Small curvature laminations in hyperbolic 3-manifolds (Q1017847)

It was shown by \textit{A. Zeghib} [Ann. Sci. Éc. Norm. Supér. (4) 24, No. 2, 171--188 (1991; Zbl 0738.53019)] that any totally geodesic codimension-one lamination in a closed hyperbolic 3-manifold is a finite union of disjoint closed surfaces. Relaxing the condition of totally geodesic to quasi-isometric, \textit{S. R. Fenley} [Topology 31, No. 3, 667--676 (1992; Zbl 0770.57015)] showed that such a lamination cannot be a foliation of the hyperbolic 3-manifold. It makes sense then to inquire into the nature of the boundary leaves of such a lamination. As any leaf can be made into a boundary leaf, by blowing up the leaf and removing the interior, the author focuses on laminations with no complementary region being an interval bundle over the surface. \textit{J. W. Cannon} and \textit{W. P. Thurston} [Geom. Topol. 11, 1315--1355 (2007; Zbl 1136.57009)] and \textit{S. R. Fenley} [J. Am. Math. Soc. 12, No. 3, 619--676 (1999; Zbl 0930.53024)] provided examples of quasi-isometric laminations all of whose leaves are planes, annuli or Möbius strips. Restricting attention to laminations with leaves that are close to totally geodesic (in principal curvature), the author proves that boundary leaves cannot be planes, annuli or Möbius strips. More precisely, the paper under review proves the following: Let \({\mathcal L}\) be a codimension-one lamination in a finite volume hyperbolic 3-manifold, with principal curvatures bounded in a neighbourhood of 0 (not containing 1 and -1), and with no complementary region an interval bundle over a surface. Then no boundary leaf of \({\mathcal L}\) is a plane. Furthermore, if the principal curvature lies in a suitably small neighbourhood of 0, then the boundary leaf cannot be an infinite annulus or Möbius strip. The proof uses relatively simple arguments. A plane boundary leaf \(L_0\) in a finite volume hyperbolic 3-manifold \(M\) must intersect a small ball infinitely many times. As a result, there are leaves \(\widetilde{L}_k\) arbitrarily close to its lift \(\widetilde{L}_0\) in \(\mathbb H^3\). As leaves that are not parallel in \(M\) lift to leaves that diverge in \(\mathbb H^3\), we can find a sequence of points \(\{ x_k \}\) in \(\widetilde{L}_0\) that are at a fixed distance from \(\widetilde{L}_k\). We observe that no element of \(\pi_1(M)\) fixes \(\widetilde{L}_0\). If three almost totally geodesic leaves in \(\mathbb H^3\) intersect a suitably small ball then one of them separates the other two. Observing different cases for the leaves \(\widetilde{L}_0\), \(\widetilde{L}_k\), \(\widetilde{L}_l\) and \(\gamma(\widetilde{L}_0)\) (for \(\gamma \in \pi_1(M)\)), the author obtains infinitely many balls of the same radius (tangent to \(\widetilde{L}_0\) at \(x_k\)) that remain disjoint under the action of \(\pi_1(M)\). These balls therefore project down to disjoint balls in \(M\) contradicting the fact that \(M\) has finite volume. The proof for no annuli (or Möbius strips) is similar in nature, using properties of parabolic and loxodromic isometries of the generator of the stabiliser of \(\widetilde{L}_0\).