Normal subgroups of the group of volume preserving diffeomorphisms of an open manifold
Let \(M\) be a connected open smooth \(n\)-manifold that is the interior of a compact one \(\widetilde M\), \(n \geq 3\). Assume that the boundary of \(\widetilde M\) is nonempty. Let \(\Omega\) be a volume element on \(M\), \(\text{Diff}^\Omega (M)\) the group of all \(C^\infty\)-diffeomorphisms of \(M\) preserving \(\Omega\) and \(\text{Diff}_0^\Omega (M)\) the path-component of the identity with respect to the compact-open \(C^\infty\)-topology. It is known that \(\text{Diff}_0^\Omega (M)\) is not a perfect group [\textit{D. McDuff}, Proc. Lond. Math. Soc., III. Ser. 43, 295-320 (1981; Zbl 0411.57028)]. The paper deals with the study of the nontrivial normal subgroups of \(\text{Diff}_0^\Omega (M)\). Various types of such subgroups are pointed out. The author's results are similar to those obtained by \textit{D. McDuff} [The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold, J. Lond. Math. Soc. 18, 353-364 (1978)] for the path-component of the identity of the group of all smooth diffeomorphisms of \(M\) with respect to the compact-open smooth topology. Finally, some examples are discussed. In particular, when \(M = \mathbb{R}^n\), one gets the same results as those of \textit{F. Mascaró} [Trans. Am. Math. Soc. 275, 163-173 (1983; Zbl 0512.58013)].
scientific article; zbMATH DE number 203538
scientific article