Every pmp equivalence relation from a locally-finite Borel graph with planar components is sofic, via approximation by treeable coverings using Dunwoody tracks.
Planar graphs and covers
4 Pith papers cite this work. Polarity classification is still indexing.
4
Pith papers citing it
abstract
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general result is obtained for such graphs where no restriction is put on the number of ends. It is shown that such a graph can be built up from one ended or finite planar graphs in a precise way. The results give a classification of the finitely generated groups with planar Cayley graphs.
verdicts
UNVERDICTED 4representative citing papers
Relative accessibility in graphs is defined relative to peripheral systems, characterized via Boolean ring subrings for quasi-transitive graphs, and shown to match the group-theoretic version while being quasi-isometry invariant when cosets are preserved.
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.
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.
citing papers explorer
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Tracks on planar complexes and soficity
Every pmp equivalence relation from a locally-finite Borel graph with planar components is sofic, via approximation by treeable coverings using Dunwoody tracks.
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Relative accessibility for graphs
Relative accessibility in graphs is defined relative to peripheral systems, characterized via Boolean ring subrings for quasi-transitive graphs, and shown to match the group-theoretic version while being quasi-isometry invariant when cosets are preserved.
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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.
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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.