球中的希格曼：普罗米斯洛群整环单位的模2二分法
Higman in balls: the mod-2 dichotomy for integral units of the Promislow group
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中文总结 AI 辅助
针对普罗米斯洛群P的整环单位猜想，研究者将其模2归约为两个子问题，论证其前沿为词半径4，该半径下的平凡性定理或可区分整数环与所有域。
中文摘要 AI 辅助
希格曼提出的“对于无挠群G，Z[G]仅含平凡单位”的猜想，针对普罗米斯洛（Hantzsche-Wendt）群P仍未解决，而Gardam已于2021年在该群上否定了系数为域的单位猜想。我们将P的整环单位猜想模2精确归约为两个子问题，记录基例为Craven-Pappas短长度定理的推论，并论证该整环问题的前沿是词半径4：Z[P]在该半径下的平凡性定理将是首个区分整数环Z与所有域的结论。全文结论均受球范围限制，未对完整猜想作出断言。
英文摘要
Higman's conjecture that Z[G] has only trivial units, for G torsion-free, is open for the Promislow (Hantzsche-Wendt) group P, the group over which Gardam disproved the field-coefficient unit conjecture in 2021. We introduce an exact reduction of the integral conjecture for P modulo 2 into two sub-problems, record the base cases as consequences of the Craven-Pappas small-length theorems, and argue that the frontier of the integral problem is word-radius 4: a triviality theorem for Z[P] there would be the first statement separating Z from every field. Throughout, claims are ball-limited and stated as such; we make no claim on the full conjecture.
