Triangulations of uniform subquadratic growth are quasi-trees

Abstract

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

Revised version submitted to Annales Henri Lebesgue

References in corpus (2)

• Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces
• Geodesic spaces of low Nagata dimension