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The asymptotic dimension is an invariant of metric spaces introduced byGromov in the context of geometric group theory. In this paper, we study theasymptotic dimension of metric spaces generated by graphs and their shortestpath metric and show their applications to some continuous spaces. Theasymptotic dimension of such graph metrics can be seen as a large scalegeneralisation of weak diameter network decomposition which has beenextensively studied in computer science.  We prove that every proper minor-closed family of graphs has asymptoticdimension at most 2, which gives optimal answers to a question of Fujiwara andPapasoglu and (in a strong form) to a problem raised by Ostrovskii andRosenthal on minor excluded groups. For some special minor-closed families,such as the class of graphs embeddable in a surface of bounded Euler genus, weprove a stronger result and apply this to show that complete Riemanniansurfaces have Assouad-Nagata dimension at most 2. Furthermore, our techniquesallow us to prove optimal results for the asymptotic dimension of graphs ofbounded layered treewidth and graphs of polynomial growth, which are graphclasses that are defined by purely combinatorial notions and properly containgraph classes with some natural topological and geometric flavours.

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces