Disproof of the tree product conjecture via the Heisenberg group

Abstract

Product structure theory aims to understand complex graphs by embedding them into products of simpler graphs. In this direction, Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2022) put forth the conjecture that all graphs of degree-$d$ polynomial growth (i.e., where balls of radius $r$ have $\mathcal{O}(r^d)$ vertices) can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size clique. In this paper, we disprove this conjecture for $d = 4$. The counterexamples are finite subgraphs of a Cayley graph of the discrete $3$-dimensional Heisenberg group $\mathbb{H}(\mathbb{Z})$. These graphs were first proposed by Huang and McCarty as potential counterexamples to the conjecture. A key technical tool of our proof is the ''quantitative central collapse'' theorem due to Cheeger, Kleiner and Naor (2011), guaranteeing that every Lipschitz map from the continuous Heisenberg group $\mathbb{H}$ to the function space $L_1$ collapses along a central line.