An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture

Abstract:

Let $\mathcal {R}$ be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let $V$ be the set of vertices, and for every $v \in V$, let $\kappa (v)$ denote the (Gaussian) curvature of $v$: $2 \pi$ minus the sum of incident polygon angles. Descartes showed that $\sum _{v \in V} \kappa (v) = 4 \pi$ whenever $\mathcal {R}$ may be realized as the surface of a convex polytope in $\mathbb {R}^3$. More generally, if $\mathcal {R}$ is made of finitely many polygons, Euler’s formula is equivalent to the equation $\sum _{v \in V} \kappa (v) = 2 \pi \chi (\mathcal {R})$ where $\chi (\mathcal {R})$ is the Euler characteristic of $\mathcal {R}$. Our main theorem shows that whenever $\sum _{v \in V : \kappa (v) < 0} \kappa (v)$ converges and there is a positive lower bound on the distance between any pair of vertices in $\mathcal {R}$, there exists a compact closed 2-manifold $\mathcal {S}$ and an integer $t$ so that $\mathcal {R}$ is homeomorphic to $\mathcal {S}$ minus $t$ points, and further $\sum _{v \in V} \kappa (v) \le 2 \pi \chi (\mathcal {S}) - 2 \pi t$. In the special case when every polygon is regular of side length one and $\kappa (v) > 0$ for every vertex $v$, we apply our main theorem to deduce that $\mathcal {R}$ is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless $\mathcal {R}$ is a prism, antiprism, or the projective planar analogue of one of these that $|V| \le 3444$. This resolves a recent conjecture of Higuchi.