Accessibility, planar graphs, and quasi-isometries
Group Theory
2026-05-14 v6 Combinatorics
Metric Geometry
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.
Cite
@article{arxiv.2310.15242,
title = {Accessibility, planar graphs, and quasi-isometries},
author = {Joseph Paul MacManus},
journal= {arXiv preprint arXiv:2310.15242},
year = {2026}
}
Comments
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)'