Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map

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 $G_n^{\text{geom}}(f)$. Furthermore, we describe exactly for which $a\in k$ the Arboreal Galois group $G_a(f)$ and $G^{\text{arith}}(f)$ are equal.