Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds

Abstract

For $n\geq3$, let $\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\overline P_n=P_n/[P_n,P_n]\cong\mathbb Z^N$, where $N=\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\overline B_n$. More precisely, for every $n\geq n_M$, where $n_M\in\{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $Γ_n(M)\leq\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong\mathbb Z^{N-3}$ consisting of translations such that $$ Γ_n(M)/K_n(M)\congπ_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.