Finite index theorems for iterated Galois groups of preperiodic points for unicritical polynomials

TL;DR

This paper proves that for certain unicritical polynomials over number fields, the Galois groups of preperiodic points have finite index in the generic Galois group, extending understanding of arboreal Galois representations.

Contribution

It establishes finite index theorems for iterated Galois groups of preperiodic points in unicritical polynomial dynamics over number fields.

Findings

Finite index of Galois groups for preperiodic points

Extension of Galois group structure understanding

Results apply to non-post-critically finite polynomials

Abstract

Let K be a number field and let f(x) = x^q + c where q is a prime power, c is in K, and f is not post-critically finite. We show that for any strictly preperiodic b in K, the iterated Galois group at b with respect to f has finite index in the generic iterated Galois group for f.