Triangulating surfaces quasi-isometrically

TL;DR

This paper proves that complete Riemannian surfaces quasi-isometric to bounded degree graphs can be triangulated so that their 1-skeletons are quasi-isometric to the original surfaces, establishing a deep geometric connection.

Contribution

It establishes a precise condition under which Riemannian surfaces can be triangulated to reflect their quasi-isometric properties with graphs.

Findings

Surfaces quasi-isometric to bounded degree graphs admit compatible triangulations.

Identification of the exact condition for the triangulation to preserve quasi-isometry.

Analysis of variants of the triangulation problem and their equivalence.

Abstract

We prove that if a complete Riemannian surface $(\Sigma,d_\Sigma)$ is quasi-isometric to some bounded degree graph $G$, then $\Sigma$ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric. We study several variants of the problem, and identify the right condition making it an if and only if statement.