The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups
Abstract
Nielsen and Soelberg proved that a finite subset A of a torsion-free group with A· A having no unique product satisfies |A|≥ 8, and exhibited two groups, here G1 and G2, attaining the bound. Nothing quantitative was known about these extremal configurations. We construct exact, independently verified models of both groups and compute the first quantitative invariants at the global minimum. In G1 no $8$-element symmetric witness lies in the radius-$6$ ball ($933$ elements, certified infeasible), while the Nielsen-Soelberg witness lies in the radius-$7$ ball: the global minimum is spread out. In G2, with its natural eight-generator metric, the witness and its inverse are the only two non-UP $8$-sets in the radius-$1$ ball, and the unique-product staircase takes the value $0$ at n=8 but $1$ at n=9 -- the first known minimizer whose square has exactly one uniquely represented element, so the simultaneous failure of t.u.p. and u.p. seen in the Promislow group is not universal. No (7,9) two-sided witness exists in the searched balls, so the Nielsen-Soelberg profile bound may not be sharp. Finally we treat the universal group G3. Its structure is known -- Soelberg's thesis identifies an index-$8$ Heisenberg subgroup of step $8$ and proves torsion-freeness, and Gardam, studying the same group as an amalgam of Klein bottle groups, shows it to be virtually nilpotent but not virtually abelian -- and what we add is a model in search coordinates in which balls can be enumerated. In it we reproduce the Nielsen-Soelberg two-sided pair and exhibit a symmetric $15$-element witness whose trivial-coset singleton generates the centre of that Heisenberg subgroup. It is rigid and rare: within B(5) the size $15$ is exactly minimal, the coset profile is forced, and exactly four such witnesses exist in B(4), one orbit. Hence m1(G3)∈[8,15] against m2(G3)=16.
Related
- The domain question for the Nielsen-Soelberg group rings: the commutative case, with certified ball checks
- On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs
- Minimum Cardinalities of Multipartite Unextendible Product Bases
- Eberlein convolution as an inner product and quantitative Bombieri--Taylor results
- On the Nielsen realisation problem for cyclic groups of prime order
BibTeX
@misc{tabei2026quantitative,
title = {{The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups}},
author = {Moe Tabei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.19687},
eprint = {2607.19687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19687},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups
AU - Moe Tabei
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2607.19687
UR - https://arxiv.org/abs/2607.19687
AB - Nielsen and Soelberg proved that a finite subset A of a torsion-free group with A· A having no unique product satisfies |A|≥ 8, and exhibited two groups, here G_1 and G_2, attaining the bound. Nothing quantitative was known about these extremal configurations. We construct exact, independently verified models of both groups and compute the first quantitative invariants at the global minimum. In G_1 no $8$-element symmetric witness lies in the radius-$6$ ball ($933$ elements, certified infeasible), while the Nielsen-Soelberg witness lies in the radius-$7$ ball: the global minimum is spread out. In G_2, with its natural eight-generator metric, the witness and its inverse are the only two non-UP $8$-sets in the radius-$1$ ball, and the unique-product staircase takes the value $0$ at n=8 but $1$ at n=9 -- the first known minimizer whose square has exactly one uniquely represented element, so the simultaneous failure of t.u.p. and u.p. seen in the Promislow group is not universal. No (7,9) two-sided witness exists in the searched balls, so the Nielsen-Soelberg profile bound may not be sharp. Finally we treat the universal group G_3. Its structure is known -- Soelberg's thesis identifies an index-$8$ Heisenberg subgroup of step $8$ and proves torsion-freeness, and Gardam, studying the same group as an amalgam of Klein bottle groups, shows it to be virtually nilpotent but not virtually abelian -- and what we add is a model in search coordinates in which balls can be enumerated. In it we reproduce the Nielsen-Soelberg two-sided pair and exhibit a symmetric $15$-element witness whose trivial-coset singleton generates the centre of that Heisenberg subgroup. It is rigid and rare: within B(5) the size $15$ is exactly minimal, the coset profile is forced, and exactly four such witnesses exist in B(4), one orbit. Hence m_1(G_3)∈[8,15] against m_2(G_3)=16.
ER -