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