Medvedev degrees of subshifts on groups

The paper proves the dichotomy for virtually polycyclic G—MSFT(G)={0M} if G is virtually cyclic, and otherwise all Π0^1 degrees—by combining (i) commensurability invariance to reduce the virtually cyclic case to Z, and (ii) a subgroup-to-group transference principle (Corollary 4.5) from Z^2≤G together with Simpson’s theorem and an upper Π0^1 bound (Observation 3.5). This is stated precisely in Theorem 5.4 and its proof. The candidate solution reaches the identical classification via an explicit construction: it directly recodes along a finite-index Z-subaction in the virtually cyclic case, and in the non-virtually-cyclic case induces an SFT on G that simulates a Z^2-SFT on each left coset, then shows Medvedev equivalence using computable coset representatives (available in polycyclic groups). The approaches differ in packaging—abstract transfer results in the paper versus a hands-on coset-wise construction in the model—but the conclusions and required hypotheses coincide. Minor clarifications the model could add are: nearest-neighbor recoding for 2D SFTs and explicitly invoking that the Π0^1 pullback bound needs recursive presentability (which polycyclic groups have). Overall, both are correct and consistent with the paper’s framework. See Theorem 5.4, Corollary 4.5/Proposition 4.7, Theorem 5.1, and Observation 3.5 in the paper .