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](http://en.wikipedia.org/wiki/Growth_rate_%28group_theory%29)).


When $\sigma(a)=\begin{pmatrix}2&1\\1&1\end{pmatrix}$, it was mentioned [here](https://mathoverflow.net/questions/56092/amenable-exponential-growth) 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](http://en.wikipedia.org/wiki/Polycyclic_group), my question is:


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



growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?
To answer your 1st question see the paper of Milnor, "Growth of finitely generated solvable groups." 


To answer your 2nd question, simply generalize your second example to higher dimensions, e.g. take $\mathbb{Z}^3\rtimes\_{\sigma} \mathbb{Z}$ where $\sigma \in SL\_3(\mathbb{Z})$ has an eigenvalue not on the unit circle.


