arXiv subjects

Moe Tabei

Publications and source records attributed to Moe Tabei.

7 recordsLinked to original sources

arXiv2026

The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups

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 $m_2(G_3)=16$.

math.GR↗

arXiv2026

Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives

Gardam's counterexample to the Kaplansky unit conjecture is a unit of $\mathbb{F}_2[P]$, $P$ the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over $\mathbb{F}_2$ the unit equation is a parity condition on a pair of supports, searchable over word balls. We upgrade two statements to machine-checked certificates: no support pair of sizes $\ge2$ lies in the radius-$3$ ball (a DRAT proof of a ball form of the Craven--Pappas theorem), and every nontrivial unit with supports in the radius-$4$ ball has total support at least $42$, confirming Gardam's expectation in ball-limited form. The relatives $H_4=F(3,4)$ and the Nielsen--Soelberg groups are swept with certificates; $G_3$ is Gardam's amalgam $S$, whose unit we localize into its radius-$4$ ball. $H_4$ resists the twisted-unitary ansatz through radius $6$ for every non-identity dihedral twist. An effective localization principle makes the least support of a nontrivial unit of $\mathbb{F}_2[P]$ computable in principle.

math.GR↗

arXiv2026

Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate

Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).

math.GR↗

arXiv2026

The domain question for the Nielsen-Soelberg group rings: the commutative case, with certified ball checks

Nielsen and Soelberg exhibited three torsion-free groups G_1, G_2, G_3 carrying 8-element sets without unique products, and asked whether any of the group rings R[G_i], R a domain, is a domain. We record that for every commutative domain R the answer is affirmative for all three groups: each G_i is virtually nilpotent, so the theorem of Kropholler, Linnell and Moody applies over every field, in every characteristic, and commutative coefficients reduce to the fraction field. Every ingredient is in the literature except the nilpotent structure of the relevant finite-index subgroup of G_2, which is supplied by the certified computational model of our companion paper (arXiv:2607.19687). The question therefore remains open exactly for noncommutative coefficient domains, where the unique-product mechanism -- the only known ring-independent one -- is precisely what these groups are constructed to lack. As a complement we report machine-checkable, DRAT-certified verifications that F_2[G_i] has no zero divisors with both supports in explicit balls of the defining generating sets, obtained by propositional reasoning alone, independent of the K-theoretic machinery; we state precisely what these certificates do and do not add.

math.GR↗

arXiv2026

Higman in balls: the mod-2 dichotomy for integral units of the Promislow group

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.

math.GR↗

arXiv2026

A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group

Let G be a torsion-free group whose real group algebra R[G] has no zero divisors, and let u -> u^{*theta} be an l2-isometric anti-involution of R[G] (a composition of the inversion involution with a ring automorphism and a sign character). We prove a compactness theorem: for every finite *theta-closed support window W the constant mu*(W) = min{ ||w^{*theta}w||_2 : ||w||_2 = 1, supp(w) in W } is strictly positive, and every real theta-unitary element (u^{*theta}u = 1) supported in W satisfies ||u||_2 <= mu*(W)^{-1/2}. In particular the integer theta-unitary elements supported in W form a finite, effectively enumerable set: the a priori infinite "height" direction of the unit search collapses to a single real constant. For the Promislow (Hantzsche-Wendt) group P and the window B(4) that hosts Gardam's counterexample to the unit conjecture over F_2, we combine this with exact SAT-certified height ladders (heights <= 31 per stratum, DRAT-certified master cell at larger heights), a radius-free depth-tail theorem, and a mirror symmetry between the +- strata, reducing the vanishing of all nontrivial theta-unitary units u = +-1 (mod 2) with supp(u) in B(4) to a single certified lower bound on mu*(B(4)). We report numerical estimates mu*(B(4)) ~ 1.4e-3, well above the required threshold, and prove three structural results about the remaining certification problem: no linear (Cauchy-Schwarz) dual certificate exists, because P carries theta-anti-unitary trivial elements; and, numerically, the level-2 sum-of-squares relaxation is boundary-pinned with an explicit slope, both in the direct and in the ideal-multiplier formulation -- the obstruction being a spurious pseudo-moment that no measure can realize. We also contrast the mechanism with Z[D_infinity], where torsion produces zero divisors, mu* = 0, and genuinely unbounded unipotent families of twisted unitaries.

math.GR↗

arXiv2026

The one-sided unit count of F_2[P] at radius four, and an integral separation theorem

Let P be the Hantzsche-Wendt (Promislow) group and B(4) the radius-four ball in its standard word metric. Dietrich, Lee, Nies and Vinyals determined the two-sided count: exactly 36 nontrivial units u of F_2[P] with both supp(u) and supp(u^{-1}) in B(4). We determine the one-sided count: exactly 52 nontrivial units with supp(u) in B(4) and no constraint on the inverse. The 16 new units have inverses supported at radius exactly 5; they form two orbits of size 8 under the symmetry group fixing the generating set, and all 52 units have support size 21 on both sides. Completeness is a single propositional unsatisfiability, certified by a DRAT proof checked with drat-trim. As an arithmetic consequence we prove: no unit of Z[P] with support in B(4) has nontrivial reduction modulo 2 -- with no bound on the coefficients or on the support of the inverse. Since F_2[P] has 52 nontrivial units on that ball, this separates, in the untwisted setting, the integral group ring from its characteristic-two quotient at the first radius where the unit conjecture fails over a field.

math.GR↗