Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

Maciej Malicki

REVIEW 2 major objections 4 minor 12 references

Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every countable structure $M$ and every Polish group $G$ of its permutations, a comeager $n$-diagonal conjugacy class exists in $G$ exactly when the family $K_{G,n}$ of $n$-tuples of $G$-extendable bijections between finitely…

desk verdict A promising generalization of Kechris–Rosendal whose main proof has a genuine countability gap; the ultrametric applications are solid. read the letter →

arxiv 1908.09494 v3 pith:YA6DNOWV submitted 2019-08-26 math.LO math.GR

Maciej Malicki This is my paper · ORCID

classification math.LOmath.GR MSC 03E1554H11

keywords weakamalgamationjointembeddingpropertydiagonalconjugacyclassesamplegenericsPolishpermutationgroupsnon-archimedeanultrametricspaceshomogenizablestructures

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The pith

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

The reading

This paper establishes a direct dictionary between large conjugacy classes in Polish permutation groups and the combinatorics of finite partial maps. For a countable structure $M$ and a Polish group $G$ of permutations of $M$, a comeager $n$-diagonal conjugacy class exists exactly when the family $K_{G,n}$ of $n$-tuples of $G$-extendable bijections between finitely generated substructures satisfies the joint embedding property and the weak amalgamation property. This widens a characterization that was previously available only for automorphism groups of classical limits of finite structures, and it also yields a dense-class analogue. The paper then applies the criterion to groups of ball-preserving bijections of ordered ultrametric spaces: these groups always have a comeager conjugacy class but never a comeager $2$-diagonal one. It also characterizes homogenizability of weak limits and gives an example of a weak limit that is not homogenizable.

What carries the argument

The load-bearing object is $K_{G,n}$, the family of $n$-tuples of partial bijections between finitely generated substructures of $M$ that extend to elements of $G$, equipped with embeddings that are themselves $G$-extendable. Chains in this family are the finite approximations of group elements, and the paper's dictionary converts conjugacy in $G$ into isomorphism of chains. The main equivalence passes through three intermediate stations: JEP plus WAP for $K_{G,n}$; the existence of a weakly $K_{G,n}$-injective tuple in $G$, built with a Rasiowa-Sikorski-style generic chain; and a winning strategy for Odd in the associated infinite game, which is equivalent to the target conjugacy class being comeager. For the ultrametric results, partial ball-preserving bijections are coded as bijections between finite families of balls, and the proofs of the cofinal amalgamation property use encompassing and monotone orbits, while the failure of weak amalgamation for pairs is engineered with words in a free group.

What would settle it

Find a countable structure $M$ and a Polish group $G \le \mathrm{Sym}(M)$ for which $K_{G,1}$ satisfies JEP and WAP but $G$ has no comeager conjugacy class; such an example would refute the central equivalence as stated. A more immediate check is whether $K_{\mathrm{Sym}(\mathbb{N})}$ for the successor structure on $\mathbb{N}$ is uncountable; if so, the generic-selection step of the proof needs an additional countability argument before Theorem 3.10 can cover that case.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.10: if $M$ is countable, $G \le \mathrm{Sym}(M)$ is Polish, and $n \ge 1$, then $G$ has a comeager $n$-diagonal conjugacy class if and only if $K_{G,n}$ has JEP and WAP. The proof treats $K_{G,n}$ as a category whose chains are finite approximations of elements of $G$; isomorphic chains correspond to conjugate group elements, and a weakly injective chain obtained by a generic construction supplies a comeager class. The dense version replaces WAP by JEP alone, and taking all $n$ together characterizes ample generics for $\mathrm{Aut}(M)$. In the ultrametric part, the paper proves that partial ball-preserving bijections of the ordered rational $N$-ultrametric Urysohn spaces have the cofinal amalgamation property, while pairs of them fail weak amalgamation; consequently $\mathrm{BP}(X)$ has a comeager conjugacy class but no comeager $2$-diagonal conjugacy class for every ordered ultrahomogeneous Polish ultrametric space $X$.

Load-bearing premise

The proof of the main equivalence assumes the family $K_{G,n}$ is countable when it applies a generic-selection lemma, and the paper does not verify that countability; for structures with function symbols, finitely generated substructures can be infinite, so $K_{G,n}$ can be uncountable.

