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Triangulations of uniform subquadratic growth are quasi-trees
Itai Benjamini, Agelos Georgakopoulos · 2021-06-11 · via math.PR updates on arXiv.org

It is known that for every $α\geq 1$ there is a planar triangulation in which every ball of radius $r$ has size $Θ(r^α)$. We prove that for $α<2$ 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.

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