A crystallographic quotient of the framed braid group and associated flat manifolds

Abstract

We study the quotient of the framed braid group $FB_n$ by the commutator subgroup of the framed pure braid group $FP_n$. We prove that $$ \frac{FB_n}{[FP_n,FP_n]}\cong\mathbb Z^n\rtimes\frac{B_n}{[P_n,P_n]}, $$ and, for $n\geq3$, that this quotient is a crystallographic group of dimension $n(n+1)/2$ with holonomy group $S_n$. We investigate its finite-order elements and conjugacy classes, prove that it has no elements of finite even order, and construct Bieberbach subgroups with prescribed finite $2$-group holonomy. Starting from suitable Bieberbach subgroups of $\frac{B_n}{[P_n,P_n]}$, we then construct Bieberbach subgroups of the framed quotient whose associated compact flat manifolds are torus bundles over the corresponding unframed flat manifolds. We describe these bundles through the generalized Calabi construction and determine their orientability, first Betti numbers, and the centers of their fundamental groups. As an application, we study a family with cyclic holonomy of odd order, obtaining a criterion for the existence of flat Kähler structures, and indicate how the same construction applies to cyclic holonomy of $2$-power order.