Subshifts on groups and computable analysis

Nicanor Carrasco-Vargas

Subshifts on groups and computable analysis

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Pith's one-line read Every effective dynamical system on a recursively presented finitely generated group has a computable zero-dimensional Cantor extension, which extends simulation results to non-symbolic systems and yields new Medvedev degree classifications.

desk verdict A serious thesis with a genuinely new main theorem, but the proof of Theorem 4.2 rests on an unproven algorithmic uniformity claim in Lemma 4.17. read the letter →

arxiv 2505.14247 v1 pith:B6YGS23G submitted 2025-05-20 math.DS cs.ITmath.GRmath.ITmath.LOmath.MG

Nicanor Carrasco-Vargas This is my paper · ORCID

classification math.DScs.ITmath.GRmath.ITmath.LOmath.MG

keywords groupssubshiftsmathbbsystemsconnectiondegreesdynamicaleffective

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Subshifts are sets of colorings of a group that respect a translation action. A subshift of finite type is defined by finitely many forbidden local patterns. A long line of simulation results shows that, on certain groups, subshifts of finite type can simulate any computable action on the Cantor set, the space of infinite binary sequences. These simulations have mostly been limited to zero-dimensional spaces such as Cantor space, and it was unclear how to handle systems like rotations of a circle or matrix actions on a torus.

This thesis introduces a broad class called effective dynamical systems: systems that are topologically conjugate to a computable action on a recursively compact subset of a computable metric space. The main theorem says that when the acting group is finitely generated and recursively presented, every such system is a topological factor of a computable action on a zero-dimensional effectively closed subset of Cantor space. In other words, the zero-dimensional restriction is not an obstacle: the system can be lifted to a symbolic-like extension while preserving computability.

The thesis also studies Medvedev degrees, a measure of how hard it is to compute a configuration of a subshift, and develops transfer tools through commensurability, quotients, translation-like actions, and quasi-isometries. It classifies the possible Medvedev degrees of SFTs for virtually polycyclic groups, direct products, branch groups, and groups quasi-isometric to the hyperbolic plane. Finally, it proves every connected locally finite infinite graph admits a translation by Z, with transitivity exactly when the graph has one or two ends.

Extended reading notes

Core claim

Theorem 4.2: for any effective dynamical system G acting on X, where G is finitely generated and recursively presented, there exists an effectively closed zero-dimensional space X~ subset of {0,1}^N and a computable action G on X~ such that G acting on X is a topological factor of G acting on X~. If true, every effective dynamical system on such groups has a computable Cantor-space extension, and after combining with simulation results, many non-symbolic systems (torus actions, circle rotations, braid group actions, algebraic actions) are factors of SFTs.

Load-bearing premise

The main proof assumes that for every recursively compact subset X of a computable metric space, the search-based algorithm in Lemma 4.17 can actually compute, uniformly in n, effective covers P_n of X with diameter at most 2^{-n}, with decidable refinement inclusions, and with the extra partition property when X is zero-dimensional. This algorithmic uniformity is asserted after a semi-decidability argument and is the step on which the zero-dimensional extension in Theorem 4.2 rests. The theorem also depends on the group being recursively presented, used in Proposition 3.20 to make pullback fullshifts effectively closed; the abstract's 'general metric space' phrasing hides this hypothesis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a substantial thesis, and the main result is a real idea: the effective zero-dimensional extension theorem for effective dynamical systems on recursively presented groups, and the corollary that many non-symbolic systems (torus actions, circle rotations, braid group actions) become factors of SFTs. The novelty is genuine, and the framing through computable analysis is useful. The Medvedev degree material is also serious work: the transference results via commensurability, quotients, translation-like actions, and quasi-isometries are well motivated, and the classification results for virtually polycyclic groups and direct products are meaningful.

But the proof of Theorem 4.2 has a load-bearing gap. Lemma 4.17 must produce effective covers P_n uniformly, with decidable refinement inclusions. The proof asserts that it is semi-decidable whether a finite union of closures of basic balls is contained in the interior of a previous P_n element, because the interior of the elements in the partition are effectively open. That is not established. The elements of P_n are finite unions of closures of basic balls; their interiors are not shown to be effectively open, and Example 2.36 shows the analogous closure/interior assertions are not automatic in this framework. The zero-dimensional disjointness condition has the same problem. Since the inverse-limit extension in Theorem 4.2 rests on this uniform sequence, the main theorem is not fully proved as written. This is a proof gap, not a counterexample, and it might be fixable with a more careful construction, but it needs to be addressed.

There are also several deferred proofs: Theorem 4.4 is sent to [BCR24], Cohen's quasi-isometry construction in Section 5.3.4 is only summarized, and the text truncates before Chapters 7-9, so parts of the thesis cannot be verified from this submission. That is a lot of weight on external and prior papers, especially given the volume of self-citations. The self-citations are not a flaw in themselves, but the reliance on them is heavy.

