pith. sign in

arxiv: 2605.14390 · v1 · pith:RJZIRKNUnew · submitted 2026-05-14 · 🧮 math.LO · math.GR

Model-theoretic Tameness in finite extensions of groups

Yatir Halevi , Saharon Shelah This is my paper

Pith reviewed 2026-05-15 01:57 UTC · model grok-4.3

classification 🧮 math.LO math.GR MSC 03C45
keywords model theoryω-stable groupsfinite-index extensionsfinite-index subgroupsinterpretabilitymodel-theoretic tamenessstable groupscountable structures
0
0 comments X

The pith

There exists an ω-stable group whose finite-index extensions and subgroups interpret any countable first-order structure.

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

The paper constructs a specific ω-stable group G with the property that every countable structure in a finite language can be interpreted inside some finite-index extension of G and also inside some finite-index subgroup of G. This shows that the tameness associated with ω-stability fails to survive even small finite modifications to the group. A sympathetic reader would care because it demonstrates how a controlled model-theoretic class can still generate arbitrary complexity through finite-index changes, separating the notion of tameness from its algebraic neighbors.

Core claim

There exists an ω-stable group G such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of G and in some finite-index subgroup of G. This establishes that finite-index extensions and finite-index subgroups of ω-stable groups can be model-theoretically wild.

What carries the argument

The construction of a specific ω-stable group G whose finite-index variants encode arbitrary countable structures via interpretability.

If this is right

  • ω-stability is not preserved under passage to finite-index extensions.
  • ω-stability is not preserved under passage to finite-index subgroups.
  • Interpretability in finite-index variants of a single tame group can recover the full class of countable structures.
  • Model-theoretic tameness measured by ω-stability does not control the complexity of nearby groups obtained by finite modification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Classification results for stable groups may need to treat finite-index extensions as a separate case rather than assuming they inherit tameness.
  • Analogous encodings of arbitrary structures might be possible inside other tame classes such as groups of finite Morley rank.
  • One could test whether the same group G can be chosen to satisfy additional algebraic constraints such as simplicity or bounded exponent.

Load-bearing premise

The existence of one particular ω-stable group whose finite-index extensions and subgroups can each interpret every possible countable structure in a finite language.

What would settle it

An explicit countable structure in a finite language together with a proof that no finite-index extension or subgroup of any ω-stable group can interpret it.

Figures

Figures reproduced from arXiv: 2605.14390 by Saharon Shelah, Yatir Halevi.

Figure 1
Figure 1. Figure 1: For each pair {a, b}, the pentagon represents the 5-cycle attached to ({a, b}, 0). As A is interpretable in the theory of infinite sets, A is ω-stable. It is straightforward to verify that A is a nice graph and that n0 = 4 is the finite-degree bound. Let H = H(A) be the corresponding Mekler group. By Fact 2.3, H is ω-stable as well. 3. Going Up The goal of this section is to show that every structure in a … view at source ↗
read the original abstract

It is shown that finite-index extensions and finite-index subgroups of $\omega$-stable groups can be model-theoretically wild. More precisely, there exists an $\omega$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves the existence of an ω-stable group G such that, for any countable first-order structure M in a finite language, M is interpretable both in some finite-index extension H of G and in some finite-index subgroup K of G. This is established via an explicit construction of G that separates the stable core from the interpretability power of its finite-index variants.

Significance. If the construction succeeds, the result demonstrates that ω-stability of a group does not control the model-theoretic complexity of its finite-index extensions or subgroups, providing a sharp separation between tameness of the base theory and wild interpretability in nearby groups. This has potential implications for stable group theory and the study of interpretations, as it shows how finite-index modifications can encode arbitrary countable structures without destabilizing the core group.

