Geodesic planes in a geometrically finite end and the halo of a measured lamination (Q6611886)

In the paper under review, \(M\) is a geometrically finite hyperbolic 3-manifold of infinite volume. The authors obtain a classification of the closures of geodesic planes outside the convex core of \(M\), in terms of their topological behavior. They show that this behavior depends on whether exotic roofs exist or not, where an exotic roof is a geodesic plane contained in an end \(E\) of \(M\) which limits on the convex core boundary \(\partial E\) but cannot be separated from the core by a support plane of \(\partial E\). The existence of exotic roofs depends on the existence of exotic rays for the bending lamination of the convex core boundary, that is, a geodesic ray that has finite intersection number with a measured lamination \(L\) but is not asymptotic to any leaf nor eventually disjoint from \(L\). The authors show that exotic rays for the bending lamination \(L\) exist if and only if \(L\) is not a multicurve. They also give a sufficient condition for the existence of geodesic rays, namely, a condition phrased only in terms of the hyperbolic surface \(\partial E\) and the bending lamination. They deduce that geometrically finite ends with exotic roofs exist in every genus, and in genus 1, when the end is homotopic to a punctured torus, a generic one contains uncountably many exotic roofs.