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I am interested in the growth rate of the poly-$\mathbb{Z}$ group. 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 $\phi_{i}$ need to have for $G$ to have polynomial growth?

Thoughts: we know that $G$ can only have exponential or polynomial growth. According to this paper, every $\phi_{i}$ corresponds to a matrix $M_i$ in $ \operatorname{GL}(m_i, \mathbb{Z})$. Also, from this post, when $\phi_1, \dots, \phi_{n-2}$ represent the identity automorphisms, we can determine the growth of $G$ by looking at the eigenvalue of $M_{n-1}$. I was wondering if there are any known results that allow us to tell the growth rate of general poly-$\mathbb{Z}$ group $G$ by looking at the matrices $M_i$?

Any references for this question would be really appreciated.

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    $\begingroup$ Probably it's of polynomial growth iff each $\phi_i$ has only roots of unity as eigenvalues. However I'm not familiar with this way to describe polycyclic groups so I'm not 100% sure. $\endgroup$
    – YCor
    Commented Aug 31, 2022 at 13:10

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