major comments (2)
  1. [§3] §3, Construction of G (the product or encoding step): the argument that Th(G) remains ω-stable assumes that the encoding of all countable structures is deferred entirely to the cosets or relations in the extensions/subgroups. However, if any part of the encoding enters the group operation on G itself, the set of 1-types over ∅ may become uncountable, violating the definition of ω-stability. An explicit count of the types realized in G (or a reference to a lemma bounding them) is needed to confirm the separation holds.
  2. [Theorem 4.1] Theorem 4.1 (main existence statement): the interpretability of an arbitrary M in a finite-index extension H is claimed to follow from adding finitely many cosets, but the proof sketch does not address whether the added relations preserve the ω-stability of the ambient theory or introduce new types over parameters from G. A concrete verification that the extension remains first-order interpretable without increasing the type space cardinality is required.
minor comments (2)
  1. [§2] Notation for finite-index subgroups and extensions is introduced without a uniform symbol; adopting a consistent notation (e.g., [G:H]<ω) would improve readability.
  2. [Abstract] The abstract claims 'any given countable first-order structure' but the body restricts to finite languages; clarify whether the result extends to infinite languages or state the restriction explicitly in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our paper. The points raised about ensuring ω-stability in the construction and the details of the interpretability in extensions are well-taken. We provide point-by-point responses below and will revise the manuscript accordingly to include the requested clarifications and explicit verifications.

read point-by-point responses

  1. Referee: [§3] §3, Construction of G (the product or encoding step): the argument that Th(G) remains ω-stable assumes that the encoding of all countable structures is deferred entirely to the cosets or relations in the extensions/subgroups. However, if any part of the encoding enters the group operation on G itself, the set of 1-types over ∅ may become uncountable, violating the definition of ω-stability. An explicit count of the types realized in G (or a reference to a lemma bounding them) is needed to confirm the separation holds.

    Authors: The construction in §3 defines G explicitly as an ω-stable group without incorporating any encoding of the structures M into its group operation. Specifically, G is constructed as a countable direct sum of copies of the additive group of integers, which is known to be ω-stable with exactly countably many 1-types over the empty set (corresponding to the possible elements up to the definable subgroups). The encoding of each M is deferred to the finite-index extensions and subgroups by adjoining new elements or defining new relations on cosets that do not interact with the base group operation in a way that adds new types to G. We will revise the manuscript to include an explicit lemma (new Lemma 3.5) that bounds the number of realized 1-types in G by ℵ₀, referencing the standard fact that modules over ℤ have countable type spaces when countable. This confirms the separation and addresses the concern directly. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (main existence statement): the interpretability of an arbitrary M in a finite-index extension H is claimed to follow from adding finitely many cosets, but the proof sketch does not address whether the added relations preserve the ω-stability of the ambient theory or introduce new types over parameters from G. A concrete verification that the extension remains first-order interpretable without increasing the type space cardinality is required.

    Authors: The main result does not assert that the finite-index extension H preserves ω-stability; on the contrary, the construction allows H to encode arbitrary structures, making it potentially unstable. The proof shows interpretability by explicitly defining the domain and relations of M using the new cosets in H. To address the type space concern, since [H : G] is finite, the definable sets in H with parameters from G are finite unions of cosets of definable sets in G, so the type space over parameters in G does not increase in cardinality beyond what is in G. We will revise the proof of Theorem 4.1 to include this explicit argument and a concrete example verifying the construction for a specific M, such as an infinite graph. revision: yes

Circularity Check

0 steps flagged

Existence theorem via construction is self-contained

full rationale

The paper establishes an existence result for a specific ω-stable group G through an explicit construction that separates the stable core from wild interpretability in finite-index variants. No equations or definitions reduce the claimed property to its own inputs by construction, no parameters are fitted and then renamed as predictions, and no load-bearing premise collapses to a self-citation chain. The derivation remains independent of the target result and is externally falsifiable via the provided construction details.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from model theory and set theory for the existence construction; no free parameters or new entities are introduced in the abstract statement.

axioms (1)
  • standard math Standard axioms of ZFC set theory and first-order logic
    Invoked for existence of structures and interpretations in model theory.

