Holonomy groups of Bieberbach groups with finite outer automorphism groups (Q1894892)

The paper is a first step towards the classification of holonomy groups \(G\) of Bieberbach groups \(\Gamma\) with finite outer automorphism group \(\text{Out}(\Gamma)\). In particular the authors completely classify all such abelian groups. Here a Bieberbach group of dimension \(n\) (the fundamental group of a compact connected flat manifold \(X\)) is a torsion free group \(\Gamma\) defined by a short exact sequence \(0\to L\to\Gamma\to G\to 1\), where \(G\) is finite (holonomy group of \(\Gamma\)), and \(L\) is a free maximal abelian subgroup of \(\Gamma\) of rank \(n\). In particular, the group \(\text{Aff}(X)\) of affine self equivalences of \(X\) is finite if and only if \(\text{Out}(\Gamma)\) is finite and the first Betti number of \(X\) is zero. See also \textit{H. Brown, J. Neubüser} and \textit{H. Zassenhaus} [Math. Comput. 27, 167-182 (1973; Zbl 0255.20031)].