## Kodai Mathematical Journal

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

Hiroshi Yamamoto

#### 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 $l_{R_1} (C)$.

#### Article information

Source
Kodai Math. J., Volume 26, Number 1 (2003), 69-84.

Dates
First available in Project Euclid: 16 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1050496649

Digital Object Identifier
doi:10.2996/kmj/1050496649

Mathematical Reviews number (MathSciNet)
MR2004b:30077

Zentralblatt MATH identifier
1059.30033

#### Citation

Yamamoto, Hiroshi. On the multiplicity of the image of simple closed curves via holomorphic maps between compact Riemann surfaces. Kodai Math. J. 26 (2003), no. 1, 69--84. doi:10.2996/kmj/1050496649. https://projecteuclid.org/euclid.kmj/1050496649