Triangulations of uniform subquadratic growth are quasi-trees

Benjamini, Itai; Georgakopoulos, Agelos

Annales Henri Lebesgue, Tome 5 (2022), pp. 905-919.

On sait que pour tout $\alpha \ge 1$ il existe une triangulation planaire dans laquelle toute boule de rayon $r$ a une taille $\Theta \left(r^{\alpha }\right)$. Nous prouvons que pour $\alpha <2$ une telle triangulation est quasi-isométrique à un arbre. Le résultat s’étend aux $2$-variétés riemanniennes de genre fini, et aux graphes simplement connexes à grande échelle. Nous prouvons également que toute triangulation planaire de dimension asymptotique 1 est quasi-isométrique à un arbre.

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.

Classification : 05C10, 05C12, 57M15, 57M50, 51F30, 60G99
Mots clés : planar triangulation, 2-manifold, uniform volume growth, quasi-tree, asymptotic dimension

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