The simple complexity of a Riemann surface

Abstract

\noindent Given a Riemann surface $M$, the \emph{complexity} of a branched cover of $M$ to the Riemann sphere $S^2$, of degree $d$ and with branching set of cardinality $n \geq 3$, is defined as $d$ times the hyperbolic area of the complement of its branching set in $S^2$. A branched cover $p \colon M \to S^2$ of degree $d$ is \emph{simple} if the cardinality of the pre-image $p^{-1}(y)$ is at least $d-1$ for all $y \in S^2$. The \emph{(simple) complexity} of $M$ is defined as the infimum of the complexities of all (simple) branched covers of $M$ to $S^2$. We prove that if $M$ is a closed, connected, orientable Riemann surface of genus $g \geq 1$, then: (1) its simple complexity equals $8\pi g$, and (2) 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 $p \colon M \to S^2$.