Residual subsets in the space of finitely generated groups of diffeomorphisms of the circle (Q1947764)

This paper deals with finitely generated groups of diffeomorphisms of the circle. The author proves that the set of such groups contains elements that act freely on the orbits of almost every point of the circle. The complete group of all orientation preserving homeomorphisms of \(S^1\) is denoted by \(Homeo_+ (S^1)\) and the complete group of all coverings of \(S^1\) is denoted by \(\widetilde{Homeo}_+ (S^1)\), i.e. for every \(q \in \widetilde{Homeo}_+ (S^1)\) and \(t \in \mathbb{R}\), \(q(t+1)=q(t)+1\). Given the group \(Q=\langle q_1, q_2, \ldots , q_s\rangle \subset \widetilde{Homeo}_+ (S^1)\), the extended group \(\mathcal{J}_Q\) is given by \(\mathcal{J}_Q = \langle q_1, q_2, \ldots , q_s, \widehat{q} \, \rangle \) with \(\widehat{q}(t)=t+1\). Diffeomorphisms \(q_1\) and \(q_2\) of \(\mathbb{R}\) or \(\mathbb{S}^1\), with \(q_1 \neq q_2\) are said to be mutually transversal if for every \(t\) for which \(q_1(t) \neq q_2 (t)\), it holds that \(\dot{q}_1 (t) \neq \dot{q}_2 (t).\) The author mentions that a group \(Q=\langle q_1, q_2, \ldots , q_s\rangle\) with generators mutually transversal acts freely on the orbit \(Q(t)\) except for countably many points. In the second section of the paper the author states the following theorems. {Theorem A.} The set of free groups of diffeomorphisms \(Q=\langle q_1, q_2, \ldots , q_s\rangle\) with \(q_j \in Diff^2 (\mathbb{R} \cap \widetilde{Homeo}_+ (S^1)\) (\(j =1, \ldots s\)) with a fixed number \(s\) of generators for which the elements of the extended \(\mathcal{J}_Q = \langle q_1, q_2, \ldots , q_s, \widehat{q} \, \rangle \) are mutually transversal, in the metric of the space \(\bigotimes_s Diff^2 (\mathbb{R} )\cap \widetilde{Homeo}_+ (S^1)\), is a countable intersection of open everywhere dense subsets (a residual set). Theorem A is reformulated in terms of the group of diffeomorphisms of the circle. Each group \(\langle q_1, q_2, \ldots , q_m\rangle\) with generators \(g_j \in Diff^2(S^1)\) (\(j=1, \ldots , m)\) is regarded as an element in \(\bigotimes_m Diff^2 (S^1)\). {Theorem B.} The set of free groups of diffeomorphisms \(Q=\langle q_1, q_2, \ldots , q_s\rangle\) with \(q_j \in Diff^2 (\mathbb{R}) \cap \widetilde{Homeo}_+ (S^1)\) (\(j =1, \ldots s\)) with a fixed number \(s\) of generators for which the elements of the extended \(\mathcal{J}_Q = \langle q_1, q_2, \ldots , q_s, \widehat{q} \, \rangle \) are mutually transversal, in the metric of the space \(\bigotimes_s Diff^2 (\mathbb{R}) \cap \widetilde{Homeo}_+ (S^1)\), is a countable intersection of open everywhere dense subsets (a residual set). The remainder of the paper is devoted to the proof of these two results.