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 q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q ≤ 2 is well-studied as such surfaces are (half-)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 q ≥ 3 and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if q ≥ 3 and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The proof is then used to characterize the flat metrics for which the embedded cylinder curves have infinite diameter.