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| Main Author: | |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1905.05123 |
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Table of Contents:
- Let $n\geq 3$. 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 $Γ_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group $Γ_{G}$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of $Γ_{G}$ and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold ${\cal X}_{Γ_{G}}$ with fundamental group the Bieberbach group $Γ_{G}\leq B_n/[P_n,P_n]$.