On flat submaps of maps of nonpositive curvature

Olshanskii, A.; Sapir, M.

doi:10.1090/tran/7487

On flat submaps of maps of nonpositive curvature
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by A. Yu. Olshanskii and M. V. Sapir PDF

Trans. Amer. Math. Soc. 371 (2019), 4869-4894 Request permission

Abstract:

We prove that for every $r>0$ if a nonpositively 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 nonflat vertices and faces. We also generalize Ivanov and Schupp’s result to a much larger class of maps, namely to maps with angle functions.

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