Surfaces without quasi-isometric simplicial triangulations

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.

Figures (3)

• Figure 1: The embedded graph $F$ consists of the dashed edges and the boundary of $D$ is the thick black edges around $F$. For the centre vertex $v$, we illustrate the two cycles $C_v$ and $C_v'$ of the Eulerian decomposition that cross at $v$ in red and blue. Also featured are the curves $\rho_{e_1}, \rho_{e_2}, \rho_{e_3}, \rho_{e_4}$ crossing their corresponding edges incident to $v$. Together with part of the boundary of $D$, they bound the subset $D(v)$ of $D$, which is purple in the figure.
• Figure 2: An illustration of (two parts of) the of closed walk $C_i^*$ (orange) of $G^{(1)}$ and $p_{C_i}$ given the cycle $C_i$ (red) of the Eulerian decomposition of $F$. We have that $p_{C_i}(u_1)=v_1$, $p_{C_i}(u_2)= p_{C_i}(u_3) = p_{C_i}(u_4) = p_{C_i}(u_5) = p_{C_i}(u_6) = v_2$, and $p_{C_i}(u_{737})=v_{9105}$.
• Figure 3: An illustration of $\gamma_1$ (orange) and $\gamma_2$ (purple) contained within $D^*$. They intersect at a point $z$ coinciding with a vertex $u$ of the closed walk $C^*_1$ with $p_{C_1}(u)=v$.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3: Erdős, Sachs erdos1963regulare
• Theorem 4
• proof