Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G{\ast }_{\psi }$ 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 ${\mathcal{A}}^{G{\ast }_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{B}\mathrm{S}(1,N)$, $N\ge 2$, for which our results imply that a $\mathrm{B}\mathrm{S}(1,N)$-SFT which contains a configuration with period $a^{N^{\mathrm{\ell }}}$, $\mathrm{\ell }\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge 3$,ofthe(right)Cayleygraphof$$\mathrm{B}\mathrm{S}$(1,N)$, estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm{B}\mathrm{S}(1,N)$ 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.