Surfaces without quasi-isometric simplicial triangulations

Abstract

We construct a complete Riemannian surface $\hat{I}\pounds$ that admits no triangulation $G\subset \hat{I}\pounds$ such that the inclusion $G^{(1)}\hookrightarrow \hat{I}\pounds$ 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 $G\subset \hat{I}\pounds$ such that $G^{(1)}\hookrightarrow \hat{I}\pounds$ is a quasi-isometry. This answers a question of Georgakopoulos.

9 pages, 3 figures