On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces (Q1420221)

Given a compact Riemann surface \(R\) and a nontrivial closed curve \(C\) on \(R\), the multiplicity of \(C\) is defined to be the positive integer \(r= N_R(C)\) such that \(C\) is freely homotopic to the \(r\)-fold iterate \(C^r_0\), for some primitive closed geodesic \(C_0\) on \(R\). The author proves that if \(f\) is a holomorphic map between the compact Riemann surfaces \(R_1\) and \(R_2\), and \(g_1= \text{genus}(R_1)\geq g_2=\text{genus}(R_2)\geq 2\), and \(l_{R_1}(C)\) is the hyperbolic length of a simple closed geodesic \(C\) of \(R_1\), then \(N_{R_2}(f (C))\) is upperly bounded by a function depending only on \(l_{R_1}(C)\), \(g_1\) and \(g_2\). The author shows also that there is no such upper bound involving just the genera \(g_1\) and \(g_2\).