horospherical group actions on homogeneous spaces are famously known to be extremely rigid. in finite volume homogeneous spaces, it is a special case of ratner's theorems that all horospherical orbit closures are homogeneous. rigidity further extends in rank-one to infinite volume but geometrically finite spaces. the geometrically infinite setting is far less understood. we consider $\mathbb{z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. in particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. based on joint work with james farre and yair minsky.