We prove that for a parabolic subgroup $\Gamma$ of $\mathrm{Aut}{𝔹}^n$ the fixed points sets of all elements in $\Gamma \setminus \left\{{\mathrm{id}}_{{𝔹}^n}\right\}$ are the same. This result, together with a deep study of the structure of subgroups of $\mathrm{Aut}{𝔹}^n$ acting freely and properly discontinuously on ${𝔹}^n$, entails a generalization of the so called weak Hurwitz’s theorem: namely that, given a complex manifold $X$ covered by ${𝔹}^n$ and such that the group of deck transformations of the covering is “sufficiently generic”, then ${\mathrm{id}}_X$ is isolated in $\mathrm{Hol}(X,X)$.

An explicit formula is developed for Nevanlinna class functions whose behaviour at the boundary is “sufficiently rational” and is then used to deduce the uniqueness of the factorization of such inner functions. A generalization of a theorem of Frostman is given and the above results are then applied to the construction of good and/or irreducible inner functions on a polydisc.