  For an ascending HNN-extension $G*_{\psi}$ of a finitely generated abelian
group $G$, we study how a synchronization between the geometry of the group and
weak periodicity of a configuration in $\mathcal{A}^{G*_{\psi}}$ forces global
constraints on it, as well as in subshifts containing it. A particular case are
Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results
imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period
$a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with
monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge
3$, of the (right) Cayley graph of $\mathrm{BS}(1,N)$, estimating the entropy
of the associated subshift together with its mixing properties. We prove that
$\mathrm{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$. We
finally suggest generalizations of the latter results to $n$-colorings of
ascending HNN-extensions of finitely generated abelian groups.
