1 Introduction

Abstract

1.1 Motivation from dynamics–a brief sketch 1.2 Thurston’s Characterization and Rigidity Theorem. Standard definitions 1.3 Examples 1.3.1 A realizable mating 1.3.2 An obstructed mating 1.3.3 An obstructed expanding Thurston map 1.3.4 A subdivision rule 1.4 Summary of this work 1.5 Survey of previous results 1.5.1 Enumeration 1.5.2 Combinations and decompositions 1.5.3 Parameter space 1.5.4 Combinations via quasiconformal surgery 1.5.5 From p.f. to geometrically finite and beyond 1.6 Analogy with three-manifolds 1.7 Connections 1.7.1 Geometric Galois theory 1.7.2 Gromov hyperbolic spaces and interesting groups 1.7.3 Cannon’s conjecture 1.8 Discussion of combinatorial subtleties 1.8.1 Overview of decomposition and combination 1.8.2 Embellishments. Technically convenient assumption. 1.8.3 Invariant multicurves for embellished map of spheres. Thurston linear map. 1.9 Tameness assumptions KeywordsMapping Class GroupRigidity TheoremCombination ProcedureSiegel DiskHyperbolic ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.