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Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets (2501.14067v2)

Published 23 Jan 2025 in math.DS, math.DG, and math.GT

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.

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