On balanced planar graphs, following W. Thurston

TL;DR

This paper explores balanced planar graphs on the sphere, which are used to characterize certain branched covering maps of the sphere, providing detailed insights into Thurston's combinatorial approach and related examples.

Contribution

It offers a detailed account of Thurston's characterization of balanced planar graphs associated with branched coverings, including examples and an appendix on Hurwitz numbers.

Findings

Balanced planar graphs characterize branched coverings with 2d-2 critical values.

Thurston's combinatorial framework is elaborated with detailed discussion and examples.

Connections to Hurwitz numbers are explored in the appendix.

Abstract

Let $f:S^2\to S^2$ be an orientation-preserving branched covering map of degree $d\geq 2$, and let $\Sigma$ be an oriented Jordan curve passing through the critical values of $f$. Then $\Gamma:=f^{-1}(\Sigma)$ is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such $\Gamma$, where $f$ has $2d-2$ distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.