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

TL;DR

It is proved 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.

Abstract

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.

Figures (5)

• Figure 1: A snapshot of the setting of Lemma \ref{['tree_extension']} and the notations used in the proof. Note that by Claim \ref{['claim_Zbasic']} we know that we can assume that $Z=N_G^{\leqslant 3\ell}(X_{t^*})$, but we only include the original setting of this lemma in the picture.
• Figure 2: (a) The two 2-components $A_1$ and $A_2$ of a subset $A$, and (b) an example of a set $B=\{u,v\}$ which has one 4-component but two $(4,3)$-components.
• Figure 3: A $q$-fat $K_3$-minor.
• Figure 4: The construction of a $q$-fat $K_{2,p}$-minor with $p=3$ in the proof of Lemma \ref{['lem:thetaminor']}. The figure is for illustrating the notation only and should not reflect any level of generality: for instance $r$ can be either much larger or much smaller than $q+\kappa$, and the vertices in $A$ or $B$ closest to $a_i$ or $b_i$ are not necessarily in $P_i$.
• Figure 5: A local view of the $(k,p)$-stretch of a graph, for $k=5$ and $p=1$.

Theorems & Definitions (93)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Corollary 1.10