Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

Key Points

The closure of a horocycle in these manifolds is identified as a properly immersed submanifold, extending previous findings.
Orbit closures of the 1-dimensional horocycle flow are classified for hyperbolic 3-manifolds with round Sierpiński limit sets.
This work allows a better understanding of the geometry of geometrically finite and acylindrical manifolds with convex core boundaries.
The results build upon prior studies, enhancing the framework of horocycle dynamics in complex geometric settings.

Abstract

Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpiński gasket, that is, 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.

Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

Key Points

Abstract

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