Triangulations of uniform subquadratic growth are quasi-trees
Annales Henri Lebesgue, Volume 5 (2022), pp. 905-919.

Keywordsplanar triangulation, 2-manifold, uniform volume growth, quasi-tree, asymptotic dimension

### Abstract

It is known that for every $\alpha \ge 1$ there is a planar triangulation in which every ball of radius $r$ has size $\Theta \left({r}^{\alpha }\right)$. 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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