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A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I

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Pith's one-line read Sets of tuples in a free group with a single-ended Makanin-Razborov diagram have a canonical minimal-complexity diagram, yielding pseudo closures, ordinal ranks, and canonical envelopes that transfer to semigroups and algebras.

desk verdict Important but unfinished: the canonicality theorem for arbitrary sets of tuples is plausible and significant, yet the proof of the main uniqueness theorem skips a non-trivial step, and the paper leans on unpublished preprints. read the letter →

arxiv 2505.22755 v1 pith:MYKT4FID submitted 2025-05-28 math.GR math.LOmath.RA

Z. Sela This is my paper
classification math.GRmath.LOmath.RA MSC 20F6520F1020E0620M0516S1017B0103C45
keywords Makanin-RazborovdiagramsfreegroupssemigroupsassociativealgebrasLiesingleendedpseudotopologycanonicalenvelope
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

Every set of $\ell$-tuples in a free group gives rise to a Makanin-Razborov (MR) diagram: a finite collection of resolutions that records how all the tuples sit inside the free group, but the diagram depends on choices and is not canonical. This paper shows that when a set admits a single-ended MR diagram — one whose abelian decompositions contain no free products — there is a minimal-complexity diagram, and it is essentially unique: any two such diagrams have the same number of resolutions and isomorphic trimmed reduced completions. Why this matters is that the canonical diagram turns a set of tuples into an invariant object, and over it the author defines a pseudo closure, a well-founded ordinal rank, and a canonical envelope for definable sets, all without invoking model theory. The same uniqueness is then transferred to sets of tuples in free semigroups, free associative algebras over countable fields, and free Lie algebras, so single-ended varieties in those categories inherit a canonical diagram.

What carries the argument

The load-bearing object is a modeled resolution: a countable resolution of limit groups augmented by subtowers that sit over QH (closed-surface) vertex groups, together with a weak test sequence of homomorphisms from the given tuple set. Associated to a modeled completion is its reduced completion, obtained by collapsing each subtower to its bottom QH vertex group, and then the trimmed reduced completion, which keeps only the QH vertex groups, edge groups, and the embedded limit group, with abelian vertex groups cut down to the minimal ranks needed. Complexity is a lexicographically ordered tuple of integers: the Euler characteristics and genera of the QH vertex groups at the bottoms of subtowers, followed by the sum of ranks of abelian vertex groups not inside subtowers, followed by their number; this makes complexities well-ordered, so a minimal complexity diagram exists. The uniqueness proof runs by taking a weak test sequence of a maximal-complexity resolution, forming framed closures, and using retractions and complexity bounds to show the two trimmed reduced completions are isomorphic.

What would settle it

Find a set of tuples in a free group that admits single-ended MR diagrams but for which two minimal-complexity collections have trimmed reduced completions with different numbers of resolutions or non-isomorphic QH or abelian vertex group structure; if such a set exists, Theorem 1.16 is false. A less direct check is whether every sequence of homomorphisms from the set really has a subsequence factoring through a countable modeled resolution with a weak test sequence, the property imported from [Se9].

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Extended reading notes

Core claim

The central claim is Theorem 1.16: if a set $U \subset F_k^\ell$ has two minimal complexity single-ended finite collections of countable modeled resolutions satisfying Definition 1.15, the two collections contain the same number of resolutions and, after reordering, the trimmed reduced completions are similar. There are isomorphisms between them that map QH vertex groups, abelian vertex groups, and the embedded images of the associated limit groups onto the corresponding objects. In other words, among all single-ended MR diagrams that encode the set, the one of least complexity is a genuine invariant of the set, not just of its Zariski closure. The paper then builds a pseudo closure $pcl(S)$ from the canonical diagram, proves the family of pseudo closed sets is Noetherian, defines a countable ordinal rank, and constructs canonical envelopes for definable sets of minimal rank; in the later sections these constructions are carried over to free semigroups, free associative algebras, and free Lie algebras.

Load-bearing premise

The whole machinery assumes as a black box that the unpublished predecessor preprint [Se9] really constructs, for every set of tuples, finitely many countable modeled resolutions with weak test sequences; if that construction fails, Theorem 1.7 and everything built on it collapse.

