Surfaces have (asymptotic) dimension 2

Abstract

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale version of weak diameter colourings (also known as weak diameter network decompositions), i.e. colourings in which each monochromatic component has small weak diameter.In this paper, we prove that for any p, the class of graphs excluding K3,p as a minor has asymptotic dimension at most 2. This implies that the class of all graphs embeddable on any fixed surface (and in particular the class of planar graphs) has asymptotic dimension 2, which gives a positive answer to a recent question of Fujiwara and Papasoglu. Our result extends from graphs to Riemannian surfaces. We also prove that graphs of bounded pathwidth have asymptotic dimension at most 1 and graphs of bounded layered pathwidth have asymptotic dimension at most 2. We give some applications of our techniques to graph classes defined in a topological or geometrical way, and to graph classes of polynomial growth. Finally we prove that the class of bounded degree graphs from any fixed proper minor-closed class has asymptotic dimension at most 2. This can be seen as a large scale generalization of the result that bounded degree graphs from any fixed proper minor-closed class are 3-colourable with monochromatic components of bounded size. This also implies that (infinite) Cayley graphs avoiding some minor have asymptotic dimension at most 2, which solves a problem raised by Ostrovskii and Rosenthal.M. Bonamy and N. Bousquet are supported by ANR Projects DISTANCIA (ANR-17-CE40-0015) and GrR (ANR-18-CE40-0032). L. Esperet and F. Pirot are supported by ANR Projects GATO (ANR-16-CE40-0009-01) and GrR (ANR-18-CE40-0032).

Minor exclusion in quasi-transitive graphs

“…Bonamy et al [1] proved that all locally finite Cayley graphs of finitely generated groups of asymptotic dimension at least 3 are not minor excluded, moreover from their discussion follows that all quasi-transitive locally finite graphs of asymptotic dimension at least 3 are not minor excluded. On the other side, quasi-transitive locally finite graph that are quasi-isometric to trees have asymptotic dimension 1.…”

Asymptotic dimension of minor-closed families and beyond

“…Ostrovskii and Rosenthal [15] proved a bound depending on the graph H: for every graph H, the class of H-minor free graphs has asymptotic dimension at most 4 |V (H)| − 1. Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott [5] proved the case for bounded maximum degree graphs: for every integer ∆ and graph H, the class of H-minor free graphs of maximum degree at most ∆ has asymptotic dimension at most 2. Bonamy et al [5] used their result to answer a question of Ostrovskii and Rosenthal [15] by showing that for every finitely generated group Γ with a symmetric finite generating set S, if there exists a graph H such that H is not a minor of the Cayley graph for (Γ, S), then the asymptotic dimension of the Cayley graph for (Γ, S) is at most 2.…”

“…Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott [5] proved the case for bounded maximum degree graphs: for every integer ∆ and graph H, the class of H-minor free graphs of maximum degree at most ∆ has asymptotic dimension at most 2. Bonamy et al [5] used their result to answer a question of Ostrovskii and Rosenthal [15] by showing that for every finitely generated group Γ with a symmetric finite generating set S, if there exists a graph H such that H is not a minor of the Cayley graph for (Γ, S), then the asymptotic dimension of the Cayley graph for (Γ, S) is at most 2. Bonamy et al [5] also proved the case H = K 3,t for every positive integer t and the case that H is an apex-forest in Question 1.1.…”

“…Bonamy et al [5] used their result to answer a question of Ostrovskii and Rosenthal [15] by showing that for every finitely generated group Γ with a symmetric finite generating set S, if there exists a graph H such that H is not a minor of the Cayley graph for (Γ, S), then the asymptotic dimension of the Cayley graph for (Γ, S) is at most 2. Bonamy et al [5] also proved the case H = K 3,t for every positive integer t and the case that H is an apex-forest in Question 1.1.…”

“…[5]. It also solves a question [5,Question 9] about the difference between forbidding minors and forbidding "fat minors" and a question [5,Question 5] about forbidding K s,t -minors in a strong sense. Theorem 1.2.…”