Curvature and geometry of tessellating plane graphs

Abstract

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesies holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where (p, q) ∈ {(3, 6), (4, 4), (6, 3)}.