Finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs; k-planar coarsely simply connected quasi-transitive graphs are quasi-isometric to planar graphs.
arXiv preprint arXiv:2312.08902 , year=
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Connected locally finite quasi-transitive graphs quasi-isometric to planar graphs are accessible, classifying such finitely generated groups as virtually free products of free and surface groups that admit planar Cayley graphs.
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Almost planar finitely presented groups
Finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs; k-planar coarsely simply connected quasi-transitive graphs are quasi-isometric to planar graphs.
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Accessibility, planar graphs, and quasi-isometries
Connected locally finite quasi-transitive graphs quasi-isometric to planar graphs are accessible, classifying such finitely generated groups as virtually free products of free and surface groups that admit planar Cayley graphs.