Finite subgraphs of the discrete Heisenberg group Cayley graph have O(r^4) growth but do not embed into any strong product of four linear-growth trees and a constant clique.
Wood (2022)
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Disproof of the tree product conjecture via the Heisenberg group
Finite subgraphs of the discrete Heisenberg group Cayley graph have O(r^4) growth but do not embed into any strong product of four linear-growth trees and a constant clique.