REVIEW 4 minor
Some results on NIP groups and their Ellis groups
T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read For any NIP group, the Ellis group has at most 2^|T| points, regardless of which model M is chosen, bounding a previously unbounded object and giving a concrete step toward proving its isomorphism type is model-independent.
desk verdict Strong, carefully written paper that makes real progress on the Ellis group question; the main theorem rests on one deep black box (Hausdorffness of the τ-topology) that a referee should check carefully. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Wideness: a definable set is wide if some finite union of its left translates by elements of the minimal model M0 is strong f-generic; the non-wide sets form an S1 ideal (Theorem 3.20). This ideal is the substitute for translation-invariant S1 ideals, which generally do not exist in non-definably amenable NIP groups. The proof also relies on the retraction map F_M from M-invariant global types to types finitely satisfiable in M, and on the black-box theorem that the τ-topology on the Ellis group is Hausdorff, which makes the Ellis group a compact Hausdorff group and lets uniqueness of τ-limits force q = q'.
What would settle it
Take a NIP group G, a model M0, and an elementary extension M; if there exist two distinct global types q, q' in the same Ellis group of G(M) with q|M0 = q'|M0, Theorem 4.17 fails. Equivalently, find any NIP group and model whose Ellis group has more than 2^|T| elements.
Extended reading notes
Core claim
The central discovery is Theorem 4.17: for any idempotent u in a minimal ideal of the Ellis semigroup of G(M), the restriction map from the Ellis group uI to the type space S_G(M0) is injective. Consequently |uI| ≤ 2^|T|, independent of M. The proof transfers 'wide' global types across models via a retraction map from invariant to finitely-satisfiable types, and uses the Hausdorffness of the τ-topology to identify limits; injectivity of restriction forces any coinciding pair q,q' to agree. This is the paper's step toward the open question of whether the Ellis group's isomorphism type is independent of M.
Load-bearing premise
The argument treats as a black box the theorem that the τ-topology on the Ellis group is Hausdorff; if that theorem were false, the uniqueness-of-limits step that yields injectivity (and hence the size bound) would not go through.
Editorial extensions
If this is right
- The Ellis group of G(M) has at most 2^|T| elements for every model M, so it cannot grow arbitrarily as M varies; outside NIP it does grow.
- For any elementary substructure M0 ≼ M, the restriction map from the Ellis group of G(M) to S_G(M0) is injective, giving structural information beyond the size bound.
- If T and M0 are countable and T has bounded VC-codensity, the Ellis group is profinite-by-Lie-by-profinite: an inverse limit of compact Lie groups of dimension at most (4δ)^2.
- For any bi-invariant NIP formula φ, the quotient G/G^{00}_φ has finite Archimedean rank: an inverse limit of compact Lie groups of dimension at most (4δ)^2.
- Gismatullin's theorem that G^∞ exists in NIP groups receives a new proof, with the stronger statement X_{M0} ⊆ X_M^G.
Reading between the lines
- The size bound alone does not answer whether Ellis groups of different models are isomorphic; a natural next test is whether the injective restriction maps can be upgraded to a functorial comparison of Ellis groups across M0 and M.
- Because the wide-type ideal exists in every NIP group, the techniques may transfer to NIP approximate groups (an extension the author says is in progress) and possibly to NTP2 groups if the paper's Question 3.25 has a positive answer.
- The local theorem suggests removing the global NIP assumption: if local analogues of Hausdorffness and Borelness of clopen traces existed for local Ellis groups, the same compact-group argument could answer the paper's Question 7.3 without global NIP.
- The paper's reliance on an unverified black-box theorem—the Hausdorffness of the τ-topology—is a point to watch; if that theorem were false, the uniqueness-of-limits step yielding injectivity would need a different proof.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a theory of piecewise (strong) f-genericity in NIP groups. It defines three notions: piecewise f-generic, piecewise strong f-generic, and piecewise strong f-generic with witnesses in M0 (the last called 'wide'). The main theorem of Section 3 states that the non-wide definable sets form an S1 ideal, and that the non-piecewise-strong-f-generic sets form a translation-invariant ideal. These tools are then applied to Ellis groups of NIP groups. Theorem 4.17 shows that, for any model M extending M0, the restriction map uI → S_G(M0) is injective on the Ellis group uI of S^fs_G(C,M), so |uI| ≤ 2^{|T|} independent of M; this is a substantial step toward the open question whether the isomorphism type of the Ellis group is independent of M. Theorem 6.5 shows that, under countability and bounded VC-codensity assumptions, the Ellis group has finite Archimedean rank. Theorem 7.11 proves a local version for G/G^{00}_phi with uniformly bounded Lie dimensions in terms of the VC-codensity of phi. Along the way the paper proves a self-contained theorem on VC-sets in compact Hausdorff groups (Theorem 5.10).
