Horocycles in hyperbolic 3-manifolds with round Sierpi\'nski limit sets
Dynamical Systems
2025-06-24 v2 Differential Geometry
Geometric Topology
Abstract
Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpi\'nski gasket, i.e. M is geometrically finite and acylindrical with a compact, totally geodesic convex core boundary. In this paper, we classify orbit closures of the 1-dimensional horocycle flow on the frame bundle of M. As a result, the closure of a horocycle in M is a properly immersed submanifold. This extends the work of McMullen-Mohammadi-Oh where M is further assumed to be convex cocompact.
Cite
@article{arxiv.2501.14067,
title = {Horocycles in hyperbolic 3-manifolds with round Sierpi\'nski limit sets},
author = {Dongryul M. Kim and Minju Lee},
journal= {arXiv preprint arXiv:2501.14067},
year = {2025}
}
Comments
40 pages, 1 figure, Final version, To appear in Groups, Geometry, and Dynamics