Surfaces have (asymptotic) dimension 2

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 colorings (also known as weak diameter network decompositions), i.e. colorings in which each monochromatic component has small weak diameter. In this paper, we prove that for any $p$, the class of graphs excluding $K_{3,p}$ 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-colorable 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.Comment: 35 pages, 4 figures - v3: correction of the statements of Theorem 5.2, Corollary 5.3 and Theorem 5.9. Most of the results in this paper have been merged to arXiv:2012.0243

Assouad-Nagata dimension of minor-closed metrics

Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$ in a fixed minor-closed family such that the underlying space of $M$ is the vertex-set of $G$, and the metric of $M$ is the distance function in $G$. Minor-closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge-deletion and edge-contraction. In this paper, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results for the asymptotic dimension of $H$-minor free unweighted graphs and the Assouad-Nagata dimension of some 2-dimensional continuous spaces (e.g.\ complete Riemannian surfaces with finite Euler genus) and their corollaries.Comment: arXiv admin note: text overlap with arXiv:2007.0877

Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2

Asymptotic dimension and Assouad-Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy et al. [J. Eur. Math. Society] showed that any proper minor-closed class has asymptotic dimension 2, dropping to 1 only if the treewidth is bounded. We improve this result by showing it also holds for the stricter Assouad-Nagata dimension. We also characterise when subdivision-closed classes of graphs have bounded Assouad-Nagata dimension.Comment: 30 pages, 1 figur

Approximating spaces of Nagata dimension zero by weighted trees

We prove that if a metric space $X$ has Nagata dimension zero with constant $c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$, that is, if $X$ is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for $0$-hyperbolic proper metric spaces. This generalizes a result of Gupta.Comment: Revised version. We have corrected many minor inaccuracies in the tex

Triangulations of uniform subquadratic growth are quasi-trees

It is known that for every $\alpha \geq 1$ there is a planar triangulation in which every ball of radius $r$ has size $\Theta(r^\alpha)$. We prove that for $\alpha <2$ every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree

Comparison of metric spectral gaps

Let $A=(a_{ij})\in M_n(\R)$ be an $n$ by $n$ symmetric stochastic matrix. For $p\in [1,\infty)$ and a metric space $(X,d_X)$, let $\gamma(A,d_X^p)$ be the infimum over those $\gamma\in (0,\infty]$ for which every $x_1,...,x_n\in X$ satisfy $$\frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n d_X(x_i,x_j)^p\le \frac{\gamma}{n}\sum_{i=1}^n\sum_{j=1}^n a_{ij} d_X(x_i,x_j)^p.$$ Thus $\gamma(A,d_X^p)$ measures the magnitude of the {\em nonlinear spectral gap} of the matrix $A$ with respect to the kernel $d_X^p:X\times X\to [0,\infty)$. We study pairs of metric spaces $(X,d_X)$ and $(Y,d_Y)$ for which there exists $\Psi:(0,\infty)\to (0,\infty)$ such that $\gamma(A,d_X^p)\le \Psi(\gamma(A,d_Y^p))$ for every symmetric stochastic $A\in M_n(\R)$ with $\gamma(A,d_Y^p)<\infty$. When $\Psi$ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if $n\in \N$ and $p\in (2,\infty)$ then for every $f_1,...,f_n\in L_p$ there exist $x_1,...,x_n\in L_2$ such that {equation}\label{eq:p factor} \forall\, i,j\in \{1,...,n\},\quad \|x_i-x_j\|_2\lesssim p\|f_i-f_j\|_p, {equation} and $$\sum_{i=1}^n\sum_{j=1}^n \|x_i-x_j\|_2^2=\sum_{i=1}^n\sum_{j=1}^n \|f_i-f_j\|_p^2.$$ This statement is impossible for $p\in [1,2)$, and the asymptotic dependence on $p$ in \eqref{eq:p factor} is sharp. We also obtain the best known lower bound on the $L_p$ distortion of Ramanujan graphs, improving over the work of Matou\v{s}ek. Links to Bourgain--Milman--Wolfson type and a conjectural nonlinear Maurey--Pisier theorem are studied.Comment: Clarifying remarks added, definition of p(n,d) modified, typos fixed, references adde