On flat submaps of maps of non-positive curvature

TL;DR

This paper establishes bounds on the area of non-positively curved maps lacking flat submaps of a certain radius and characterizes when such infinite maps are quasi-isometric to the Euclidean plane, extending previous results.

Contribution

It proves new area bounds for non-positively curved maps without flat submaps and characterizes Euclidean plane tessellations among infinite maps with angle functions.

Findings

Area of maps without flat submaps is linearly bounded by radius and size.

Infinite tessellating maps are quasi-isometric to the Euclidean plane iff finitely many non-flat vertices and faces.

Generalizes previous results to maps with angle functions.

Abstract

We prove that for every $r>0$ if a non-positively curved $(p,q)$-map $M$ contains no flat submaps of radius $r$, then the area of $M$ does not exceed $Crn$ for some constant $C$. This strengthens a theorem of Ivanov and Schupp. We show that an infinite $(p,q)$-map which tessellates the plane is quasi-isometric to the Euclidean plane if and only if the map contains only finitely many non-flat vertices and faces. We also generalize Ivanov and Schupp's result to a much larger class of maps, namely to maps with angle functions.