Elementary planes in the Apollonian orbifold

In this paper, we study the topological behavior of elementary planes in theApollonian orbifold $M_A$, whose limit set is the classical Apollonian gasket.The existence of these elementary planes leads to the following failure ofequidistribution: there exists a sequence of closed geodesic planes in $M_A$limiting only on a finite union of closed geodesic planes. This contrasts withother acylindrical hyperbolic 3-manifolds analyzed in [MMO1, arXiv:1802.03853,arXiv:1802.04423]. On the other hand, we show that certain rigidity still holds: the area of anelementary plane in $M_A$ is uniformly bounded above, and the union of allelementary planes is closed. This is achieved by obtaining a complete list ofelementary planes in $M_A$, indexed by their intersection with the convex coreboundary. The key idea is to recover information on a closed geodesic plane in$M_A$ from its boundary data; requiring the plane to be elementary in turn putsrestrictions on these data.