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

Abstract

For an ascending HNN-extension G * ψ 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 A G * ψ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups BS(1, N ), N ≥ 2, for which our results imply that a BS(1, N )-SFT which contains a configuration with period a N ℓ , ℓ ≥ 0, must contain a strongly periodic configuration with monochromatic Z-sections. Then we study proper n-colorings, n ≥ 3, of the (right) Cayley graph of BS(1, N ), estimating the entropy of the associated subshift together with its mixing properties. We prove that 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.