The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?



Example of a polycyclic group which is not of polynomial growth?
Semi-direct product of $\mathbb Z$ and $\mathbb Z^2$ determined by the matrix
$\left( \begin{array}{ccc}
2 & 1 \\
1 & 1 \end{array} \right)$.


