Triangulations of uniform subquadratic growth are quasi-trees

TL;DR

The paper proves that planar triangulations with subquadratic growth are quasi-isometric to trees, extending the result to Riemannian surfaces and large-scale simply connected graphs, and also to triangulations with asymptotic dimension 1.

Contribution

It establishes that all planar triangulations with growth rate less than quadratic are quasi-trees, broadening understanding of their large-scale geometric structure.

Findings

Triangulations with growth rate less than quadratic are quasi-isometric to trees.

The result extends to Riemannian 2-manifolds of finite genus.

Planar triangulations with asymptotic dimension 1 are quasi-isometric to trees.

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.