Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points

TL;DR

This paper investigates the structure of arboreal Galois groups associated with a specific postcritically finite quadratic rational function, providing explicit descriptions, criteria for maximality, and insights into their constant fields.

Contribution

It offers explicit recursive descriptions of the geometric iterated monodromy group and a criterion to determine when the arboreal Galois group is maximal, including computationally accessible checks.

Findings

Hausdorff dimension of the arithmetic iterated monodromy group is zero

Maximality can be verified at level four

Determined the intersection of the constant field with $k(9_{2^{d}})$

Abstract

We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function $f(x)=\frac{2}{(x-1)^2}$ defined over a number field $k$, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of $f$ and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values $a\in k$ for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. Finally, we determine the intersection of the constant field of the arithmetic iterated monodromy group with $k(\mu_{2^{\infty}})$, providing the first full study of a PCF quadratic map with non-abelian constant field.