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The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In V-minimal, T-convex, and p-adically closed valued fields, every infinite interpretable group $G$ gains a canonical type-definable infinitesimal subgroup $\nu(G)$, the product of four commuting infinitesimal pieces.

desk verdict Substantial advance in the infinitesimal-subgroup program, with the main theorem's injectivity resting on a quoted foreignness result that a referee should ask to be proved or precisely sourced. read the letter →

arxiv 2502.08163 v1 pith:GN2WEKF3 submitted 2025-02-12 math.LO

Yatir Halevi , Assaf Hasson , Ya'acov Peterzil This is my paper · ORCID
classification math.LO MSC 03C4503C6012L12
keywords dp-minimalvaluedfieldsinfinitesimalsubgroupsinterpretablegroupsV-minimalT-convexp-adicallyclosedtype-definableeliminationofimaginaries
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

The paper proves that in V-minimal, power-bounded $T$-convex, and p-adically closed valued fields of characteristic 0, every infinite interpretable group $G$ carries a canonical type-definable infinitesimal subgroup $\nu(G)$. This subgroup is generated by four infinitesimal pieces $\nu_D(G)$, one for each distinguished sort: the field $K$, the residue field $k$, the value group $\Gamma$, and the coset space $K/\mathcal{O}$. The main theorem shows the four pieces commute, so $\nu(G)$ is definably isomorphic to the direct product $\nu_K \times \nu_\Gamma \times \nu_{K/\mathcal{O}} \times \nu_k$. The construction is uniform across the three settings and, except for the $K/\mathcal{O}$ sort in the p-adically closed case, avoids passing to a finite quotient.

What carries the argument

The central object is the infinitesimal vicinity $\nu_X(a)$: the partial type consisting of all definable generic neighborhoods of a generic point $a$ in a set $X$ that is almost strongly internal to a distinguished sort $D$. From these one builds the infinitesimal subgroup $\nu_D(G)$ as the partial type $\{XX^{-1} : X \subseteq G \text{ is almost } D\text{-critical}\}$, and the notion of a $D$-balanced group ensures that this is a type-definable subgroup with the correct dp-rank. The mechanism that makes the direct product work is foreignness of distinct distinguished sorts—no definable infinite set can project finite-to-one onto two different sorts—which forces $\nu_{D_1} \cap \nu_{D_2} = \{e\}$; a local analysis proposition then upgrades mutual normalization to full commutation.

What would settle it

Exhibit, in one of the three classes of valued fields, an infinite definable set $Z$ contained in the product of two distinct distinguished sorts whose projection to each factor is finite-to-one; this would violate the foreignness lemma and break the injectivity step of Theorem 6.1. Alternatively, in a concrete group such as the semidirect product $K/\mathcal{O} \rtimes \mathcal{O}^\times$ of Example 5.6, find $a \in \nu_{K/\mathcal{O}}(\hat{K})$ and $b \in \nu_K(\hat{K})$ with $ab \neq ba$.

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Extended reading notes

Core claim

The central claim is Theorem 6.1: for $K$ an expansion of a valued field of characteristic 0 that is V-minimal, power-bounded $T$-convex, or p-adically closed, and $G$ an infinite interpretable group, the subgroup of $G(\hat{K})$ generated by the four infinitesimal subgroups $\nu_D(\hat{K})$ is type-definable and definably isomorphic to $\nu_K(\hat{K}) \times \nu_\Gamma(\hat{K}) \times \nu_{K/\mathcal{O}}(\hat{K}) \times \nu_k(\hat{K})$. The proof shows that the $\nu_D(\hat{K})$ commute pairwise. The paper further shows that $\nu(G_1 \times G_2) = \nu(G_1) \times \nu(G_2)$, that $\nu(G_2)$ is relatively definable in $\nu(G_1)$ whenever $G_2 \leq G_1$ is definable, and that a definable surjective homomorphism with finite kernel maps $\nu(G_1)$ onto $\nu(G_2)$. It also constructs an analytic expansion of $\mathrm{ACVF}_{0,0}$ with $\mathrm{dp\text{-}rk}(\nu(G)) < \mathrm{dp\text{-}rk}(G)$, yielding a short proof that such fields do not eliminate imaginaries down to the geometric sorts.

Load-bearing premise

The direct-product decomposition rests on the quoted fact that any two distinct distinguished sorts are foreign: no infinite definable set can be mapped finite-to-one into both sorts. If that fact failed, the infinitesimal pieces could overlap and the map from the direct product onto $\nu(G)$ would not be injective.