Overall: this is a serious thesis that deserves a serious referee. I would send it to peer review, and ask the author to either fill the gap in Lemma 4.17 or state it as a hypothesis, and to provide stable references for the deferred results. The ideas are worth engaging with.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters fitted to data. The paper's claims are parameter-free theorems; the only numeric constants (quasi-isometry constants, cover diameters, radii) are auxiliary bounds chosen in proofs, not fitted. The axioms listed are the external theorems and background frameworks on which the main results rest. The new notions EDS and WEDS are definitions within existing frameworks, and no new particles, forces, dimensions, or constants are introduced. The Medvedev degree invariant predates the paper.

assumptions (5)

standard math ZFC and standard computability theory (Turing machines, type-2 effectivity)
Background for all definitions in Chapters 2-9; not questioned.
domain assumption The simulation theorems cited in Theorem 4.24 (Hochman, Barbieri, Salo, etc.) are correct
Theorem 4.25 depends on these external simulation results to turn zero-dimensional extensions into factors of SFTs.
domain assumption Simpson's theorem: MSFT(Z^d) for d at least 2 equals all Pi0-1 Medvedev degrees
Used in Theorem 5.37 and Theorem 5.38 to transfer all degrees to virtually polycyclic and direct product groups.
domain assumption Seward's theorem: every infinite finitely generated group admits a translation-like action by Z
Used in Corollary 5.24 and Theorem 5.40; the thesis later proves an effective version for decidable word problem groups in Chapter 9.
domain assumption Cohen's QI subshift construction and the Jeandel-Kari-Hooper immortal Turing machine encodings
Section 5.3.4 and Theorem 5.42 rely on these external constructions; they are cited but not reproduced fully.

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Pith. "Pith review of Subshifts on groups and computable analysis." pith.science (2026). https://pith.science/paper/B6YGS23G

@misc{pith2026250514247,
  author       = {Pith},
  title        = {Pith review of: Subshifts on groups and computable analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6YGS23G}},
  note         = {Machine review of arXiv:2505.14247}
}

abstract

The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and recursion theory. Different questions about the dynamics of these systems have been answered in recursion-theoretical terms. In this work we further explore this connection. We use the framework of computable analysis to explore the class of effective dynamical systems on metric spaces, and relate these systems to subshifts of finite type (SFTs) on groups. We prove that every effective dynamical system on a general metric space is the topological factor of an effective dynamical system with topological dimension zero. We combine this result with existing simulation results to obtain new examples of systems that are factors of SFTsWe also study a conjugacy invariant for subshifts on groups called Medvedev degree. This invariant is a complexity measure of algorithmic nature. We develop the basic theory of these degrees for subshifts on arbitrary finitely generated groups. Using these tools we are able to classify the values that this invariant attains for SFTs and other classes of subshifts on several groups. Furthermore, we establish a connection between these degrees and the distribution of isolated points in the space of all subshifts. Motivated by the study of Medvedev degrees of subshifts, we also consider translation-like actions of groups on graphs. We prove that every connected, locally finite, and infinite graph admits a translation by $\mathbb{Z}$, and that this action can be chosen transitive exactly when the graph has one or two ends. This generalizes a result of Seward about translation-like actions of $\mathbb{Z}$ on finitely generated groups.

Figures

Figures reproduced from arXiv: 2505.14247 by the authors.

**Figure 1.1.** Many decorative tilings can be seen as Z 2 -SFTs. Given a finite set of decorated square tiles of the same size, we can define an SFT whose alphabet is the set of tiles, and with the local rule that adjacent tiles preserve the intended decoration. Conversely, every SFT on Z 2 is topologically conjugate to one defined by square decorated tiles. This identification between SFTs and tilings is also valid on finitely ge… view at source ↗

**Figure 4.1.** A representation of the actions Z ↷ X and Z ↷ Y . Now let A ⊂ N \ {0} be some infinite set and let Y ⊂ C be given by Y = {0} ∪ z ∈ C : |z| = 1 n for some n ∈ A . Similarly, the map S : Y → Y given by S(z) = z exp (2πi|z|) is a homeomorphism which induces an action Z ↷ Y . 41 [PITH_FULL_IMAGE:figures/full_fig_p048_4_1.png] view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Decision Problems for Groups and Semigroups

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[BMV09] O. Bogopolski, A. Martino, and E. Ventura. “Orbit Decidability and the Conju- gacy Problem for Some Extensions of Groups”. In:Transactions of the American Mathematical Society362.4 (Nov. 2009), pp. 2003–2036. [BMV10] Oleg Bogopolski, Armando Martino, and Enric Ventura. “Orbit decidability and the conjugacy problem for some extensions of groups”. I...

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Effective Dy- namical Systems beyond Dimension Zero and Factors of SFTs

[BCR24] Sebastián Barbieri, Nicanor Carrasco-Vargas, and Cristóbal Rojas. “Effective Dy- namical Systems beyond Dimension Zero and Factors of SFTs”. In:Ergodic Theory and Dynamical Systems(Oct. 2024), pp. 1–41. [Car24a] Nicanor Carrasco-Vargas. Infinite Eulerian Trails Are Computable on Graphs with Vertices of Infinite Degree.Jan

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On a Rice theorem for dynamical properties of SFTs on groups

[Car25] Nicanor Carrasco-Vargas. “On a Rice theorem for dynamical properties of SFTs on groups”. In:Archiv der Mathematik124.591-603 (6 2025). [Car24b] Nicanor Carrasco-Vargas. “Translation-like Actions by Z, the subgroup membership problem, and Medvedev degrees of effective subshifts”. In:Groups, Geometry, and Dynamics (Aug. 2024). [CRR24] Nicanor Carras...

work page 2025