The complexity of Riemann surfaces and the Hurwitz existence problem

Authors

  • Aldo-Hilario Cruz-Cota Grand Valley State University
  • Teresita Ramirez-Rosas Grand Valley State University

Keywords:

Complexity of Riemann surfaces, branched covers between Riemann surfaces, Hurwitz existence problem

Abstract

\noindent The \emph{complexity} of a branched cover of a Riemann surface $M$ to the Riemann sphere $S^2$ is defined as its degree times the hyperbolic area of the complement of its branching set in $S^2$. The \emph{complexity} of $M$ is defined as the infimum of the complexities of all branched covers of $M$ to $S^2$. We prove that if $M$ is a (connected, closed, orientable) Riemann surface of genus $g \geq 1$, then its complexity equals $2\pi(m_{\text{min}}+2g-2)$, where $m_{\text{min}}$ is the minimum total length of a branch datum realizable by a branched cover $p \colon M \to S^2$. DOI: 10.1017/S0004972712000469

Published

2012-12-17

Issue

Section

Articles