Translation-like actions by $\mathbb{Z}$, the subgroup membership problem, and Medvedev degrees of effective subshifts

TL;DR

The paper shows that every infinite, connected, locally finite graph admits a translation-like action by $\mathbb{Z}$ with $d(v,v\ast1)\le 3$, and that such an action is transitive precisely when the graph has 1 or 2 ends. It then develops computable versions of translation-like actions, proving that any finitely generated infinite group with decidable word problem carries a computable translation-like action by $\mathbb{Z}$ with decidable orbit membership; for groups with more than two ends this arises from a computable $\mathbb{Z}$-subgroup via Stallings-type decompositions. Finally, it applies these results to symbolic dynamics by showing that, for such groups, effective subshifts realize all $\Pi_{1}^{0}$ Medvedev degrees, generalizing Miller’s classification from $\mathbb{Z}$ to a broad class of groups. The work provides a unified framework linking geometric group actions, computability, and the complexity of subshifts, with potential implications for Hamiltonicity questions and the transfer of dynamical properties across groups via translation-like actions.

Abstract

We show that every infinite, locally finite, and connected graph admitsa translation-like action by $\mathbb{Z}$, and that this action can be takento be transitive exactly when the graph has either one or two ends.The actions constructed satisfy $d(v,v\ast 1)\leq3$ for every vertex$v$. This strengthens a theorem by Brandon Seward. We also study the effective computability of translation-like actionson groups and graphs. We prove that every finitely generated infinitegroup with decidable word problem admits a translation-like actionby $\mathbb{Z}$ which is computable, and satisfies an extra condition whichwe call decidable orbit membership problem. As a nontrivial application of our results, we prove that for everyfinitely generated infinite group with decidable word problem, effectivesubshifts attain all $Π_{1}^{0}$ Medvedev degrees. This extends a classification proved by Joseph Miller for $\mathbb{Z}^{d},$ $d\geq1$.

Figures (2)

• Figure 1: Representation of some orbits of a translation-like action in $T_{2}(\mathbb{Z},\mathbb{Z}^{2})$, or alternatively, a finite pattern in a configuration in $X_{2}(\mathbb{Z},\mathbb{Z}^{2})$. In this case, $\mathbb{Z}^{2}$ is endowed with the set of four generators $S=\{(\pm1,0),(0,\pm1)\}$.
• Figure 2: Representation of a finite pattern in a of configuration in $Y[X_{2}(\mathbb{Z},\mathbb{Z}^{2})]$. Here $A$ is the alphabet $\{\text{circle},\text{square},\text{rhombus}\}$, and $Y\subset A^{\mathbb{Z}}$ is the subshift of all sequences that alternate circle, square, and rhombus in that order.

Theorems & Definitions (80)

• Theorem 1.1: Geometric Burnside's problem, seward_burnside_2014
• Theorem 1.2: seward_burnside_2014
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Definition 2.2