We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement.
@article{10_37236_12661,
author = {Louis Esperet and Ugo Giocanti},
title = {Coarse geometry of quasi-transitive graphs beyond planarity},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/12661},
zbl = {1543.05033},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12661/}
}
TY - JOUR
AU - Louis Esperet
AU - Ugo Giocanti
TI - Coarse geometry of quasi-transitive graphs beyond planarity
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/12661/
DO - 10.37236/12661
ID - 10_37236_12661
ER -
%0 Journal Article
%A Louis Esperet
%A Ugo Giocanti
%T Coarse geometry of quasi-transitive graphs beyond planarity
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/12661/
%R 10.37236/12661
%F 10_37236_12661
Louis Esperet; Ugo Giocanti. Coarse geometry of quasi-transitive graphs beyond planarity. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/12661
