Surfaces without quasi-isometric simplicial triangulations
2026-03-27 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors build a special kind of curved surface that cannot be covered nicely by a network of triangles where the network closely matches the distances on the surface. They show that no matter how you try to place a graph (a collection of points connected by lines) on the surface, the graph won't serve as a good approximation of the surface's shape. Additionally, their surface has no edges or borders and can have very large "loops" that can't be shrunk. This work answers a previously open question posed by another researcher, Georgakopoulos.
Riemannian surfacetriangulationquasi-isometrysimplicial complex1-skeletonsystoleembedded graphgeometric group theorymetric geometry
Authors
James Davies
Abstract
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.