Cylinder curves in finite holonomy flat metrics

TL;DR

This paper investigates the behavior of embedded cylinders in flat surfaces with finite holonomy, revealing conditions under which their core curves form finite or infinite diameter sets in the curve complex and characterizing their accumulation points.

Contribution

It provides a detailed analysis of embedded cylinder curves in flat surfaces with finite holonomy, especially for q > 2, and characterizes when these curves have finite or infinite diameter in the curve complex.

Findings

Embedded cylinder curves form finite diameter sets if the surface is fully punctured and q > 2.

Embedded cylinder curves can only have infinite diameter under specific metric conditions.

The paper characterizes when these curves accumulate on a point in the Gromov boundary.

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.