A Finite Index Conjecture for Iterated Galois Groups

Let f be a rational function of degree greater than 1 defined over a number field k and let x in k.  Iterating f, we obtain
a sequence of field extensions K_n generated by the inverse images of x under f^n. Passing to the inverse limit one obtains an iterated Galois group G_x.  There is a natural "generic" Galois group G attached to f in which G_x sits. Conjecturally, G_x will have finite index in G unless x is "special" in some fashion.  While in its earliest form, this conjecture (originally made in a special case by Boston and Jones) was made in analogy with the Serre finite index theorem, it seems to have a very different flavor.  We will discuss various results towards this conjecture, the diophantine techniques used in these results, a more general higher-dimensional conjecture, and a very simple conjecture on the irreducibility of iterates of polynomials that is completely unresolved.

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