Surfaces without quasi-isometric simplicial triangulations

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