Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

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.