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

## TL;DR

This paper constructs Bieberbach groups with prescribed finite abelian holonomy from Artin braid groups, providing explicit holonomy representations and exploring geometric structures like Anosov diffeomorphisms and Kähler geometry.

## Contribution

It introduces a method to realize Bieberbach groups with specific abelian holonomy from braid groups and describes their holonomy representations explicitly.

## Key findings

- Existence of Bieberbach subgroups with given abelian holonomy
- Explicit description of holonomy representations
- Conditions for Kähler and Anosov structures on associated flat manifolds

## Abstract

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 $\Gamma_{G}$ 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 K\"ahler geometry of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group $\Gamma_{G}\leq B_n/[P_n,P_n]$.