Undecidability of the block gluing classes of homshifts

2025-07-28

AI Summary

This paper addresses the decidability of the Θ(n)-block gluing property for homshifts—d-dimensional shifts of finite type defined by graph homomorphisms. Resolving a long-standing conjecture on the undecidability of this mixed dynamical-combinatorial property, the authors introduce algebraic topology into block covering analysis for the first time, establishing an equivalence between block-gluing classes and the finiteness of certain finitely presented groups. By integrating tools from graph homomorphisms, multidimensional symbolic dynamics, group theory, and homological algebra, they construct a critical counterexample and deliver a rigorous proof. The main result shows that determining whether an arbitrary homshift satisfies the Θ(n)-block gluing property is Turing-undecidable. This work settles a central open problem in the field and uncovers a novel source of undecidability rooted in group representation theory—not in classical symbolic dynamical mechanisms.

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Abstract

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.

Problem

• Determining undecidability of block gluing in homshifts
• Relating homshift properties to finitely presented groups
• Topological interpretation of square cover in homshifts

Innovation

• Interpreting square cover topologically
• Relating block gluing to group finiteness
• Proving undecidability via algebraic topology