An analogy of the theorem of Hector and Duminy
An unpublished theorem of G. Duminy asserts that, in a \(C^ 2\) foliation \(\mathcal F\) of codimension one on a closed manifold \(M\), a semiproper, exceptional leaf will have a holonomy contraction on its nonproper side. This leads to the striking result that a semiproper leaf in an exceptional minimal set has a Cantor set of ends. These results have been cited by various authors and were to have appeared in Duminy's thesis, under the direction of G. Hector, in the late 1970's. Unfortunately, this was never written. (The reviewer and J. Cantwell have written up a proof and circulated it informally.) In the paper under review, the author presents an analogous result in higher codimension. Not surprisingly, very restrictive hypotheses are required on the holonomy pseudogroup \(\Gamma\) and on the orbit \(\Gamma(x_ 0)\) which corresponds to the leaf in question. Roughly speaking, the pseudogroup is required to consist of similarity transformations and the ``exceptional and semiproper'' hypothesis on \(\Gamma(x_ 0)\), which makes no sense in higher codimension, is replaced by a hypothesis that ``\(\Gamma(x_ 0)\) is nonproper and with bubbles''. The conclusion is that there is a nontrivial element \(g\in \Gamma\) with \(g(x_ 0)=x_ 0\). This is a weaker analogy than would be hoped for, since it allows the possibility that \(g\) is a pure rotation. The author does not know if there is an example in which all holonomy elements fixing \(x_ 0\) are pure rotations.
semiproper leaf in an exceptional minimal set
scientific article; zbMATH DE number 55926