Almost-crystallographic groups with many outer automorphisms

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Almost-crystallographic groups arise as a natural generalization of the well known crystallographic groups. Automorphisms and outer automorphisms of crystallographic groups were treated in [l] (see also [2]). In [10], we developed a method to describe Aut(E) and Out(E) for an almost-crystallographic group E in terms of commutative diagrams with calculable entries. Here, we continue this study from a different point of view, namely we search for almost-crystallographic groups admitting many (outer) automorphisms. The aim of this paper is to indicate situations in which we can explicitly construct free abelian groups of (outer) automorphisms. Moreover, in many cases, these groups are normal in the full group of (outer) automorphisms. The results obtained are not only of pure algebraic interest, as it is well known that for K(E, l)-manifolds, Out(E) has a relevant geometrical meaning