Classify geodesic-plane orbit closures for geometrically finite manifolds with Fuchsian ends

Determine the orbit closures of geodesic planes, and extend the orbit-closure results used for convex cocompact manifolds with Fuchsian ends to general geometrically finite hyperbolic manifolds with Fuchsian ends, in order to enable the paper’s recurrence and measure-rigidity strategy in that setting.

Background

The paper proves arithmeticity for infinite-volume convex cocompact hyperbolic manifolds with Fuchsian ends. It explains that extending the result to geometrically finite manifolds with Fuchsian ends would require handling cusps and the degenerations they produce in recurrence and nonconcentration arguments.

A prior obstacle is that the relevant orbit closures are not known in the geometrically finite setting. In particular, the classification of geodesic-plane orbit closures is unresolved, including in the setting of Apollonian circle packings. Establishing such orbit-closure results is therefore identified as a prerequisite for extending the paper’s method.