bibkey: mohan2026smalldoubling authors: Mohan, Neetu year: 2026 title: On small doubling in right-ordered groups and Baumslag-Solitar groups - II doi: 10.48550/arXiv.2607.11194 url: https://arxiv.org/abs/2607.11194v1 claim: Section 6 prints Conjectures 6.1, 6.2 and 6.3 for finite subsets of torsion-free groups. strata_touched:

• D5/S0/Certificates/Groups/MohanNeetuSmallDoublingRefutation license: citation-only triage: anchor

Three printed small-doubling conjectures

Section 1 defines an abelian set as a set whose generated subgroup is abelian, and defines the product set by S^2 = {s_1 s_2 : s_1, s_2 in S}. Section 5 uses the coordinate law (r,n)(s,m) = (r + q^n s,n+m) on Z[1/q] semidirect Z. Section 6, “Future Problems”, prints Conjectures 6.1, 6.2 and 6.3 on page 26.

At q = -1, the coordinate group is the Klein bottle group. The formal results use two explicit three-element subsets of this group to refute all three conjectures as printed. The paper’s theorems about BS(1,q) exclude q = -1; the three conjectures quantify over torsion-free groups without that exclusion.

The arXiv record lists only version 1, submitted 2026-07-13. An arXiv API search through 2026-09-19 found no later item matching both “small doubling” and “torsion-free”, and no item matching both “Klein bottle” and “small doubling”. A Crossref title query returned the authors’ 2025 paper without the “II” suffix and did not identify a journal DOI for this preprint.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2607.11194
• URL: https://arxiv.org/abs/2607.11194v1
• Version and scope: https://arxiv.org/pdf/2607.11194v1, printed pages 2, 22 and 26; notation, the coordinate group law, and Conjectures 6.1-6.3.