Accessibility, planar graphs, and quasi-isometries

Abstract

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such a group is virtually a free product of free and surface groups, and thus virtually admits a planar Cayley graph.

Major revision with several corrections and streamlining changes. 64 pages, 17 figures. Some results related to 'relative accessibility' have been moved to a standalone article on the topic; see 'Relative accessibility for graphs (2026)'