Cylinder curves in finite holonomy flat metrics

Abstract

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