Examples of Lie foliations (Q2372643)

\textit{D. Tischler} [Topology 9, 153--154 (1970; Zbl 0177.52103)] has shown that every \(G\)-foliation, where \(G\) is an abelian Lie group, can be deformed into a \(G\)--foliation with discrete holonomy. \textit{D. Lehmann} [C. R. Acad. Sci., Paris, Sér. A 286, 251--254 (1978; Zbl 0379.57004)], showed this result is false if \(G\) is the Heisenberg group. In this paper, the authors generalize Lehmann's counterexample showing that, for any simply-connected non-abelian nilpotent group \(G\), there exists a \(G\)-foliation which cannot be deformed into a \(G\)-foliation with trivial holonomy. Similarly, they prove that for any (almost) simple Lie group \(G\), there exists a \(G\)-foliation which has dense holonomy and is structurally stable. Finally, if \(G\) is the Heisenberg group they show that Lehmann's example can be obtained by deforming an \({\mathbb R}^3\)-foliation.