Chun-Hung Liu (劉俊宏), Asymptotic dimension of minor-closed families and beyond

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.