Horocycles in hyperbolic 3-manifolds with round Sierpi\'nski limit sets

TL;DR

This paper classifies the orbit closures of horocycle flows in a specific class of hyperbolic 3-manifolds with Sierpiński gasket limit sets, extending previous convex cocompact results to more general geometrically finite cases.

Contribution

It provides a classification of horocycle orbit closures in geometrically finite hyperbolic 3-manifolds with Sierpiński gasket limit sets, broadening the scope beyond convex cocompact manifolds.

Findings

Horocycle orbit closures are properly immersed submanifolds.

Extension of previous convex cocompact results to geometrically finite manifolds.

Classification of orbit closures in manifolds with Sierpiński gasket limit sets.

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.