Triangulating surfaces quasi-isometrically

Abstract

We prove that if a complete Riemannian surface $(\hat{I}\pounds ,d_{\hat{I}}\pounds )$ is quasi-isometric to some bounded degree graph $G$, then $\hat{I}\pounds$ 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.

26 pages, 6 figures. Reorganized the introduction and Section 3