Significance. If the results are correct, they constitute a clear advance in the model-theoretic study of Ellis groups of NIP groups. The uniform size bound independent of the model is new and structurally stronger than previous partial results. The paper is careful in detail: it corrects an error in [31, Prop. 2.18] before using it (Fact 2.18), proves the needed topological-dynamics lemma (Fact 4.15, Lemma 12 of [4]) in full, and isolates a general VC-set theorem that is likely of independent interest. It also gives a new proof of Gismatullin's theorem on the existence of G^infinity. The main caveat is that Theorem 4.17 depends on Fact 2.15, the Hausdorffness of the τ-topology, proved for tame minimal flows in [10] and [5]. The author explicitly states in the AI Declaration that this is used as a black box and that the proof has not been learned. This is an honest and transparent acknowledgment of reliance on a deep external theorem, not an internal inconsistency; it does not undermine the internal derivation, but it is the principal correctness risk of the paper.
minor comments (4)
- [Section 2.5 and Lemmas 5.8, 6.4] The paper defines VC-density as an infimum, but Lemmas 5.8 and 6.4 assume an exact bound of the form π_F(n) ≤ C n^δ. If 'VC-density at most δ' is used in the infimum sense, a short limiting argument is needed to justify the exact polynomial bound. The intended reading should be clarified, or the statements should explicitly assume the stronger growth bound.
- [Lemma 7.4] In the proof, p is chosen as 'any complete global type finitely satisfiable in M and extending the shared phi-type' of g and h. To make the later contradiction valid, p should be chosen as a global coheir of tp(g/M) (which exists since every type over a model is finitely satisfiable in M); otherwise the phi-type of a Morley-sequence element may contain additional phi-formulas not in the shared phi-type. The intended argument is sound, but the wording is imprecise.
- [References] A few entries in the bibliography (e.g. [2] and [64]) appear not to be cited in the body of the paper. Please check that all listed references are used or remove them.
- [Fact 2.15 / Section 2.4] Since Fact 2.15 is the paper's main black box and is used for the size bound in Theorem 4.17, it would be helpful to add one sentence in Section 2.4 explicitly identifying the τ-topology constructed there with the topology for which [10] and [5] prove Hausdorffness, and noting that the relevant flows S^fs_G(C,M) are tame by [15]. This would make the transfer fully explicit for the reader.
Circularity Check
No significant circularity: the Ellis-group bound is a forward derivation from NIP machinery plus an external Hausdorffness theorem, not a reduction of the conclusion to its own inputs.
full rationale
Walking the derivation chain, the central results are forward proofs from stated hypotheses. Section 3 proves Theorems 3.18-3.21 from Fact 2.6, Fact 2.5, Fact 2.7, and Fact 2.8, with no target Ellis-group statement used as an input. Theorem 4.12 is distilled from Lemmas 4.8-4.11, and Lemma 4.13 plus Lemma 4.16 feed directly into the injectivity proof of Theorem 4.17. The only genuinely load-bearing external input is Fact 2.15, the Hausdorffness of the tau-topology, which the paper explicitly uses as a black box, citing [10] and [5]; the authors' AI Declaration and disclaimer that they have not learned the details of its proof is a transparency/robustness caveat, not a circularity. The paper's self-citations ([70], [60]) appear only in context passages, and the text explicitly says of the summarized equivalences 'none of which we need' (Fact 3.2), so they are not load-bearing. Sections 5-7 are also forward applications: Theorem 5.10 is self-contained, Theorem 6.5 combines Fact 2.16, Theorem 4.17, and Theorem 5.10, and Theorem 7.11 follows from Lemmas 7.9-7.10 together with Facts 2.15-2.16. No equation makes a claimed prediction equal to a fitted or defined quantity, no uniqueness claim is imported from the author's own prior work, and no known result is merely renamed. The derivation is therefore self-contained apart from standard external theorems, and no circular step is exhibited.
Assumptions & free parameters
assumptions (10)
- domain assumption Fact 2.15: in NIP theories the τ-topology on the Ellis group of G(M) is Hausdorff, so the Ellis group is a compact Hausdorff topological group
- domain assumption Fact 2.6: NIP theories have bounded weight with respect to dp-rank (a |T|^+-length Morley sequence contains an element independent from any given tuple)
- domain assumption Fact 2.5: Kim's lemma for NIP (forking over M is witnessed by inconsistency along a strictly-non-forking Morley sequence)
- standard math Facts 2.2–2.3: over models in NIP, non-forking = invariance and forking = dividing
- domain assumption Fact 2.8: definable (p,q)-theorem in NIP theories
- domain assumption Fact 2.16: for NIP, [φ(x,b)] ∩ uI is Borel in the τ-topology
- standard math Fact 2.18: corrected ε-net/VC-theorem for finite measurable families
- standard math Facts 2.12–2.14, 2.19–2.21: Ellis semigroup facts; Peter-Weyl; compact group structure; finite Archimedean rank characterization
- domain assumption dcl(∅) is a model, denoted M0, assumed throughout
- domain assumption Fact 7.2: for bi-invariant NIP φ, G^00_φ exists, is normal, type-definable by countably many φ-formulas; G/G^00_φ is compact Hausdorff in the logic topology
Cite this review
Pith. "Pith review of Some results on NIP groups and their Ellis groups." pith.science (2026). https://pith.science/paper/VLJSBPS2
@misc{pith2026260726265,
author = {Pith},
title = {Pith review of: Some results on NIP groups and their Ellis groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/VLJSBPS2}},
note = {Machine review of arXiv:2607.26265}
}
abstract
This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications. Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupi\'nski and Basso-Zucker that the $\tau$-topology on the Ellis group is Hausdorff. Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $\phi(x,y)$, the group $G/G^{00}_\phi$ has finite Archimedean rank. More precisely, if the VC-codensity of $\phi(x,y)$ is at most $\delta$, then $G/G^{00}_\phi$ is an inverse limit of compact Lie groups of dimension at most $(4\delta)^2$. This connects to, though is different than, a question of Hrushovski's.
Reviewed August 1, 2026 · model on record in the stance chip above.
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