论文标题
彻底固定的瑟斯顿地图：分类，识别和扭曲
Critically fixed Thurston maps: classification, recognition, and twisting
论文作者
论文摘要
如果$ f $修复了其每个关键点，则定向覆盖的覆盖地图$ f \ colon s^2 \ to s^2 $被称为急切固定的瑟斯顿地图。最近显示，迫切固定有理图的Möbius共轭类别和平面嵌入式连接图的同构类别之间存在明确的一对一对应关系。在本文中，我们将此结果推广到整个批判性固定的瑟斯顿地图。 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 $(G,φ)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $φ$ is an orientation-preserving homeomorphism of $S^2$ that fixes each vertex of $ g $。该结果使我们能够提供批判性固定瑟斯顿地图的组合等效类别的分类。我们还开发了一种算法，该算法将重建（直至同位素）与与急切固定的瑟斯顿地图$ f $相关的$（g，φ）$。最后，我们解决了通过吹起对$（g，\ mathrm {id} _ {s^2}）$获得的一些扭曲问题的特殊实例。
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 Möbius 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 $(G,φ)$, where $G$ is a planar embedded graph in $S^2$ without isolated vertices and $φ$ 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 $(G,φ)$ 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 $(G, \mathrm{id}_{S^2})$.
