The Free Orthogonal Quantum Group O_N^+ Is Topologically Generated by SO_N and the Dual of a Free Group Exactly When N ≥ 4: On a Question of Freslon and the Nilpotent Residual of O(O_N^+)

In the problem session of the Oberwolfach workshop "Quantum Groups – Algebra, Analysis and Category Theory" (Report 44/2021), Freslon, reporting on joint work with Franz and Skalski, asked three questions about the nilpotent residual I^∞ = ⋂_n I^n of the augmentation ideal I of a Hopf algebra A: for which A it vanishes; what it is for the free orthogonal quantum group, A = O(O_N^+); and, as the final problem, whether there is a discrete group Γ, residually torsion-free nilpotent, such that O_N^+ is topologically generated by SO_N and the dual of Γ. We answer the final question for every N ≥ 2. The answer is no for N = 2 and N = 3: no torsion-free group has this property, for any embedding of its dual (for N = 2 the negative answer was known from the work of Franz, Freslon and Skalski). The answer is yes for N ≥ 4, with the free group F_2, whose dual is embedded by two rotation blocks. As a consequence we determine the nilpotent residual of O(O_N^+) for every N ≥ 2: it is zero for N ≥ 4, so that O_N^+ is strongly connected, and it is the kernel of the map onto O(SO_N) for N = 2 (known) and N = 3. For N = 4 the proof uses the model C^4 = C^2 ⊗ C^2, a lowest-weight argument for sl_2 and words in a free group, and needs only standard facts. For N ≥ 5 it is an induction which relies on one published theorem, due to Brannan, Collins and Vergnioux; its proof is recalled in an appendix. Two auxiliary statements are cases of generation theorems of Chirvasitu, for which we give new proofs. The general question, for which Hopf algebras the nilpotent residual vanishes, is a programme and stays open; the Gaussian part of O_N^+ is not determined for N ≥ 4; and we do not know which torsion-free quotients of F_2 could replace F_2. This is an unrefereed note. Version 1.1 (10 October 2026) adds one sentence to Remark 4.12(iii), which records a personal communication of A. Chirvasitu, and a paragraph on the version history. The theorems and proofs are unchanged. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-8415347-019 (Freslon, with Franz and Skalski, Residual nilpotence of the augmentation ideal, Oberwolfach Rep. 18 (2021), no. 3, pp. 2452–2453).