Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Let $n\ge 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n\mathrm{/}[P_n,P_n]$ has Bieberbach subgroups ${\mathrm{\Gamma }}_G$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group ${\mathrm{\Gamma }}_G$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of ${\mathrm{\Gamma }}_G$ and determine the existence of Anosov diffeomorphisms and K"ahler geometry of the flat manifold ${\mathcal{X}}_{{\mathrm{\Gamma }}_G}$ with fundamental group the Bieberbach group ${\mathrm{\Gamma }}_G\le B_n\mathrm{/}[P_n,P_n]$.