String graphs are quasi-isometric to planar graphs

We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that $$\frac{1}{23660800}{d}_S(u,v)\le d_G(u,v)\le 162d_S(u,v)$$ for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the distance between $u$ and $v$ in $S$ and $G$ respectively. In other words, string graphs are quasi-isometric to planar graphs. This theorem lifts a number of theorems from planar graphs to string graphs, we give some examples. String graphs have Assouad-Nagata (and asymptotic dimension) at most 2. Connected, locally finite, quasi-transitive string graphs are accessible. A finitely generated group $\mathrm{\Gamma }$ is virtually a free product of free and surface groups if and only if $\mathrm{\Gamma }$ is quasi-isometric to a string graph. Two further corollaries are that countable planar metric graphs and complete Riemannian planes are also quasi-isometric to planar graphs, which answers a question of Georgakopoulos and Papasoglu. For finite string graphs and planar metric graphs, our proofs yield polynomial time (for string graphs, this is in terms of the size of a representation given in the input) algorithms for generating such quasi-isometric planar graphs. We further extend our techniques to show that every complete Riemannian surfaces $\mathrm{\Sigma }$ of bounded Euler genus has a triangulation $G\subset \mathrm{\Sigma }$ such that $G^{(1)}\hookrightarrow \mathrm{\Sigma }$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$.