Editorial extensions

If this is right

A purely combinatorial condition on finite partial maps decides whether a Polish permutation group has a comeager $n$-diagonal conjugacy class, so no information outside $K_{G,n}$ is needed.
For automorphism groups of countable structures, ample generics is equivalent to $K_{\mathrm{Aut}(M),n}$ having JEP and WAP for every $n$, making a single failed WAP for some $n$ a uniform obstruction to ample generics.
The ordered ultrametric examples give a family of non-archimedean groups with a generic element but no generic pair: ball-preserving bijection groups of ordered rational $N$-ultrametric Urysohn spaces and generalized ordered Ważewski dendrites.
The homogenizability criterion identifies when a weak Fraïssé limit can be expanded by finitely many definable relations into an ultrahomogeneous structure with the same automorphism group.

Reading between the lines

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

Editorial extension: the proof's countability step for $K_{G,n}$ is a genuine gap for structures with function symbols; passing to a relational expansion that preserves $G$ would make finitely generated substructures finite and appears to repair it, but the paper does not state this.
Editorial extension: the free-group-word obstruction to weak amalgamation for pairs of ball-preserving maps may be a template for other hierarchically ordered Fraïssé-like classes, suggesting that 'generic element but no generic pair' is common among tree-like structures.
Editorial extension: the dense version of the criterion means that JEP alone can certify a dense orbit in the space of $n$-tuples of group elements; this could be used to detect non-classifiable orbit equivalence relations in non-archimedean groups without computing full amalgamation.

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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

The paper generalizes Kechris-Rosendal to arbitrary Polish permutation groups of countable structures, and the ultrametric applications are genuinely interesting. The central theorem may be true, but as written the proof has a real gap.

The new content is mostly in Theorem 3.10: for any Polish G ≤ Sym(M), a comeager n-diagonal conjugacy class is equivalent to JEP+WAP for K_{G,n}. That extends the earlier relational/finitely-generated setting, and the Krawczyk-Kubis game formalism is a natural fit. The homogenizability part (Theorem 4.3 and the example) is a clean application of Ahlman's criterion. The ultrametric Section 5 contains new results on BP(U_N^≺): comeager conjugacy class but no comeager 2-diagonal one. These recover older theorems from the author's own paper in a shorter way, and the ball-preserving perspective is worthwhile.

The soft spot is Theorem 3.3. The proof invokes Rasiowa-Sikorski with P = K_G. The version the paper states requires P countable; the more general standard lemma does not require P countable, but it still needs a countable family of dense subsets. Here the sets E_S and D_{S,T,U} are indexed by all S,T,U in K_G. When M has function symbols, finitely generated substructures can be infinite, and K_G can contain uncountably many mappings between two infinite substructures (e.g., Sym(N) acting on (N,S)). No countable subfamily of K_G is identified that would make the construction go through. This is not a cosmetic gap: the (1)⇒(2) direction of Theorem 3.10 is exactly what needs that construction. The rest of the paper seems to avoid the issue because Section 5 works with relational ultrametric structures, where finitely generated means finite and K_G is countable. I could not find an explicit countability assumption or a workaround in the paper, so the main theorem is not fully proved as stated.

I would still send this to a serious referee. The generalization is significant enough, and the applications are solid enough, that the paper deserves engagement. A referee should ask the author to either state a countability/finiteness assumption or repair the argument. This is a revamp, not a reject.

If I were to bring it to a reading group, it would be to discuss the gap and whether the theorem has a hidden counterexample. I'd cite the ultrametric results cautiously.

Referee Report

2 major / 4 minor

Summary. The paper develops a general framework, based on Krawczyk-Kubis games, for characterizing when a Polish group of permutations of a countable structure has a comeager (or dense) n-diagonal conjugacy class. The main announced result, Theorem 3.10, states that for a countable structure M, a Polish group G <= Sym(M), and n >= 1, G has a comeager n-diagonal conjugacy class if and only if the family K_{G,n} of n-tuples of G-extendable bijections between finitely generated substructures has JEP and WAP. The paper also studies homogenizability of limits of weak Fraisse classes and applies the abstract results to ball-preserving bijections of ordered ultrametric spaces, recovering and extending results on ordered boron trees and Wazewski dendrites.

