Iterated monodromy group of a PCF quadratic non-polynomial map

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