On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces

Abstract

Every non-trivial closed curve $C$ on a compact Riemann surface $R$ is freely homotopic to the $r$-fold iterate ${C_0}^r$ of some primitive closed geodesic $C_0$ on $R$. We call $r$ the multiplicity of $C$, and denote it by $N_{R} (C)$. Let $f$ be a non-constant holomorphic map of a compact Riemann surface $R_1$ of genus $g_1$ onto another compact Riemann surface $R_2$ of genus $g_2$ with $g_1 \geq g_2 > 1$, and $C$ a simple closed geodesic of hypebolic length $l_{R_1} (C)$ on $R_1$. In this paper, we give an upper bound for $N_{R_2} (f(C))$ depending only on $g_1$, $g_2$ and $l_{R_1} (C)$.