Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\phi_3} \mathbb{Z} \dots ) \rtimes_{\phi_{n-1}} \mathbb{Z}$$ What are the necessary and sufficient conditions $\phi_{n-1}$ needs to have for $G$ to have polynomial growth?
Thoughts: we know that $G$ can only have exponential or polynomial growth. According to this paper, $\phi_{n-1}$ corresponds to a matrix $M$ in $ \operatorname{GL}(m, \mathbb{Z})$ for some $m$. I was wondering if there are any known results that allow us to tell the growth of G by looking at this matrix $M$?
Any references for this question would be really appreciated, thanks for reading.
Let's write $G = G_{n-1} \rtimes_{\phi_{n-1}} \mathbb{Z}$. Letting $M$ be the identity matrix, equivalently $\phi_{n-1} : \mathbb Z \to \text{Aut}(G)$ is the trivial homomorphism, we obtain a direct product $G = G_{n-1} \times \mathbb Z$. From this it follows that the growths of $G$ and $G_{n-1}$ are equivalent: $G$ has polynomial growth if and only if $G_{n-1}$ does; and $G$ has exponential growth if and only if $G_{n-1}$ does.
Therefore $\phi_{n-1}$ does not determine the growth. To put it another way, there do not exist any conditions on $\phi_{n-1}$ alone which are necessary and sufficient for $G$ to have exponential growth.