# Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic

I'm studying the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain (link at ProjectEuclid) and I cannot see a claim made in a proof. I don't know if the my question is appropriate to this site, I apologize if not.

Theorem 1.3. Let $$G$$ be a compact $$p$$-adic analytic group. Then any word $$w$$ of a free group $$F$$ has finite width in $$G$$.

Definition. We say that $$w$$ is a $$\cal{N}_p$$-word if for every finitely generated pro-$$p$$ group $$H$$, $$H/\overline{\omega(H)}$$ is nilpotent-by-finite, where $$\overline{\omega(H)}$$ denotes the closure of $$\omega(H)$$ in $$H$$.

Theorem 3.1. Let $$\omega$$ be a $$\cal{N}_p$$-word and $$G$$ a finitely generated pro-$$p$$ group. Then $$\omega(G)$$ is closed.

Set $$d = d(G)$$ and $$H$$ a free pro-$$p$$ group on generators $$x_1,...,x_d,z$$. We have that $$\gamma_n(H^{p^t}) \leq \overline{\omega(H)}$$ since $$\omega$$ is a $$\cal{N}_p$$-word. The generators $$y_1,...,y_s$$ of $$\overline{\langle x_1,...,x_d \rangle^{p^t}}$$ are pro-$$p$$ words in $$x_i$$. So he applies the theorem 1.3 to get $$k$$ such that, for every $$i_1,...,i_n \in \{1,...,s\}$$, $$[z,y_{i_1},...,y_{i_n}] \equiv v_{i_1,...,i_n} \pmod{\gamma_{n+2}(H^{p^t})}$$ where $$v_{i_1,...,i_n}$$ is a product of at most $$k$$ $$\omega$$-values in $$H$$.

I suppose that he works on the quotient group, but I cannot see the conditions to apply the theorem 1.3. Whence the question:

Why is every nilpotent-by-finite finitely generated pro-p-group always $$p$$-adic analytic?

• What's the condition you can't see. Why the quotient is $p$-adic analytic?
– YCor
Dec 10, 2020 at 17:05
• @YCor exactly. I cannot see why the quotient is $p$-adic analytic. Dec 10, 2020 at 17:12
• So, to make it self-contained, the question is: why is every nilpotent-by-finite finitely generated pro-$p$-group always $p$-adic analytic.
– YCor
Dec 10, 2020 at 19:21
• @YCor it's right! I've edited Dec 10, 2020 at 19:47

This is indeed true: every finitely generated nilpotent-by-finite (= virtually nilpotent) pro-$$p$$-group is $$p$$-adic analytic.
Since every finitely generated nilpotent group has all its subgroup finitely generated, a finitely generated nilpotent profinite group $$G$$ has a composition series by closed subgroups in which each successive quotient is procyclic. If $$G$$ is moreover pro-$$p$$, it follows that each successive quotient is either isomorphic to $$\mathbf{Z}_p$$, or a quotient thereof (which is finite). So these successive quotients are $$p$$-adic analytic. Since being $$p$$-adic analytic is stable under taking extensions, it follows that every finitely generated nilpotent pro-$$p$$-group is $$p$$-adic analytic, and the same follows with "nilpotent-by-finite".