Editorial extensions

If this is right

  • A set of tuples with a single-ended diagram has an invariant canonical diagram: minimal-complexity collections agree in the number of resolutions and in the isomorphism type of every trimmed reduced completion.
  • The pseudo closure of a single-ended set refines the Zariski topology: every variety with a single-ended diagram is pseudo closed, intersections and unions of pseudo closed sets again have canonical diagrams, and the family of pseudo closed sets is Noetherian.
  • With each such set one can associate a countable ordinal rank, defined purely from diagram complexity, that plays the role of Shelah and Lascar ranks without using model theory.
  • Every minimal-rank definable set with a single-ended diagram has a canonical envelope, unique up to isomorphism of trimmed reduced completions.
  • The same uniqueness, closure, and rank machinery applies to sets of tuples in free semigroups, free associative algebras over countable fields, free Lie algebras, and torsion-free hyperbolic groups, whenever a single-ended diagram exists.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not compare its geometric rank with Lascar rank on the stable theory of free groups; if the two coincide on definable sets, it would give a purely combinatorial way to compute model-theoretic ranks, but this comparison is not made here.
  • For associative algebras, the canonical diagram is attached to the top homogeneous part of a variety; a natural extension is to iterate the same construction on the lower-degree strata to obtain a full filtration of non-homogeneous varieties.
  • The failure of the pseudo closure to be a genuine topology, together with the Noetherianity result, suggests that the complexity of the canonical diagram may be a sharper invariant than Zariski closure for distinguishing definable sets with the same closure; testing this on explicit pairs of sets would clarify how far the pseudo topology goes beyond the Zariski topology.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a canonical Makanin-Razborov diagram for arbitrary sets of tuples in a non-abelian free group, provided the set admits a single-ended MR diagram. The construction passes through countable modeled resolutions equipped with weak test sequences (Theorem 1.7), a lexicographic complexity for such resolutions, and a minimality argument: Theorem 1.16 claims that all minimal-complexity single-ended diagrams for a given set are equivalent up to isomorphism of trimmed reduced completions. The paper then defines a pseudo-closure operation and a rank for sets of tuples (Section 2), a canonical envelope for definable sets, and transfers the main theorem to free semigroups (Theorem 3.2), free associative algebras, and free Lie algebras (Section 4).

Significance. If correct, the main theorem would be a genuine advance: it turns the MR diagram, previously a non-canonical invariant of a variety, into a canonical invariant of a set of tuples, and it provides the first complexity-based uniqueness statement for single-ended diagrams. The idea of using countable modeled resolutions with weak test sequences, and the well-ordering of the complexity, is natural and elegant. The applications to pseudo-closure, rank, and envelopes are plausible and likely to be useful in the program on varieties over semigroups and algebras. The paper builds on deep established tools (Guirardel's theorem on R-tree actions, JSJ decompositions, accessibility, d.c.c. for limit groups) and is clearly the product of a mature research program. However, the proof of the central theorem is incomplete at a key passage, and the paper depends on an unpublished preprint for its main construction.

