Non-unique-product sets in the Promislow group: a centring theorem, and bounds by word diameter

Abstract

A finite subset A of a group is non-UP if no element of A · A is a product ab with a,b ∈ A in exactly one way. In the Promislow group P — the orientable Hantzsche–Wendt Bieberbach group of dimension three, where Promislow found a 14-element non-UP set in 1988 and where Gardam refuted the unit conjecture in 2021 — the least size of a non-UP set is not known; it lies between 8, the bound valid in every torsion-free group, and 14. What has narrowed the question is a statement about a ball: the minimum over non-UP subsets of the word-metric ball B(r) is 14 for 3 ≤ r ≤ 6, and a ball computation cannot be promoted to a statement about the group, because the one-set non-UP condition is not translation invariant. We give a normal form that is a conjugation rather than a translation, and therefore is a symmetry of the problem. For each of the three coordinate directions r let σᵣ be the diagonal character of the point group, Nᵣ=kerσᵣ the corresponding index-two subgroup, and Psiᵣ the r-th translation coordinate. We prove that a non-UP set A satisfies maxPsiᵣ(A ∩ Nᵣ)=-minPsiᵣ(A ∩ Nᵣ)=:hᵣ ≥ 0 — an absolute constraint, with no translation freedom left — that Psiᵣ(Asetminus Nᵣ) occupies a window of width exactly 2hᵣ, that A meets at least three of the four point-group fibres, and consequently that some conjugate of A by a translation lies in an explicit box U(h) determined by h=(h₁,h₂,h₃) alone. A search of U(h) is then a statement about all of P. Sweeping h over finite ranges gives three such statements: no non-UP subset of P, of any cardinality, has word diameter at most 3; no non-UP subset of P of cardinality at most 13 has word diameter at most 5; and a non-UP subset of P whose three spreads are all at most 2 has spread vector a cyclic rotation of Promislow's own (1,2,1) and at least 14 elements. The last is a complete classification of 27 profiles. All three are sharp against Promislow's witness, which has spread vector (1,2,1) and word diameter exactly 6. Every theorem below is machine-checked; the underlying refutations are 43 unsatisfiability results, 13 discharged inside the proof kernel and 30 by replaying clausal proofs.

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