Measurable versions of Whitney's 2-isomorphism theorem are established for locally finite graphings by defining weak isomorphisms that preserve edge measures, cycles, and hyperfinite subgraphs, with rigidity for weakly 3-connected infinitely-ended cases and implementation via countable measurable Wh
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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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Whitney's 2-isomorphism theorem for graphings
Measurable versions of Whitney's 2-isomorphism theorem are established for locally finite graphings by defining weak isomorphisms that preserve edge measures, cycles, and hyperfinite subgraphs, with rigidity for weakly 3-connected infinitely-ended cases and implementation via countable measurable Wh
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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.