major comments (4)
  1. [§1, proof of Theorem 1.16, paragraph after Lemma 1.18] The passage from trivial modeled structures to arbitrary modeled structures is asserted rather than proved. After proving Lemmas 1.17 and 1.18 under the assumption that every modeled subtower contains only a base QH vertex group, the proof states that 'by the same argument' these lemmas remain valid for general single ended resolutions with non-trivial modeled structures. No argument is given that the map ν : Red(Comp(Res_i)) → FCl(Red(Comp(Res_1))), obtained after composing with the retractions of the modeled subtowers, still supports the surviving-surface rearrangement used in the proof of Lemma 1.18, nor that the complexity comparisons remain valid when the modeled subtowers are countable and non-trivial. Since Definition 1.15 and Theorem 1.7 explicitly allow arbitrary countable modeled structures, Theorem 1.16 covers this general case, and the missing argument is load-bearing. The same 'by the same argument' gap occurs in the generalization of Lemmas 1.9/1.10 to Lemmas 1.12/1.13.
  2. [§1, Theorem 1.7 and surrounding discussion] The existence and the weak-test-sequence property of the countable modeled resolutions are taken as a black box from the unpublished preprint [Se9]. The paper does not state or prove the needed results from [Se9]; it only refers to 'the construction of resolutions described in [Se9]'. Consequently, the proof of Theorem 1.16 and all its consequences (Theorems 3.2, the pseudo-closure and rank results, and the algebra applications) rest on an unpublished external input. If [Se9] is incomplete or contains an error, the present paper's central claims are unsupported. The dependence should be made explicit, and either the needed statements should be proved here or the paper should be conditional on the availability of [Se9].
  3. [§2, Definition 2.1, paragraph after the definition] The claim that pcl(pcl(S)) = S is not a consequence of Definition 2.1. The definition of pcl(S) is the set of all tuples that factor through the trimmed reduced completions of the canonical diagram of S. If ∆ is the canonical diagram of S, then ∆ is also an MR diagram for pcl(S); by Theorem 1.16 the canonical diagram of pcl(S) is isomorphic to ∆. Hence the correct conclusion is pcl(pcl(S)) = pcl(S), not equality with S. The stated equality is generally false, for instance when S is not pseudo-closed. This is an internal error in Section 2 and should be corrected; it does not affect Theorem 1.16 but it affects the exposition of the pseudo-topology.
  4. [§3, Theorem 3.2] The proof of Theorem 3.2 is given as 'Identical to the proof of theorem 1.16', with no account of the additional structure of pair homomorphisms and the semigroup-theoretic complications from [Se9]. Since the proof of Theorem 1.16 is incomplete at the passage to general modeled structures, Theorem 3.2 inherits that gap. Moreover, the transfer from group homomorphisms to pair homomorphisms requires showing that the trimmed reduced completion uniqueness holds in the category of limit pairs, which is not obvious from the group case. The theorem should either be proved in detail or reduced to explicitly stated results in [Se9].
minor comments (6)
  1. [Throughout] There are numerous typos: 'the the' in the introductory paragraph of §4, 'defer' for 'differ' in the paragraph after Lemma 1.18, 's,c,c,' for 's.c.c.' in several places, and 'T rimRed(Comp(Res1 i ))' missing a closing parenthesis in Theorem 1.16 and Theorem 3.2.
  2. [§2, Theorem 2.5] The sentence 'With the assumptions and the notation of theorem 2.1' should refer to Definition 2.1, not Theorem 2.1.
  3. [§2, proof of Theorem 2.2(2)] In the phrase 'the complexity of T rimRed(Comp(Res1))', the symbol Res1 should be Res, since no resolution Res1 has been introduced in that part of the proof.
  4. [§1, Definition 1.14] The phrase 'replaced with a minimal possible subgroup' is vague; the existence and uniqueness of such a minimal subgroup are not established. Since the trimmed reduced completion is central to Theorem 1.16, a precise definition would improve the paper.
  5. [§3, Theorem 3.2] The hypothesis says 'with U it is possible to associate a single ended MR diagram that satisfies the properties that are listed in theorem 3.1', but Theorem 3.1 states existence of a diagram, not single-endedness; the hypothesis should be stated explicitly (all resolutions single ended and satisfying the weak-test-sequence property).
  6. [§1–§2, notation] The paper would benefit from a table of notation: modeled structure, reduced completion, trimmed reduced completion, framed closure, and weak test sequence are defined in quick succession and used heavily throughout.

Circularity Check

2 steps flagged · score 2.0 of 10

No circular reduction of the central canonicality theorem: Theorem 1.16's uniqueness argument runs through complexity minimality, framed closures and weak test sequences and does not restate its own conclusion.

  1. self definitional [§2, note after Definition 2.1]
    "Note that by definition ∆, which is the canonical diagram of S is also the canonical diagram of pcl(S), so pcl(pcl(S)) = S."

