Iterated monodromy group of a PCF quadratic non-polynomial map

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