离散的Syn不等式与三角化曲面的分解
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
摘要：
需要多少切割来简化曲面的拓扑？我们提供了这个问题的几个实例的界限，拓扑非平凡的闭曲线，裤子分解，并与一个给定的组合映射在三角组合曲面（或其对偶交叉度量对应）的切割图的最小长度。我们的工作是建立在黎曼收缩不等式的基础上的，该不等式以曲面的亏格和面积来约束非平凡闭曲线的最小长度。我们首先描述了一个系统的方法来翻译黎曼收缩不等式的离散设置，反之亦然。这意味着一个猜想Przytycka和Przytycki从1993年，一些新的收缩不等式的离散设置，以及事实上，一个定理的哈钦森边宽度的三角曲面和格罗莫夫的收缩不等式的表面是本质上等价的。我们还讨论了这些证明如何推广到更高的维度。然后重点讨论曲面的拓扑分解。利用Buser的思想，证明了任意亏格为g的n个三角形的组合三角剖分曲面存在长度为O（g^{3/2}n^{1/2}）的Pants分解，并给出了计算这种分解的O（gn）时间算法.最后，我们考虑的问题嵌入一个割图（或更一般的细胞图）与给定的组合映射在一个给定的表面上。使用随机三角剖分，我们证明（本质上），对于任何选择的组合地图，有一些表面上的任何细胞嵌入与组合地图的三角形的三角形的数量具有长度超线性的组合表面。对于嵌入在多面体三角剖分上的图也有类似的结果。
英文摘要：
How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g^{3/2}n^{1/2}) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
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