The simple complexity of a Riemann surface

TL;DR

This paper introduces a measure called complexity for branched covers of Riemann surfaces, establishing exact formulas for simple and general cases based on genus and minimal branch data, respectively.

Contribution

It provides explicit formulas for the complexity of Riemann surfaces, linking geometric properties with branch data and extending understanding of branched covers.

Findings

Simple complexity of genus g surfaces is 8πg.

Complexity depends on minimal total length of realizable branch data.

Formulas connect geometric complexity with algebraic branch data.

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 $p \colon M \to S^2$.