Abstract

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