Generic subgroups of Aut ${𝔹}^n$

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 $\mathrm{Hol}(X,X)$.