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 \(I^\infty\)\({}=\bigcap_nI^n\) of the augmentation ideal \(I\) of a Hopf algebra \(A\): for which \(A\) it vanishes; what it is for the free orthogonal quantum group, \(A=\mathcal{O}(O_N^+)\); and, as the final problem, whether there is a discrete group \(\Gamma\), residually torsion-free nilpotent, such that \(O_N^+\) is topologically generated by \(SO_N\) and the dual \(\widehat{\Gamma}\). We answer the final question for every \(N\ge2\). 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 \(N\ge4\), with the free group \(\mathbb{F}_2\), whose dual is embedded by two rotation blocks. As a consequence we determine the nilpotent residual of \(\mathcal{O}(O_N^+)\) for every \(N\ge2\): it is zero for \(N\ge4\), so that \(O_N^+\) is strongly connected, and it is the kernel of the map onto \(\mathcal{O}(SO_N)\) for \(N=2\) (known) and \(N=3\). For \(N=4\) the proof uses the model \(\mathbb{C}^4\)\({}=\mathbb{C}^2\)\({}\otimes\mathbb{C}^2\), a lowest-weight argument for \(\mathfrak{sl}_2\) and words in a free group, and needs only standard facts. For \(N\ge5\) 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 \(N\ge4\); and we do not know which torsion-free quotients of \(\mathbb{F}_2\) could replace \(\mathbb{F}_2\). This is an unrefereed note.