Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric holonomy order [Formula: see text], the set maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when text] is well-studied as such surfaces are (semi-)translation surfaces. Not only always infinite, core curves form infinite diameter subset curve complex. In this paper we focus on and construct examples illustrating range behaviors for cylinder curves. We prove that if fully punctured, then same analysis shows have has very specific form. Using characterize precisely accumulate point in Gromov boundary.