Critically fixed Thurston maps: classification, recognition, and twisting

TL;DR

This paper extends the classification of critically fixed Thurston maps by linking them to planar graphs and homeomorphisms, providing a reconstruction algorithm and addressing twisting problems.

Contribution

It generalizes the classification of critically fixed Thurston maps using planar graphs and homeomorphisms, and introduces an algorithm for their reconstruction.

Findings

Established a correspondence between critically fixed Thurston maps and planar graphs.

Developed an algorithm to reconstruct maps from graph-homeomorphism pairs.

Solved specific twisting problems for certain critically fixed Thurston maps.

Abstract

An orientation-preserving branched covering map $f\colon S^2 \to S^2$ is called a critically fixed Thurston map if $f$ fixes each of its critical points. It was recently shown that there is an explicit one-to-one correspondence between M\"obius conjugacy classes of critically fixed rational maps and isomorphism classes of planar embedded connected graphs. In the paper, we generalize this result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map $f$ is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair $(G,\varphi)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $\varphi$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $G$. This result allows us to provide a classification of combinatorial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair $(G,\varphi)$ associated with a critically fixed Thurston map $f$. Finally, we solve some special instances of the twisting problem for the family of critically fixed Thurston maps obtained by blowing up pairs $(G, \mathrm{id}_{S^2})$.