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 study Z-covers of compact hyperbolic surfaces and provide the first description of all possible horocycle orbit closures in this category. Surprisingly, the topology and Hausdorff dimension of these non-homogeneous orbit closures delicately and discontinuously depends on the choice of a hyperbolic metric on the covered compact surface. Nevertheless, some rigidity is preserved in the form of integer Hausdorff dimension of all orbit closures. Based on an ongoing series of works together with James Farre and Yair Minsky.