Horocycle orbit closures in periodic surfaces

(Joint work with James Farre and Or Landesberg)

Horocyclic (and generally horospherical) orbit closures in hyperbolic manifolds are not very well understood in the infinite-volume setting. For finite volume it is classical that they are either closed (associated to cusps) or dense. In the case of Z-covers of compact manifolds there is an interesting connection between such closures and the geometry of "tight" circle valued functions on the compact manifolds and their maximal-stretch laminations. In dimension 2, this makes it possible to describe the orbit closures in great detail. In particular, although nontrivial orbit closures are fractal in some sense, they always have integral Hausdorff dimension.