    The pseudo-closure pcl(S) is defined in Definition 2.1 as the set of tuples factoring through the trimmed reduced completions of the canonical diagram ∆(S) of S. From the asserted fact that the same ∆ is the canonical diagram of pcl(S), the definition yields only pcl(pcl(S)) = pcl(S), i.e., idempotence of the closure operator; the printed equality pcl(pcl(S)) = S would require S = pcl(S), which is precisely what the note claims to obtain 'by definition'. For a non-pseudo-closed set the sentence is false, and the equality is in effect built into the definition rather than derived from it. The paper itself later proves S = pcl(S) only for varieties (Theorem 2.2(4)), confirming that the note's inference is not a definitional consequence.

  2. self citation load bearing [§1, discussion preceding Theorem 1.7]
    "By the construction of resolutions that is described in [Se9], the sequence of homomorphisms from which the countable resolutions are constructed are weak test sequences for these countable resolutions. ... Therefore, for the rest of this section we will continue to work with these countable resolutions"

    Theorem 1.7 is the load-bearing premise of the section: it guarantees, for every set of tuples U, finitely many countable modeled resolutions through whose reduced completions all tuples of U factor, each resolution admitting a weak test sequence from U, and every later claim — well-ordered complexities, existence of minimal-complexity collections, and the uniqueness theorems 1.8 and 1.16 — sits on top of it. That premise is imported from the author's own unpublished preprint [Se9], which is neither machine-checked nor reproduced in the paper; the text only asserts that the [Se9] construction 'works easier for set of tuples in groups'. This is load-bearing self-citation under pattern 3.

full rationale

The paper's central claim — that two minimal-complexity single-ended modeled MR diagrams for the same set of tuples have the same number of resolutions and, up to order, isomorphic trimmed reduced completions (Theorem 1.16) — is not circular. The uniqueness argument compares two minimal collections directly: it builds maps ν from the reduced completion of a resolution in one diagram to a framed closure of a resolution in the other (via formal solutions from [Se2]), applies the surviving-surface rearrangement from [Se3]/[Se4], and derives a contradiction from any strict drop in the complexity defined in Definition 1.4. None of these inputs contains Theorem 1.16's conclusion, and the conclusion never appears among the assumptions; the isomorphism between completions is argued from minimality and weak test sequences, not from the equivalence relation or from the definitions of complexity or trimmed reduced completion. The heavy self-citation ([Se1]–[Se9], [Ja-Se]) is therefore not circular in the operative sense, with the following caveats, which the reviewing rule requires me to weigh. (1) Theorem 1.7, the existence of finite collections of countable modeled resolutions with weak test sequences from the tuples of U, is imported from the author's own unpublished preprint [Se9]; if [Se9] is incomplete or wrong, Theorems 1.8, 1.16, 3.2 and the algebra results in §4 are unsupported. This is load-bearing same-author citation, but it is dependency rather than an identity between input and output, because [Se9] does not claim uniqueness. (2) The proof of Theorem 1.16 states Lemmas 1.17 and 1.18 only for trivial modeled structures and then asserts 'by the same argument, lemmas 1.17 and 1.18 remain valid for general single ended resolutions'; since Definition 1.15 and Theorem 1.7 allow arbitrary countable modeled structures, this unproved sentence is exactly the load-bearing step for the general uniqueness claim. That is an omitted proof, not a circular reduction, but it means Theorem 1.16 is not fully demonstrated as printed, and Theorem 3.2 ('Identical to the proof of theorem 1.16') inherits the gap. (3) In §2, the note after Definition 2.1 claims 'by definition ∆ ... is also the canonical diagram of pcl(S), so pcl(pcl(S)) = S'; the definition of pcl(S) via ∆(S) yields at most pcl(pcl(S)) = pcl(S), and equality with S presupposes S is already pseudo-closed, which is later proved only for varieties (Theorem 2.2(4)) and is false for general sets.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper's central claim depends on a large body of prior results on JSJ decompositions, limit groups, R-tree actions, and on the author's own theory of completions and modeled resolutions. Two inputs are preprints or in preparation ([Se9], [At-Se]). No free parameters are fitted to data. No new algebraic entities are postulated; the modeled structures, trimmed reduced completions, and pseudo-closure are defined concepts, not independent entities.

