# American Institute of Mathematical Sciences

February  2019, 39(2): 881-903. doi: 10.3934/dcds.2019037

## Geodesic planes in geometrically finite manifolds

 Department of Mathematics, Ohio State University, 231 W 18th Ave., Columbus, OH 43210, USA

Received  January 2018 Revised  July 2018 Published  November 2018

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank 1 cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property played a key role in the recent breakthroughs of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtain results on the existence and isolation of proper closed immersions of elementary surfaces.

Citation: Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037
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##### References:
Apollonian circle packing (solid). Inversions through dual circles (dashed) generate a geometrically finite group containing rank-$1$ parabolic subgroups
Proof of Lemma 3.2
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