AI 中文总结
研究尼尔森-索尔伯格群中全局最小值处的定量非唯一乘积景观。构建群模型计算定量不变量，如\(G_1\)中半径为\(6\)的球内无\(8\)元对称见证，\(G_2\)有特殊性质，\(G_3\)重现双边对并展示\(15\)元对称见证，得出相关结论。
AI 中文摘要
尼尔森和索尔伯格证明，无挠群中满足\(A\cdot A\)无唯一乘积的有限子集\(A\)满足\(|A|\geq8\)，并展示了达到该界限的两个群\(G_1\)和\(G_2\)。此前对这些极值构型缺乏定量认识。我们构建了两个群的精确且经独立验证的模型，并计算了全局最小值处的首个定量不变量。在\(G_1\)中，半径为\(6\)的球内不存在\(8\)元对称见证（\(933\)个元素，经认证不可行），而尼尔森 - 索尔伯格见证位于半径为\(7\)的球内，全局最小值分布较分散。在\(G_2\)中，在其天然的八生成元度量下，见证及其逆是半径为\(1\)的球内仅有的两个非唯一乘积\(8\)集，唯一乘积阶梯在\(n = 8\)时取值\(0\)但在\(n = 9\)时取值\(1\)，这是首个已知的其平方恰好有一个唯一表示元素的极小值点，所以在普罗米斯洛群中看到的非唯一乘积和唯一乘积的同时失效并非普遍现象。在所搜索的球内不存在\((7,9)\)双边见证，所以尼尔森 - 索尔伯格轮廓界限可能不精确。最后我们研究通用群\(G_3\)。其结构已知，索尔伯格的论文识别出一个步长为\(8\)的指数为\(8\)的海森堡子群并证明其无挠性，加德姆将其作为克莱因瓶群的融合来研究，表明它几乎幂零但并非几乎阿贝尔。我们添加的是一个搜索坐标中的模型，在其中可以枚举球。在该模型中我们重现了尼尔森 - 索尔伯格双边对，并展示了一个对称的\(15\)元见证，其平凡陪集单元素生成该海森堡子群的中心。它是刚性且罕见的：在\(B(5)\)内大小\(15\)恰好是最小的，陪集轮廓是强制的，并且在\(B(4)\)内恰好存在四个这样的见证，处于一个轨道。因此\(m_1(G_3)\in[8,15]\)而\(m_2(G_3)=16\)。
英文摘要
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 $m_2(G_3)=16$.
Comments16 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
