Free subgroups of diffeomorphism groups. (English) Zbl 0666.58011

It is well known that, for all \(k\in \{1,2,...,\infty \}\), the group \(Diff^ k_ c(X)\) of all compactly supported \(C^ k\)-diffeomorphisms of the \(C^ k\)-manifold X contains elements arbitrarily close to the identity which belong to no 1-parameter subgroup. In this paper, the author strengthens this result, showing that \(Diff^ k_ c(X)\) in fact contains an arc-connected subgroup consisting of such elements. Moreover, one can assume this subgroup is freely generated by the elements \(\gamma\) (t), \(t\in (0,1)\), where \(\gamma\) : [0,1)\(\to Diff^ k_ c(X)\) is a continuous map with \(\gamma (0)=id.\)
As an intermediate step in the proof, the author shows that, given any sequence of elements in \(Diff^ k_ c(X)\), there are diffeomorphisms arbitrarily close to the given ones which are free generators of a subgroup in \(Diff^ k_ c(X)\).
Reviewer: D.McDuff


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
57R50 Differential topological aspects of diffeomorphisms
58B25 Group structures and generalizations on infinite-dimensional manifolds
Full Text: DOI EuDML