Surfaces without quasi-isometric simplicial triangulations

TL;DR

We construct a boundary-less complete Riemannian surface with arbitrarily large systole that admits no quasi-isometric simplicial triangulation or embedded graph, answering a question by Georgakopoulos.

Contribution

We provide the first example of a surface without boundary that cannot be quasi-isometrically triangulated or embedded with a quasi-isometric graph.

Findings

Constructed a boundary-less surface with no quasi-isometric triangulation.

Showed the surface has arbitrarily large systole.

Proved no embedded graph yields a quasi-isometry to the surface.

Abstract

We construct a complete Riemannian surface $\Sigma$ that admits no triangulation $G\subset \Sigma$ such that the inclusion $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry, where $G^{(1)}$ is the simplicial 1-skeleton of $G$. Our construction is without boundary, has arbitrarily large systole, and furthermore, there is no embedded graph $G\subset\Sigma$ such that $G^{(1)} \hookrightarrow \Sigma$ is a quasi-isometry. This answers a question of Georgakopoulos.