Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

Abstract

Let $\mathsf{M}$ be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpiński gasket, that is, $\mathsf{M}$ 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 $\mathsf{M}$. As a result, the closure of a horocycle in $\mathsf{M}$ is a properly immersed submanifold. This extends the work of McMullen–Mohammadi–Oh, where $\mathsf{M}$ is further assumed to be convex cocompact.