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

Abstract

Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C⁽¹⁾ smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C⁽²⁾ smoothness, was obtained by the author earlier.