# the growth rate of poly-$\mathbb{Z}$ group

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.

• 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.
– YCor
Commented Aug 31, 2022 at 13:10