Elementary planes in the Apollonian orbifold

Zhang, Yongquan

doi:10.1090/tran/8751

Elementary planes in the Apollonian orbifold
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by Yongquan Zhang;

Trans. Amer. Math. Soc. 376 (2023), 453-506

DOI: https://doi.org/10.1090/tran/8751

Published electronically: October 14, 2022

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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 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 $M_A$ 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.

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