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

Abstract

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.