子式闭族的渐近维数与曲面的Assouad-Nagata维数
Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces
AI总结：
该研究证明真子式闭图族的渐近维数至多为2，解决了相关公开问题，还将结果推广至曲面的Assouad-Nagata维数及其他组合图类。
摘要：
渐近维数是Gromov在几何群论背景下引入的度量空间不变量。本文研究由图及其最短路径度量生成的度量空间的渐近维数，并展示其在若干连续空间中的应用。这类图度量的渐近维数可视为弱直径网络分解的大尺度推广，而弱直径网络分解在计算机科学中已有广泛研究。\n 我们证明每个真子式闭图族的渐近维数至多为2，这为Fujiwara与Papasoglu的一个问题，以及（以强形式）Ostrovskii与Rosenthal提出的关于排除子式群的问题给出了最优答案。对于一些特殊的子式闭族，例如可嵌入有界欧拉亏格曲面的图类，我们证明了更强的结果，并应用该结果证明完备黎曼曲面的Assouad-Nagata维数至多为2。此外，我们的技术还能证明有界分层树宽图和多项式增长图的渐近维数的最优结果，这两类图均由纯组合概念定义，且真包含具有某些自然拓扑和几何特征的图类。
英文摘要：
The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. In this paper, we study the asymptotic dimension of metric spaces generated by graphs and their shortest path metric and show their applications to some continuous spaces. The asymptotic dimension of such graph metrics can be seen as a large scale generalisation of weak diameter network decomposition which has been extensively studied in computer science. We prove that every proper minor-closed family of graphs has asymptotic dimension at most 2, which gives optimal answers to a question of Fujiwara and Papasoglu and (in a strong form) to a problem raised by Ostrovskii and Rosenthal on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, we prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniques allow us to prove optimal results for the asymptotic dimension of graphs of bounded layered treewidth and graphs of polynomial growth, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavours.
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