On massive subsets in the space of finitely generated groups of diffeomorphisms of the line and the circle in the case of \(C^{(1)}\) smoothness (Q822653)

A subset \(Q\) of \(C^1\) diffeomorphisms of a manifold \(N\) is \textit{mutually transversal} if for any distinct \(q_1,q_2\in Q\) and \(x\in N\) such that \(q_1(x)=q_2(x)\) then the condition \(D_xq_1\neq D_xq_2\) holds. The author gives a proof of the fact that for given \(n\ge 1\), the collection of \(n\)-tuples of orientation-preserving \(C^1\) diffeomorphisms of a one-manifold \(N\in\{\mathbb R,\mathbb S^1\}\) that generate a rank-\(n\) free subgroup which is mutually transversal, forms a \(G_\delta\)-dense subset in \(\mathrm{Diff}^1_+(N)^n\), when equipped with the topology of uniform convergence.