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 < 3 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 q > 2 and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if q > 2 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.
Keywords
Cite
@article{arxiv.1909.13760,
title = {Cylinder curves in finite holonomy flat metrics},
author = {Ser-Wei Fu and Christopher J Leininger},
journal= {arXiv preprint arXiv:1909.13760},
year = {2020}
}
Comments
v2. Edits following referee's comments: The structure of a flat metric with infinite diameter set of embedded cylinder curves is made more precise. Together with a generalization of the construction of such metrics, this results in a characterization of those metrics for which the embedded cylinder curves accumulate on the Gromov boundary. Several additional cosmetic changes