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

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 $\Pi_{1}^{0}$ Medvedev degrees. This extends a classification proved by Joseph Miller for $\mathbb{Z}^{d},$ $d\geq1$.