Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds
Abstract
For n≥3, let B̅n=Bn/[Pn,Pn] be the crystallographic quotient of the Artin braid group, with translation subgroup P̅n=Pn/[Pn,Pn]≅ℤN, 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 B̅n. More precisely, for every n≥ nM, where nM∈{3,4,5} is an explicit optimal threshold depending on M, we construct a Bieberbach subgroup Γn(M)≤B̅n of dimension N and a complete normal subgroup Kn(M)≅ℤN−3 consisting of translations such that Γn(M)/Kn(M)≅π1(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.
Related
- Spherical and affine Artin groups, finite quotients, and the isomorphism problem
- Finite Quotients of Right-angled Artin Groups
- Smallest quotients and profinite rigidity of irreducible spherical type Artin groups
- Compact quotients of homogeneous spaces and homotopy theory of sphere bundles
- Crystallographic Torus Bundles: Obstructions, Elastic Sectors, and Global Acoustic Modes
BibTeX
@misc{ocampo2026crystallographic,
title = {{Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds}},
author = {Oscar Ocampo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20478},
eprint = {2610.20478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20478},
}
RIS
TY - UNPB
TI - Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds
AU - Oscar Ocampo
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20478
UR - https://arxiv.org/abs/2610.20478
AB - For n≥3, let B̅_n=B_n/[P_n,P_n] be the crystallographic quotient of the Artin braid group, with translation subgroup P̅_n=P_n/[P_n,P_n]≅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 B̅_n. More precisely, for every n≥ n_M, where n_M∈{3,4,5} is an explicit optimal threshold depending on M, we construct a Bieberbach subgroup Γ_n(M)≤B̅_n of dimension N and a complete normal subgroup K_n(M)≅ℤ^N−3 consisting of translations such that Γ_n(M)/K_n(M)≅π_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.
ER -