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Chun-Hung Liu (劉俊宏), Asymptotic dimension of minor-closed families and beyond
Thursday, October 22, 2020 @ 10:10 AM - 11:10 AM KST
The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance functions in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.