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

Abstract

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

arXiv comment16 pages

Cited by in corpus (1)

• Crystallographic groups and flat manifolds from surface braid groups