Elementary planes in the Apollonian orbifold

Abstract

In this paper, we study the topological behavior of elementary planes in the Apollonian orbifold $M_A$, whose limit set is the classical Apollonian gasket. The existence of these elementary planes leads to the following failure of equidistribution: there exists a sequence of closed geodesic planes in $M_A$ limiting only on a finite union of closed geodesic planes. This contrasts with other 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 an elementary plane in $M_A$ is uniformly bounded above, and the union of all elementary planes is closed. This is achieved by obtaining a complete list of elementary planes in $M_A$, indexed by their intersection with the convex core boundary. 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 puts restrictions on these data.