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

TL;DR

This paper investigates how the structure of ascending HNN-extensions of finitely generated abelian groups influences the properties of subshifts and colorings, revealing constraints on periodic configurations and entropy characteristics.

Contribution

It introduces new results on the relationship between group geometry and weak periodicity, including conditions for strong periodicity and entropy estimates for colorings on Baumslag-Solitar groups.

Findings

A $ ext{BS}(1,N)$-SFT with a configuration of period $a^{N^ ext{ell}}$ must contain a strongly periodic configuration.

$ ext{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$.

Results are generalized to $n$-colorings of ascending HNN-extensions of finitely generated abelian groups.

Abstract

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.