Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866909886118363136 |
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| author | Ocampo, Oscar |
| author_facet | Ocampo, Oscar |
| 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 $Γ_{G}\leq B_n/[P_n,P_n]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_05123 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups Ocampo, Oscar Geometric Topology Group Theory 20F36, 20H15, 57N16 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 $Γ_{G}\leq B_n/[P_n,P_n]$. |
| title | Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups |
| topic | Geometric Topology Group Theory 20F36, 20H15, 57N16 |
| url | https://arxiv.org/abs/1905.05123 |