Recent works [MMO1, arXiv:1802.03853, arXiv:1802.04423, arXiv:2101.08956]
We shed light on the topological behavior of the geodesic plane in the convex core of a geometrically finite hyperbolic 3-dimensional manifold $M$ of infinite volume. In this paper, we focus on the remaining cases of geodesic planes outside the convex core of $M$ and present a complete classification of their closure in $M$.
In particular, it shows different behavior depending on whether exotic roofs are present. where the exotic roof is the geodesic plane contained in the edge $E$ of $M$, bounding the convex core boundary $\partial E$, but by the support planes of $\partial E$ It cannot be separated from the core. .
A prerequisite for the existence of exotic roofs is the presence of exotic rays for bending lamination. where the exotic ray is a geodesic with a finite number of intersections with the measured laminate $\mathcal{L}$, but not asymptotic to any leaf, and eventually from $\mathcal{L}$ I will never leave. Establishes that exotic rays exist only if $\mathcal{L}$ is not multicurve. The evidence is constructive and the ideas involved are important for exotic roof construction.
We also show that the presence of exotic roofs is sufficient for the existence of geodesics satisfying the condition stronger than exotic, represented only by the hyperboloid $\partial E$ and bending stacks. Consequently, we show that geometrically finite edges with exotic roofs exist in all genera. Moreover, in species $1$, the common (in the sense of the veil category) contains an innumerable number of exotic roofs if the terminations are pitted torus and homotopic.
