Critically fixed Thurston maps: classification, recognition, and twisting

Abstract

An orientation-preserving branched covering map (Formula presented.) is called a critically fixed Thurston map if (Formula presented.) 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 this paper, we generalize the result to the whole family of critically fixed Thurston maps. Namely, we show that each critically fixed Thurston map (Formula presented.) is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair (Formula presented.), where (Formula presented.) is a planar embedded graph in (Formula presented.) without isolated vertices and (Formula presented.) is an orientation-preserving homeomorphism of (Formula presented.) that fixes each vertex of (Formula presented.). 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 (Formula presented.) associated with a critically fixed Thurston map (Formula presented.). Finally, we solve some special instances of the Twisting Problem for the family of critically fixed Thurston maps obtained by blowing up pairs (Formula presented.).