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 $A\cdot A$ having no unique product satisfies $|A|\ge 8$, 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 $m_1(G_3)\in [8,15]$ against $m_2(G_3)=16$.

16 pages. v3: Section 6 now records that R[G_i] is a domain for every commutative domain R, the ingredient not in the literature being the class-2 structure of the index-4 subgroup of G_2 supplied here; details and the noncommutative question in arXiv:2609.25016. v2: Remark 6.2 corrected using Strojnowski's theorem, as Gardam pointed out. Quantitative results unchanged