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

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.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). C.-H. Liu is partially supported by NSF under Grant No. DMS-1929851 and DMS-1954054.

Polynomial growth and asymptotic dimension

“…Krauthgamer and Lee [16] showed that graphs of polynomial growth γ(r) ≤ Cr k embed in this sense in R O(k log k) . Bonamy et al deduce from this in [4] that graphs of polynomial growth ≤ Cr k have asymptotic dimension bounded by O(k log k). It is further shown that graphs of superpolynomial growth can have infinite asymptotic dimension.…”

“…Asymptotic dimension is relevant in several contexts: in geometric group theory, as groups of finite asymptotic dimension satisfy the Novikov conjecture [21], in geometry [17], [7] and in graph theory [22], [11], [4].…”

“…From the geometric point of view it is interesting to calculate the exact asymptotic dimension of a space. This has been accomplished for several 'natural' classes of spaces: It is shown in [7] that the asymptotic dimension of a hyperbolic group G is equal to dim(∂G) + 1, it is shown in [12], [17] that n-dimensional Hadamard manifolds of pinched negative curvature have asymptotic dimension n, and in [11], [15], [4] that planar graphs (or more generally planar geodesic metric spaces) have asymptotic dimension at most 2. In this paper we extend this list to the class of spaces with polynomial growth.…”

Triangulations of uniform subquadratic growth are quasi-trees

“…Proof of Theorem 1.4. It has been proved that every planar graph has asymptotic dimension at most 2 [8,23]. Thus it only remains to show that asdimpGq " 1 implies that G is quasi-isometric to a tree when G is a planar triangulation.…”

“…Then either asdimpGq " 2, or G is quasi-isometric to a tree (in which case asdimpGq " 1). This uses a recent result that every planar graph has asymptotic dimension at most 2 [8,23]. Theorem 1.4 cannot be extended to planar graphs with arbitrarily long facial cycles as shown by [12,Example 2.4.].…”