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 C₀^r of some primitive closed geodesic C₀ 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 R₁ of genus g₁ onto another compact Riemann surface R₂ of genus g₂ with g₁≥g₂>1, and C a simple closed geodesic of hyperbolic length l_R1(C) on R₁. In this paper, we give an upper bound for N_R2(f(C)) depending only on g₁, g₂ and l_R1(C).