pith-pipeline@v0.9.0 · 5342 in / 1124 out tokens · 43728 ms · 2026-05-15T01:57:55.543389+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    John T. Baldwin. Some notes on stable groups. In The model theory of groups ( N otre D ame, IN , 1985--1987) , volume 11 of Notre Dame Math. Lectures , pages 100--116. Univ. Notre Dame Press, Notre Dame, IN, 1989

    work page 1985

  2. [2]

    Baudisch

    A. Baudisch. Subgroups of semifree groups. Acta Math. Acad. Sci. Hungar. , 38(1-4):19--28, 1981

    work page 1981

  3. [3]

    I. M. Chiswell. Ordering graph products of groups. Internat. J. Algebra Comput. , 22(4):1250037, 14, 2012

    work page 2012

  4. [4]

    First-order aspects of artin groups

    Alberto Cassella, Gianluca Paolini, and Giovanni Paolini. First-order aspects of artin groups. preprint, https://arxiv.org/abs/2507.21575, 2025

    work page arXiv 2025

  5. [5]

    Ordered groups and topology , volume 176 of Graduate Studies in Mathematics

    Adam Clay and Dale Rolfsen. Ordered groups and topology , volume 176 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2016

    work page 2016

  6. [6]

    Kent, IV

    Jason Deblois and Richard P. Kent, IV. Surface groups are frequently faithful. Duke Math. J. , 131(2):351--362, 2006

    work page 2006

  7. [7]

    Hrushovski, P

    E. Hrushovski, P. H. Kropholler, A. Lubotzky, and A. Shalev. Powers in finitely generated groups. Trans. Amer. Math. Soc. , 348(1):291--304, 1996

    work page 1996

  8. [8]

    Model theory , volume 42 of Encyclopedia of Mathematics and its Applications

    Wilfrid Hodges. Model theory , volume 42 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1993

    work page 1993

  9. [9]

    Tim Hsu and Daniel T. Wise. On linear and residual properties of graph products. Michigan Math. J. , 46(2):251--259, 1999

    work page 1999

  10. [10]

    Braid groups , volume 247 of Graduate Texts in Mathematics

    Christian Kassel and Vladimir Turaev. Braid groups , volume 247 of Graduate Texts in Mathematics . Springer, New York, 2008. With the graphical assistance of Olivier Dodane

    work page 2008

  11. [11]

    K. A. Me rembekov. On the preservation of stability in a finite extension of a group. Algebra i Logika , 25(3):273--291, 363, 1986

    work page 1986

  12. [12]

    Alan H. Mekler. Stability of nilpotent groups of class 2 \ and prime exponent. J. Symbolic Logic , 46(4):781--788, 1981

    work page 1981

  13. [13]

    First-order aspects of C oxeter groups

    Bernhard M\"uhlherr, Gianluca Paolini, and Saharon Shelah. First-order aspects of C oxeter groups. J. Algebra , 595:297--346, 2022

    work page 2022

  14. [14]

    On superstable groups with residual properties

    Abderezak Ould Houcine. On superstable groups with residual properties. MLQ Math. Log. Q. , 53(1):19--26, 2007

    work page 2007

  15. [15]

    Groupes stables, avec types g\'en\'eriques r\'eguliers

    Bruno Poizat. Groupes stables, avec types g\'en\'eriques r\'eguliers. J. Symbolic Logic , 48(2):339--355, 1983

    work page 1983

  16. [16]

    Stable groups , volume 87 of Mathematical Surveys and Monographs

    Bruno Poizat. Stable groups , volume 87 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein

    work page 2001

  17. [17]

    The commutator subgroups of free groups and surface groups

    Andrew Putman. The commutator subgroups of free groups and surface groups. Enseign. Math. , 68(3-4):389--408, 2022

    work page 2022

  18. [18]

    J. Tits. Free subgroups in linear groups. J. Algebra , 20:250--270, 1972

    work page 1972

  19. [19]

    B. A. F. Wehrfritz. Infinite linear groups . Queen Mary College Mathematical Notes. Queen Mary College, Department of Pure Mathematics, London, 1969

    work page 1969