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Critically fixed Thurston maps: classification, recognition, and twisting
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DOI：10.1112/plms.70129.png)
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Abstract
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An orientation-preserving branched covering map f & ratio;S2 -> S2 iscalled 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 & ouml;bius 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 f is obtained by applying the blow-up operation, introduced by Kevin Pilgrim and Tan Lei, to a pair (G,phi), where G is a planar embedded graph in S2 with out isolated vertices and phi is an orientation-preserving homeomorphism of S2 that fixes each vertex of G.This result allows us to provide a classification of combina torial equivalence classes of critically fixed Thurston maps. We also develop an algorithm that reconstructs (up to isotopy) the pair (G,phi) 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,idS2)
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TOPOLOGICAL CHARACTERIZATION
POLYNOMIALS
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