Coarse geometry of quasi-transitive graphs beyond planarity

Abstract

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.

14 pages, 1 figure. This version corrects two mistakes in Section 5 of the journal version of the paper (see the note at the end of the new section 5)

References in corpus (6)

• Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions
• Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces
• Surfaces have (asymptotic) dimension 2
• Graph minors and metric spaces
• Accessibility, planar graphs, and quasi-isometries
• Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2

Cited by in corpus (1)

• Coarse cops and robber in graphs and groups