Simple lacunary hyperbolic Kazhdan groups of rational cohomological dimension 16 are built with arbitrary positive real second ℓ²-Betti number.
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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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Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number
Simple lacunary hyperbolic Kazhdan groups of rational cohomological dimension 16 are built with arbitrary positive real second ℓ²-Betti number.
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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.