Elementary planes in the Apollonian orbifold

TL;DR

This paper investigates the topological and geometric properties of elementary planes in the Apollonian orbifold, revealing unique behaviors in their distribution and establishing bounds and classifications for these planes.

Contribution

It provides a complete classification of elementary planes in the Apollonian orbifold and demonstrates their bounded area and the closedness of their union, highlighting unique distribution properties.

Findings

Existence of elementary planes causes failure of equidistribution.

The area of elementary planes is uniformly bounded.

The union of all elementary planes is closed.

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.