On the structure of certain nontransitive diffeomorphism groups on open manifolds (Q2901969)

A group \(G\) is called perfect if \(G=[G,G]\), where the commutator subgroup is generated by all commutators \(\left[g_{_1},g_{_2}\right]=g_{_1}g_{_2}g_{_1}^{-1}g_{_2}^{-1}\), or if its homology group \(H_1(G)=G/[G,G]=0\). Let \((M,{\mathcal F})\) be a foliated manifold. A diffeomorphism \(f:M\to M\) is leaf preserving if \(f(L_x)=L_x\) and is foliation preserving if \(f(L_x)=L_{f(x)}\) for all \(x\in M\), where \(L_x\) is the leaf of \({\mathcal F}\) passing through \(x\). Let \(\text{Diff}^\infty_c(M,{\mathcal F})\) denote the group of leaf preserving diffeomorphisms of \(M\) which are isotopic to the identity through compactly supported isotopies of leaf preserving diffeomorphisms, and \(\text{Diff}^r(M)\) the group of all \(C^r\)-diffeomorphisms that are isotopic to the identity. For an \(n\)-dimensional connected, compact manifold \(M\), let \(M^{(k)}=M\times\mathbb R^k\) be the product \((n+k)\)-dimensional manifold endowed with the \(k\)-dimensional product foliation \({\mathcal F}_k=\{\{\text{pt}\}\times\mathbb R^k\}\). Let \(G^k=\text{Diff}^\infty(M^{(k)},{\mathcal F}_k)\) be the group of all leaf preserving \(C^\infty\)-diffeomorphisms that can be joined with the identity by smooth isotopies of leaf preserving \(C^\infty\)-diffeomorphisms, and \(G^k_c=\text{Diff}^\infty_c(M^{(k)},{\mathcal F}_k)\) the group of all leaf preserving \(C^\infty\)-diffeomorphisms that can be joined with the identity by compactly supported smooth isotopies of leaf preserving \(C^\infty\)-diffeomorphisms. A group is called bounded if it is bounded with respect to any bi-invariant metric. In [Monatsh. Math. 120, No. 3--4, 289--305 (1995; Zbl 0847.57033)], \textit{T.~Rybicki} proved that if \((M,{\mathcal F})\) is a foliated smooth manifold, then \(\text{Diff}^\infty_c(M,{\mathcal F})\) is perfect.NEWLINENEWLINEIn this interesting paper, the authors prove that the groups \(G^k\) and \(G^k_c\) are perfect and bounded. Some other properties are investigated.