assumptions (9)
  • domain assumption Guirardel's analysis of actions of f.g. groups on R-trees, theorem 4.1 in [Gu]
    Used in Section 1 to deduce from the action of the limit group L on a real tree Y either a free product splitting or a graph of actions; this is the foundation of the resolution construction.
  • domain assumption The descending chain condition (Noetherianity) for limit groups (section 5 in [Se1])
    Used to guarantee that the iterative shortening/refinement procedure terminates after finitely many steps, producing finite resolutions.
  • domain assumption Bestvina-Feighn accessibility / acylindrical accessibility (cf. [We])
    Used to prove that the sequence of proper refinements of virtually abelian decompositions terminates.
  • domain assumption Hopf property for limit groups
    Used in Lemma 1.10 to conclude that an onto map between completions is an isomorphism.
  • domain assumption The theory of completions, closures, framed closures, formal solutions and formal limit groups from [Se2]
    The construction of maps between completions and framed closures in Lemmas 1.9, 1.10, 1.17, 1.18 relies on this machinery.
  • domain assumption Existence of countable modeled resolutions with weak test sequences, from the preprint [Se9]
    Theorem 1.7 and the subsequent analysis assume resolutions constructed in Sela's preprint 'Word equations I'; this input is not yet published in a refereed venue.
  • standard math Every homogeneous element w in a free associative algebra over a field has a most refined presentation as a product of non-trivial homogeneous elements, and any two presentations have a common refinement
    Used in Section 4 to associate a word in the free semigroup F S(U) with a homogeneous tuple in F Ak.
  • domain assumption Every set of tuples U treated in Theorems 1.16 and 3.2 admits a single ended MR diagram
    This is the restriction under which the canonical minimal complexity diagram is proven; without it the paper gives no canonical diagram, deferring to the second paper in the sequence.
  • standard math A free Lie algebra embeds into a free associative algebra via [x,y] -> xy - yx
    Used in Section 4 to associate an MR diagram with a variety over a free Lie algebra by viewing it inside the associative algebra.

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Cite this review

Pith. "Pith review of A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I." pith.science (2026). https://pith.science/paper/MYKT4FID

@misc{pith2026250522755,
  author       = {Pith},
  title        = {Pith review of: A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYKT4FID}},
  note         = {Machine review of arXiv:2505.22755}
}
read the original abstract

The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of these structures over all these algebraic categories. With a variety over each of these algebraic categories we naturally associate a set of tuples in a free group. Then we show how to associate a Makanin-Razborov diagram with any set of tuples over a free group. Furthermore, in case the MR diagram that is associated with a set of tuples is single ended, we prove that there is a canonical Makanin-Razborov diagram that can be associated with such a set. This canonical diagram is a main key in studying varieties over free semigroups, associative algebras and Lie algebras, and encodes the global structure of these varieties. It enables us to define a (pseudo) closure of a set of tuples over each of the algebraic objects, associate a rank with it (analogous to Shelah and Lascar ranks), and over free groups the closure provides a canonical envelope that is essential in studying the structure and the properties of definable sets.

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Works this paper leans on

4 extracted references · 4 canonical work pages

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    Atkarskaya and Z

    [At-Se] A. Atkarskaya and Z. Sela, Monomial varieties over a free associative algebra , in preparation. [Be] G. Berk, Canonicality of Makanin-Razborov diagrams - counterexamp le, Annals Inst. Fourier (Grenoble) 70 (2020), 2027-2047. [Gu] V. Guirardel, Actions of finitely generated groups on R-trees , Annals Inst. Fourier (Grenoble) 58 (2008), 159-211. [Ja-...

    work page 2020

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    Reinfeldt and R

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    work page 2019

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    [Ra1] A. A. Razborov, On systems of equations in a free group , Math. USSR Izvestiya 25 (1985), 115-162. [Ra2] , On systems of equations in a free group , Ph.D. thesis, Steklov Math. institute,

    work page 1985

  4. [2012]

    Weidmann, On accessibility of finitely generated groups , Quarterly journal of math

    [We] R. Weidmann, On accessibility of finitely generated groups , Quarterly journal of math. 63 (2012), 211-225. 35

    work page 2012

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