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

Recent works [MMO1, MMO2, BO, Zha2] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically ﬁnite hyperbolic 3-manifolds M of inﬁnite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M , giving a complete classiﬁcation of their closures in M . In particular, we show that the behavior is diﬀerent depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M , which limits on the convex core boundary ∂E , but cannot be separated from the core by a support plane of ∂E . A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray 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 . We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs. We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased in terms of only the hyperbolic surface ∂E and the bending lamination, is suﬃcient for the existence of exotic roofs. As a result, we show that geometrically ﬁnite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.