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

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.