English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

For an ascending HNN-extension $G*_{\psi}$ of a finitely generated abeliangroup $G$, we study how a synchronization between the geometry of the group andweak periodicity of a configuration in $\mathcal{A}^{G*_{\psi}}$ forces globalconstraints on it, as well as in subshifts containing it. A particular case areBaumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our resultsimply 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 withmonochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge3$, of the (right) Cayley graph of $\mathrm{BS}(1,N)$, estimating the entropyof the associated subshift together with its mixing properties. We prove that$\mathrm{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$. Wefinally suggest generalizations of the latter results to $n$-colorings ofascending HNN-extensions of finitely generated abelian groups.

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