$. If $\tau$ is a holomorphic or an anti-holomorphic homeomorphism, then $\tau (D)$ is a principal divisor on ${S_2}$ if $D$ is a principal divisor on ${S_1}$. To what extent is the converse of this statement true? The answer to this question is provided by Theorem 1 of this paper: If $\tau (D)$ is a principal divisor on ${S_2}$ whenever $D$ is a principal divisor on ${S_1}$, then $\tau$ is either a holomorphic or an anti-holomorphic homeomorphism.