Editorial extensions

If this is right

  • Every infinite interpretable group $G$ in the three settings now has a single canonical type-definable infinitesimal subgroup $\nu(G)$, definably isomorphic to the direct product of the four $\nu_D$.
  • The construction is uniform and, except for the $K/\mathcal{O}$ sort in the p-adically closed case, eliminates the need to pass to a finite quotient before defining infinitesimals.
  • The assignment $G \mapsto \nu(G)$ respects products ($\nu(G_1 \times G_2) = \nu(G_1) \times \nu(G_2)$) and subgroups: if $G_2 \leq G_1$ is definable then $\nu(G_2)$ is relatively definable in $\nu(G_1)$, and if the dp-ranks agree then $\nu(G_1) = \nu(G_2)$.
  • A definable surjective homomorphism with finite kernel maps $\nu(G_1)$ onto $\nu(G_2)$; for unstable distinguished sorts other than the p-adic $K/\mathcal{O}$, it induces an isomorphism of the corresponding $\nu_D$ pieces.
  • The rank inequality $\mathrm{dp\text{-}rk}(\nu(G)) = \sum_D a_D\text{-rk}(G) \leq \mathrm{dp\text{-}rk}(G)$ can be strict, as shown by an analytic expansion of $\mathrm{ACVF}_{0,0}$ where $\mathrm{dp\text{-}rk}(\nu(G)) = 1 < \mathrm{dp\text{-}rk}(G) = 2$.

Reading between the lines

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

  • If the rank equality $\mathrm{dp\text{-}rk}(\nu(G)) = \mathrm{dp\text{-}rk}(G)$ turns out to be equivalent to a form of elimination of imaginaries for definable groups, then strict inequality is not a pathology but a diagnostic for unexpected imaginaries.
  • The commutation theorem suggests a general recipe in other multi-sorted dp-minimal structures: whenever distinct geometric sorts are foreign, infinitesimal subgroups built from each sort should commute and assemble into a direct product.
  • The paper leaves open whether an analogue of Example 6.4 exists in analytic p-adic expansions; if none exists, the rank equality might hold for $Q_{p,\mathrm{an}}$ even though it fails over analytic expansions of $\mathrm{ACVF}_{0,0}$.
  • The relative definability of $\nu(G_2)$ inside $\nu(G_1)$ may allow properties of infinitesimals to be transferred between a group and its definable subgroups, potentially supporting a structure theorem for interpretable groups via infinitesimal quotients.
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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

3 major / 4 minor

Summary. The paper continues the authors' program on groups interpretable in dp-minimal valued fields. For an infinite interpretable group G in a V-minimal, power-bounded T-convex, or p-adically closed valued field K, it defines a type-definable infinitesimal subgroup ν(G) generated by four subgroups νD(G) attached to the distinguished sorts K, Γ, k, and K/O. The main theorem states that these four subgroups commute and that ν(G) is definably isomorphic to their direct product. The paper also proves that ν is relatively definable on definable subgroups, that ν is additive under direct products, and that dp-rank of ν(G) is the sum of the almost D-ranks. A new example in an analytic expansion of ACVF shows that the rank equality dp-rk(ν(G)) = dp-rk(G) can fail, and this is applied to give a short proof that such fields do not eliminate imaginaries down to the 0-dimensional sorts.

Significance. If the main theorem is correct, this is a substantial improvement over the authors' earlier infinitesimal subgroup construction: it removes the need for finite quotients in most cases, provides a uniform description of νD(G) as the type generated by sets XX^{-1}, and consolidates the four infinitesimal subgroups into a single canonical type-definable object. The paper also repairs a gap in [8] concerning additivity of almost D-rank and gives an interesting new example separating dp-rk(ν(G)) from dp-rk(G), with a clean application to elimination of imaginaries. The arguments are detailed and mostly self-contained modulo the authors' published companion papers; the explicit acknowledgement that a proof in [8] had a gap and is fixed here is commendable.

major comments (3)
  1. [Section 6, proof of Theorem 6.1] The injectivity of the map τ: νK × νΓ × νK/O × νk → N rests on the assertion, made in the last two paragraphs of the proof, that 'since both K and Γ and K and k are foreign it is not hard to see that K and Γ × k are foreign', and similarly for K/O against the product of the other three sorts. This step is load-bearing: it is exactly what makes the displayed intersections finite and hence trivial by torsion-freeness. Pairwise foreignness of K with Γ and with k does not formally imply foreignness of K with the product Γ × k; a definable subset of K^n × Γ^m × k^l projecting finite-to-one onto K^n and onto Γ^m × k^l does not obviously contradict either pairwise foreignness statement. Please supply a proof (for instance, a compactness argument using Fact 5.2 and the stable embeddedness of Γ and k) or a precise reference to where this product-foreignness statement is proved.
  2. [Section 5, Lemma 5.3] The proof of Lemma 5.3 is too compressed. From the foreignness of D1 and D2 it is supposed to follow immediately that the intersection νD1(ˆK) ∩ νD2(ˆK) is finite. Since νD1 and νD2 are type-definable, one needs the additional compactness step: choose definable sets X1, X2 almost strongly internal to D1 and D2 respectively with νD1 ⊢ X1 and νD2 ⊢ X2, so that an infinite intersection would yield an infinite definable X1 ∩ X2 projecting finite-to-one onto both D1^n and D2^m, contradicting Fact 5.2. As written, the sentence 'Since any two distinct distinguished sorts are foreign, the intersection ... is finite' is not quite a complete justification for type-definable subgroups.
  3. [Remark 2.1 and Fact 5.2] The paper claims in Remark 2.1 that the results remain true in P-minimal expansions such as Qp,an, but Fact 5.2 (foreignness of distinct distinguished sorts) is quoted from [8, Proposition 9.3] without proof. If [8, Proposition 9.3] was established only for the three base settings and not for analytic expansions, then the main theorem for Qp,an is conditional on an unverified fact. Please state explicitly whether [8, Proposition 9.3] covers the analytic expansion setting, or include a proof of Fact 5.2 in that setting.
minor comments (4)
  1. [Throughout] There are several typographical errors, including 'V ALUED' in the title, 'F or' in a few places, 'wwell-defined' in the proof of Lemma 4.11, and missing spaces in expressions such as 'νD1( ˆK) andνD2( ˆK) commute'.
  2. [Example 7.5] The displayed inequality 'HG1 /lessn⋊tequalHG =HG ∩G1' appears to be a typesetting corruption; it should presumably read 'HG1 ≤ HG ∩ G1'.
  3. [Section 2.1] The sentence 'All distinguished sorts in our settings have algebraic dp-rank' is potentially confusing because K/O is explicitly said not to satisfy the Exchange Property. A one-sentence clarification that algebraic dp-rank here does not require exchange would help the reader.
  4. [Corollary 6.5] The corollary relies on the equivalence between elimination of imaginaries down to the geometric sorts of [9] and elimination down to 0-dimensional sorts, citing [1, §4.1]. Since this is a key step in the application, it would be helpful to spell out the implication rather than only cite it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from internally proved commutation results plus a cited, non-equivalent foreignness fact, not from its own conclusion.

