Asymptotic dimension of minor-closed families and Assouad-Nagata dimension of surfaces (Q6582315)

Let \((X, d)\) be a pseudometric space and let \(\mathcal{U}\) be a family of subsets of \(X\). Then \(U\) is said to be \(D\)-bounded if each set \(U \in \mathcal{U}\) has diameter at most \(D\) and \(\mathcal{U}\) is said to be \(r\)-disjoint if for any \(a, b\) belonging to different elements of \(U\) we have \(d(a,b)\geq r\). An \(n\)-dimensional control function for \((X, d)\) is a map \(D_{X}: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) such that, for any \(r> 0\), \((X, d)\) has a cover \(U=\bigcup_{i=1}^{n+1} \mathcal{U}_{i}\) such that each \(\mathcal{U}_{i}\) is \(r\)-disjoint and each element of \(\mathcal{U}\) is \(D_{X}(r)\)-bounded. \N\NThe asymptotic dimension of \((X,d)\), denoted by \(\operatorname{asdim}(X,d)\), is the least integer \(n\) such that \((X,d)\) has an \(n\)-dimensional control function (if no such integer \(n\) exists, then the asymptotic dimension is infinite). Asymptotic dimension of metric spaces was introduced in the seminal paper by \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Cambridge: Cambridge University Press. (1993; Zbl 0841.20039)] in the context of geometric group theory. Every metric space can be realized by a graph whose edges are weighted. \N\NIn the paper under review, the authors address the connections between structural graph theory and the theory of asymptotic dimension. In particular, they solve open problems in coarse geometry and geometric group theory via tools from structural graph theory. In particular, the authors prove that every proper minor-closed family of graphs has asymptotic dimension at most \(2\), which gives optimal answers to a question of \textit{K. Fujiwara} and \textit{P. Papasoglu} [Trans. Am. Math. Soc. 374, No. 12, 8887--8901 (2021; Zbl 1506.54013)] and (in a strong form) to a problem raised by \textit{M. I. Ostrovskii} and \textit{D. Rosenthal} [Int. J. Algebra Comput. 25, No. 4, 541--554 (2015; Zbl 1334.20037)] on minor excluded groups. For some special minor-closed families, such as the class of graphs embeddable in a surface of bounded Euler genus, the authors prove a stronger result and apply this to show that complete Riemannian surfaces have Assouad-Nagata dimension at most \(2\) [\textit{P. Assouad}, C. R. Acad. Sci., Paris, Sér. I 294, 31--34 (1982; Zbl 0481.54015)]. In addition, the techniques developed in this paper bring the authors to determine the asymptotic dimension of graphs of bounded layered treewidth and graphs with any fixed growth rate, which are graph classes that are defined by purely combinatorial notions and properly contain graph classes with some natural topological and geometric flavour.