Triangulations of uniform subquadratic growth are quasi-trees

Itai Benjamini; Agelos Georgakopoulos

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arXiv · 2106.06443

Triangulations of uniform subquadratic growth are quasi-trees

Itai Benjamini·Agelos Georgakopoulos

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Abstract

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.

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Itai Benjamini, Agelos Georgakopoulos. 2021-06-11. Triangulations of uniform subquadratic growth are quasi-trees. https://arxiv.org/abs/2106.06443

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