full rationale

The derivation chain is not circular. The infinitesimal subgroups νD(G) are defined internally via almost D-critical sets and the canonical finite quotient HG (Definition 4.16, Proposition 4.20); this construction does not presuppose the direct-product decomposition of Theorem 6.1. Commutation of the four νD subgroups is proved in Proposition 5.5 using the local analysis of Proposition 5.1 and does not quote the target theorem. Theorem 6.1 then builds the product isomorphism from (i) commutation, which gives the homomorphism, and (ii) injectivity via Lemma 5.3 and the foreignness Fact 5.2, quoted from the authors' published [8, Proposition 9.3]. Fact 5.2 is a separate, parameter-free statement about definable sets in the distinguished sorts; it is not an assumption equivalent to Theorem 6.1, and the paper does not fit, rename, or define any parameter in terms of the conclusion. The unproved 'not hard to see' extension of foreignness to Γ × k in the proof of Theorem 6.1, and the reliance on Fact 5.2 for analytic or p-adic expansions mentioned in Remark 2.1, are potential correctness or completeness gaps, but they are not circular reductions. The new content—commutation, relative definability of ν(G1) in ν(G), products, and the Example 6.4 / Corollary 6.5 application—is derived rather than imported.

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

The central new object ν(G) is constructed from existing data (the four νD) and proved to be type-definable, so it is not a postulated entity. The paper's load-bearing background consists of prior results from the authors' own series and standard dp-rank machinery, all explicitly cited.

assumptions (5)
  • domain assumption Fact 2.4: every infinite definable set X in K contains an infinite subset almost strongly internal to one of the distinguished sorts K, k, Γ, K/O.
    Quoted from [8, Lemmas 7.3, 7.6, 7.10]; the infinitesimal construction begins from this.
  • domain assumption Fact 5.2: any two distinct distinguished sorts are foreign, i.e. no definable infinite Z ⊆ D1^n × D2^m projects finite-to-one onto both coordinates.
    Used in Lemma 5.3 to prove νD1 ∩ νD2 = {e}, which is essential for Theorem 6.1.
  • domain assumption The distinguished sorts have algebraic dp-rank, so dp-rank equals acl-dimension and is type-definable.
    Used in Proposition 5.1 and Section 2.1; enables the type-definability arguments needed for commutation.
  • domain assumption K is an expansion of a valued field of characteristic 0 that is V-minimal, power-bounded T-convex, or p-adically closed, and is sufficiently saturated.
    This defines the scope of Theorems 1 and 2; see Section 2.1.
  • standard math The standard properties of dp-rank (sub-additivity, invariance under definable finite-to-one correspondences, etc.) are assumed.
    Used throughout; see Section 2.1.

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Pith. "Pith review of The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields." pith.science (2026). https://pith.science/paper/GN2WEKF3

@misc{pith2026250208163,
  author       = {Pith},
  title        = {Pith review of: The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GN2WEKF3}},
  note         = {Machine review of arXiv:2502.08163}
}
abstract

We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $\nu(G)$, generated by the four infinitesimal subgroups $\nu_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $\Gamma$ and $K/\mathcal{O}$. To show that $\nu(G)$ is type-definable, we show that the resulting subgroups $\nu_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $\nu(G)$. Among others, we show that $\nu(G_1\times G_2)=\nu(G_1)\times \nu(G_2)$ and that if $G_1\le G$ is a definable subgroup then $\nu(G_1)$ is relatively definable in $\nu(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(\nu(G))$ and elimination of imaginaries.

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Reference graph

Works this paper leans on

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