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Undecidability of the block gluing classes of homshifts

Nishant Chandgotia, Silvère Gangloff, Benjamin Hellouin de Menibus, Piotr Oprocha

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.

Undecidability of the block gluing classes of homshifts

Abstract

A homshift is a -dimensional shift of finite type which arises as the space of graph homomorphisms from the grid graph to a finite connected undirected graph . 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 -block gluing or not, by relating this problem to the one of finiteness for finitely presented groups.

Paper Structure

This paper contains 40 sections, 42 theorems, 35 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Elementary background and notations
  3. Graphs
  4. Shift spaces
  5. Finitely presented groups
  6. Algebraic topology background
  7. Fundamental group of a graph
  8. Graph covers and deck transformations
  9. Universal graph cover
  10. Some deck transformations of the universal cover
  11. Quotients of the universal cover
  12. Regular covers
  13. Quotienting by squares
  14. The square group
  15. Definition
  16. ...and 25 more sections

Key Result

theorem 1.1

The classes of $\Theta(n)$-block gluing and $O(\log(n))$-block gluing two-dimensional homshifts are computably inseparable. That is, it is not possible to algorithmically decide if a two-dimensional homshift is $\Theta(n)$-block gluing or $O(\log(n))$-block gluing.

Figures (17)

  • Figure 1:
  • Figure 2: Illustration for the definition of fundamental group on three examples. Left column: the graph $G$. Middle column: full lines represent the chosen spanning tree, dotted lines are the generators. Right column: presentation for the fundamental group obtained by Proposition \ref{['proposition: Free product']}.
  • Figure 3: Illustration of the function $v \mapsto p^a_T(v)$. The spanning tree $T$ has been indicated in the middle column.
  • Figure 4: Illustration for the definition of cover. For each graph $G$ in the left column is represented a cover of $\tilde{G}$ in the right column, where the labels are the images of vertices by the covering map.
  • Figure 5: Illustration for the definition of universal cover. Left column: the three graphs from Figure \ref{['figure.example.fundamental.group']}. Middle and right columns: two representations of the square cover for different base points, that are indeed isomorphic graphs.
  • ...and 12 more figures

Theorems & Definitions (103)

  • theorem 1.1
  • Remark 2.3
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • proof
  • Definition 3.2: Fundamental Group
  • Proposition 3.5
  • ...and 93 more