  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.
