Remarks on stability for semiproper exceptional leaves (Q789069)

There are two main results. Theorem A states that, for a closed, foliated 3-manifold M and a semiproper leaf L with finitely generated fundamental group, L is stable on a proper side if and only if the germinal holonomy group of L on that side is trivial. Theorem C gives a \(C^\infty\) example showing that the condition on the fundamental group is essential. For this, a method of G. Hector is used for constructing an exceptional minimal set. The author states no differentiability hypothesis in Theorem A, so one assumes that the weakest hypothesis used in this paper, \(C^ 0\) with \(C^\infty\) leaves, is intended. If the semiproper leaf L is proper, the theorem is due to T. Inaba, who assumed \(C^ 1\)-smoothness. If L is exceptional, the author shows that Inaba's proof extends without difficulty. Some remarks are in order. (1) Due to an unpublished result of G. Duminy, this extension is vacuous for \(C^ k\) foliations with \(k\geq 2\). By Duminy, in these cases a semiproper exceptional leaf L has a Cantor subset of ends. Since \(\dim(L)=2\), the fundamental group will not be finitely generated. (2) Inaba's proof used, in an essential way, Novikov's theorem that, in foliated 3-manifolds, a leaf with a vanishing cycle is compact. Novikov assumed \(C^ 2\)-smoothness, but the extension to \(C^ 1\) foliations was established by J. Franks in 1970. The \(C^ 0\) case has been proven recently by \textit{V. V. Solodov} [Mat. Sb., Nov. Ser. 119 (161), No.3 (11), 340-354 (1982; Zbl 0518.57014)]. Thus, Inaba's theorem and the author's extension are valid for \(C^ 0\)-foliated 3- manifolds.