Iterated monodromy group of a PCF quadratic non-polynomial map

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