Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map

TL;DR

This paper investigates the structure of the iterated monodromy groups associated with a specific postcritically finite non-polynomial quadratic map, revealing their properties and relationships under certain conjectures.

Contribution

It provides a detailed analysis of the geometric and arithmetic iterated monodromy groups for the map, including conditions for their equality and conjectural characterizations.

Findings

Elements of $G^{ ext{geom}}(f)$ fix roots of unity

Description of when $G_a(f)$ equals $G^{ ext{arith}}(f)$

Results depend on a conjecture about group sizes

Abstract

We study the postcritically finite non-polynomial map $f(x)=\frac{1}{(x-1)^2}$ over a number field $k$ and prove various results about the geometric $G^{\text{geom}}(f)$ and arithmetic $G^{\text{arith}}(f)$ iterated monodromy groups of $f$. We show that the elements of $G^{\text{geom}}(f)$ are the ones in $G^{\text{arith}}(f)$ that are fixing the roots of unity by assuming a conjecture on the size of $G^{\text{geom}}_n(f)$. Furthermore, we describe exactly for which $a \in k$ the Arboreal Galois group $G_a(f)$ and $G^{\text{arith}}(f)$ are equal.