Elementary planes in the Apollonian orbifold

Abstract

In this paper, we study the topological behavior of elementary planes in the Apollonian orbifold M A 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 M_A limiting only on a finite union of closed geodesic planes. This contrasts with other acylindrical hyperbolic 3-manifolds analyzed by Benoist and Oh [Ergodic Theory Dynam. Systems 42 (2002), pp. 514–553], McMullen, Mohammadi, and Oh [Invent. Math. 209 (2017), pp. 425–461] and McMullen, Mohammadi, and Oh [Duke Math. J. 171 (2022), pp. 1029–1060]. On the other hand, we show that certain rigidity still holds: the area of an elementary plane in M A 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 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 M_A from its boundary data; requiring the plane to be elementary in turn puts restrictions on these data.