Surfaces without quasi-isometric simplicial triangulations
This resolves a foundational question in geometric topology and metric geometry, with implications for understanding the structure of surfaces.
The authors tackled the problem of whether every complete Riemannian surface admits a quasi-isometric simplicial triangulation, and they constructed a surface without such a triangulation, answering a question by Georgakopoulos.
We construct a complete Riemannian surface $Σ$ that admits no triangulation $G\subset Σ$ such that the inclusion $G^{(1)} \hookrightarrow Σ$ 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 $G\subsetΣ$ such that $G^{(1)} \hookrightarrow Σ$ is a quasi-isometry. This answers a question of Georgakopoulos.