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

Metadata

DOI10.5802/ahl.139
Keywords planar triangulation, 2-manifold, uniform volume growth, quasi-tree, asymptotic dimension

Abstract

It is known that for every α≥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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