There has progress in understanding dynamics of the straight-line flow on some infinite abelian covers of translation surfaces. By analogy with other classes of systems (e.g., horocycle flows on hyperbolic surfaces), it seems reasonable to hope that our understanding will extend to some infinite nilpotent covers of translation surfaces, and there are a few results in this direction.

I will describe a sequence C_k of affinely-natural homogeneity constraints on the cylinders of a square-tiled surface. By naturality, the largest surface satisfying such a constraint C_k automatically has a Veech group of SL(2,Z) and has a deck group that acts transitively on squares. I will explain that the largest surface satisfying constraint C_4 is an infinite nilpotent cover of the famous eierlegende Wollmilchsau surface built from 8 squares. Our methods also can produce other natural infinite nilpotent covers of square tiled surfaces, but lots of open questions remain. Understanding the constraints boils down to studying group theoretic questions that are similar in spirit to the still open Burnside problems, and our understanding benefits from the interplay between geometry and algebra. This talk is on work joint with Khalid Bou-Rabee.