A homshift is a $d$-dimensional shift of finite type which arises as the space of graph homomorphisms from the grid graph $\mathbb Z^d$ to a finite connected undirected graph $G$. While shifts of finite type are known to be mired by the swamp of undecidability, homshifts seem to be better behaved and there was hope that all the properties of homshifts are decidable. In this paper we build on the work by Gangloff, Hellouin de Menibus and Oprocha (arxiv:2211.04075) to show that finer mixing properties are undecidable for reasons completely different than the ones used to prove undecidability for general multidimensional shifts of finite type. Inspired by the work of Gao, Jackson, Krohne and Seward (arxiv:1803.03872) and elementary algebraic topology, we interpret the square cover introduced by Gangloff, Hellouin de Menibus and Oprocha topologically. Using this interpretation, we prove that it is undecidable whether a homshift is $Θ(n)$-block gluing or not, by relating this problem to the one of finiteness for finitely presented groups.

翻译：同态移位是一种有限型$d$维移位，它产生于从网格图$\mathbb Z^d$到有限连通无向图$G$的图同态空间。尽管已知有限型移位深陷于不可判定性的泥沼，同态移位似乎表现更佳，人们曾期望其所有性质都是可判定的。本文基于Gangloff、Hellouin de Menibus和Oprocha的研究（arxiv:2211.04075），证明更精细的混合性质因完全不同于用于证明一般多维有限型移位不可判定性的原因而不可判定。受Gao、Jackson、Krohne和Seward的工作（arxiv:1803.03872）及初等代数拓扑的启发，我们从拓扑角度阐释了Gangloff、Hellouin de Menibus和Oprocha引入的方形覆盖。利用这一阐释，通过将该问题与有限展示群的有限性问题相关联，我们证明了同态移位是否为$Θ(n)$-块粘合是不可判定的。