I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is the generator on the right copy of $\mathbb{Z}$ and the action is just by matrix multiplication.

Here are two examples:

For $\sigma(a)=\begin{pmatrix}1&1\\0&1\end{pmatrix}$, this gives us the discrete Heisenberg group $H_3$, which is nilpotent, and hence by Gromov's theorem, it has polynomial growth rate(see here).

When $\sigma(a)=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, it was mentioned here this group has exponential growth rate.

So my first question is:

1, **Could anyone give me a reference to show the link between whether the above group $G$ has polynomial growth rate or not and the property, say eignvalue, of the matrix $\sigma(a)$?**

Note that the above $G$ is a polycyclic-by-finite group, my question is:

2, **Could anyone give me a polycyclic-by-finite group not of the type of $G$ with exponential growth rate?**