Cylinder curves in finite holonomy flat metrics

Ser-Wei Fu; Christopher J Leininger

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arXiv · 1909.13760

Cylinder curves in finite holonomy flat metrics

Ser-Wei Fu·Christopher J Leininger

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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.

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BibTeX RIS

Ser-Wei Fu, Christopher J Leininger. 2020-12-17. Cylinder curves in finite holonomy flat metrics. https://arxiv.org/abs/1909.13760

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