Thick embedding conjecture for the solvable Heisenberg extension

Let $H$) be the Heisenberg group, and let $H\rtimes_{\psi}\mathbb{R}$ be its semidirect product with $\mathbb{R}$, where the action is given by

$$\left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array}\right)\cdot\psi(t) = \left(\begin{array}{ccc} 1 & e^tx & z \\ 0 & 1 & e^{-t}y \\ 0 & 0 & 1 \end{array}\right).$$

An embedding is $\varepsilon$-thick when its image satisfies the paper's thickness condition, and its volume is measured in $H\rtimes_{\psi}\mathbb{R}$. Thick embedding conjecture. For every $d$ there exist constants $C=C(d)$ and $\varepsilon=\varepsilon(d)$ such that every finite graph $\Gamma$ with maximal degree $d$ admits an $\varepsilon$-thick embedding into $H\rtimes_{\psi}\mathbb{R}$ with volume

$$\leq C|\Gamma|\ln(1+|\Gamma|).$$

An immediate consequence of this conjecture is that the dichotomy at the heart of Hume, Mackay and Tessera's work is also detected by wiring profiles. The conjecture is presented as an open question about improving bounds for thick embeddings into rank-one symmetric-space analogues and nilpotent or solvable Lie groups.