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Algorithm for filling curves on surfaces
Geometriae Dedicata ( IF 0.5 ) Pub Date : 2020-01-10 , DOI: 10.1007/s10711-019-00509-2
Monika Kudlinska

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $\pi_1(\Sigma)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length.

中文翻译:

在曲面上填充曲线的算法

令 $\Sigma$ 是一个紧凑的、可定向的负欧拉特征曲面,让 $h$ 是 $\Sigma$ 上的一个完全双曲线度量。$\Sigma$ 中的测地曲线 $\gamma$ 正在填充,如果它将表面切割成拓扑圆盘和环。我们提出了一种有效的算法来确定测地曲线(在 $\pi_1(\Sigma)$ 的某些生成器中表示为一个单词)是否正在填充。在此过程中,我们找到了由 Dehn-Thurston 坐标给出的曲线组合长度的明确界限,即双曲长度。这为我们提供了一种生成集合的有效方法,该集合保证包含与双曲长度有界的简单测地线相对应的所有单词。
更新日期:2020-01-10
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