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
$\mathrm {BS}(1,N)$,
$N\ge 2$, for which our results imply that a
$\mathrm {BS}(1,N)$-subshift of finite type 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.