Limit sets of foliations in hyperbolic 3-manifolds
Throughout the paper, \(\mathcal F\) is a 2-dimensional foliation of a closed irreducible 3-manifold \(M\) with negatively curved (in Gromov's sense) non-elementary fundamental group, \(\widetilde{\mathcal F}\) is its lift to the universal covering \(\widetilde M\) of \(M\) and \(\mathcal H\) is the leaf space of \(\widetilde{\mathcal F}\). The limit set \(\Lambda_F\) of a leaf \(F\) of \(\widetilde{\mathcal F}\) is defined as the intersection \(\overline F\cap S^2_\infty\), where \(\overline F\) denotes the closure of \(F\) in \(\widetilde M\cup S^2_\infty\). The author obtains a number of results concerning limit sets of the leaves of \(\widetilde{\mathcal F}\), among other things: (1) If \(\mathcal F\) is Reebless (i.e. contains no Reeb components), then either \(\Lambda_F= S^2_\infty\) for every \(F\) of \(\widetilde{\mathcal F}\) or \(\Lambda_F\neq S^2_\infty\) for every such \(F\); (2) if \(\mathcal F\) has a compact leaf, then \(\mathcal H\) is Hausdorff iff \(\Lambda_F =S^2_\infty\) for every \(F\); (3) if \(\mathcal F\) is Reebless and contains a compact leaf (or one of two other, more technical, conditions is satisfied) and \(\Lambda_F\neq S^2_\infty\) for some \(F\), then the Hausdorff dimensions of all the limit sets \(\Lambda_F\) are bounded from above by a constant strictly less than 2.
scientific article; zbMATH DE number 1133070