Finite index theorems for iterated Galois groups of cubic polynomials

@article{Bridy2017FiniteIT,
  title={Finite index theorems for iterated Galois groups of cubic polynomials},
  author={Andrew Bridy and Thomas J. Tucker},
  journal={Mathematische Annalen},
  year={2017},
  volume={373},
  pages={37-72}
}
Let K be a number field or a function field. Let $$f\in K(x)$$f∈K(x) be a rational function of degree $$d\ge 2$$d≥2, and let $$\beta \in {\mathbb {P}}^1(\overline{K})$$β∈P1(K¯). For all $$n\in \mathbb {N}\cup \{\infty \}$$n∈N∪{∞}, the Galois groups $$G_n(\beta )={{\mathrm{Gal}}}(K(f^{-n}(\beta ))/K(\beta ))$$Gn(β)=Gal(K(f-n(β))/K(β)) embed into $${{\mathrm{Aut}}}(T_n)$$Aut(Tn), the automorphism group of the d-ary rooted tree of level n. A major problem in arithmetic dynamics is the arboreal… 

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