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We study geometric and topological properties of infinite graphs that arequasi-isometric to a planar graph of bounded degree. We prove that everylocally finite quasi-transitive graph excluding a minor is quasi-isometric to aplanar graph of bounded degree. We use the result to give a simple proof of theresult that finitely generated minor-excluded groups have Assouad-Nagatadimension at most 2 (this is known to hold in greater generality, but all knownproofs use significantly deeper tools). We also prove that every locally finitequasi-transitive graph that is quasi-isometric to a planar graph is $k$-planarfor some $k$ (i.e. it has a planar drawing with at most $k$ crossings peredge), and discuss a possible approach to prove the converse statement.

Coarse geometry of quasi-transitive graphs beyond planarity