Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

Abstract

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

40 pages, 1 figure, Final version, To appear in Groups, Geometry, and Dynamics