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

TL;DR

The paper constructs Bieberbach subgroups of the braid-group quotient $B_n/[P_n,P_n]$ whose holonomy can be any finite abelian group, via embeddings into $S_n$ and Reidemeister–Schreier machinery, yielding explicit holonomy representations for flat manifolds $\mathcal{X}_{\Gamma_G}$. It provides a comprehensive analysis of cyclic odd-order holonomy, giving a detailed matrix decomposition of the holonomy action, a Betti-number formula, and a sharp Anosov-diffeomorphism criterion ($q\neq 3$). Furthermore, it specializes to $\mathbb{Z}_{p^r}$-manifolds to establish a precise Kähler condition: the flat manifold is Kähler if and only if $p^r=4u+1$, in which case the manifold is Calabi–Yau with dimension $2u(4u+1)$. By bridging braid-group quotients with geometric properties of flat manifolds, the work extends Auslander–Kuranishi-type realizations and provides constructive tools for realizing flat manifolds with prescribed holonomy and rich geometric/dynamical structures.

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 $Γ_{G}$ 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 $Γ_{G}\leq B_n/[P_n,P_n]$.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition
• Lemma 4: Lemma 8 of GGO
• Remark 5
• Theorem 6: Auslander and Kuranishi
• Theorem 7: Theorem 3(a) of GGO
• Proposition 8: Proposition 12 of GGO
• proof : Proof of Theorem \ref{['th:akthm']}