Significance. If the main theorem is correct, it provides a uniform and substantial generalization of the Kechris-Rosendal characterization to arbitrary countable structures and arbitrary Polish permutation groups, not only automorphism groups of Fraisse limits. The paper supplies detailed game-theoretic arguments, identifies a useful notion of weak K-injectivity, and gives new examples of groups with a comeager conjugacy class but no comeager 2-diagonal conjugacy class. The ultrametric applications are original and connect the abstract machinery to natural topological groups. However, the central equivalence currently rests on a Rasiowa-Sikorski step whose hypotheses are not verified, so the main theorem is not fully proved as stated.

major comments (2)

[Section 3, Theorem 3.3] The Rasiowa-Sikorski step is not justified. The proof sets P = K_G and applies the lemma to the family of all sets F_m, E_S, and D_{S,T,U}. The lemma as stated in the proof requires P to be countable, and in any case the family of dense subsets must be countable. The paper does not show that K_G is countable. For a countable structure with function symbols this can fail: if M = (N, S) and G = Sym(M), the substructure N itself is finitely generated, and every permutation of N is a G-extendable bijection from N to N, so K_G has cardinality continuum. No countable cofinal subfamily of K_G is identified, and the proof of weak injectivity requires meeting D_{S,T,U} for every U extending T, not just for a countable skeleton. Consequently the equivalence (1) iff (4) in Theorem 3.10 is not proved as written for arbitrary countable structures; a restriction to a relational expansion or an additional countability/finiteness argument is needed.
[Section 5, Theorem 5.13] The final 'In particular' assertion is not supported by the preceding theorems. Theorems 5.8 and 5.9 are proved for the classes U^\prec_N, i.e., for ordered rational N-ultrametric Urysohn spaces with uniformly bounded polygon size. The theorem, however, states that for every ordered ultrahomogeneous Polish ultrametric space X, BP(X) has a comeager conjugacy class and no comeager 2-diagonal conjugacy class. No reduction of an arbitrary such space to some U^\prec_N is given, and the proofs of Theorems 5.8 and 5.9 do not obviously apply to spaces whose distance sets are not the rationals. This claim therefore needs either a proof or an explicit restriction of the statement.

minor comments (4)

[Section 3, Theorem 3.10] The statement begins 'LetG be a countable structure'; this should be 'Let M be a countable structure'.
[Section 1, Introduction] The first sentence contains a typo: 'Let us can consider' should be 'Let us consider'.
[Section 3, proof of Theorem 3.3] The dense sets are introduced as D_{S,T,U}, but a later sentence refers to 'the sets D_S,f'; this notation is undefined and should be D_{S,T,U}.
[Section 5, Theorem 5.13] The transition from the countable Fraisse limits U^\prec_N to 'Polish ultrametric spaces' is not explained: U^\prec_N with its ultrametric is not complete. Please clarify whether X is meant to be a metric completion and why the family of finite subspaces of the completion is the same as that of U^\prec_N.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central theorems are self-contained generalizations of external prior work.

full rationale

The paper's central equivalence (Theorem 3.10) is proved from scratch: weak injectivity is constructed from K_G, and the Banach-Mazur game is imported from Krawczyk-Kubiś [11]. This is not an input/output renaming: comeager n-diagonal conjugacy classes are topological properties of G, while JEP and WAP are combinatorial properties of tuples of G-extendable partial bijections, and the paper proves both directions rather than defining one in terms of the other. The only self-citations are [9] and [10]: [9]'s theorems are recovered as corollaries of Theorems 5.8 and 5.9, explicitly presented as consequences rather than assumptions, and [10] is a side fact used in an example. Neither is load-bearing for the main characterization. The countability gap in Theorem 3.3, where Rasiowa-Sikorski is applied to a potentially uncountable K_G, is a real correctness concern but not a circularity: it concerns whether the required dense family can be made countable, not whether a result is being assumed as its own conclusion. Thus there is no circular dependence to report.

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

No free parameters: this is a pure mathematics paper with no empirical fitting. The axioms are standard background (Fraisse theory, Rasiowa-Sikorski, external theorems from [1] and [11]) plus one ad hoc countability assumption in Theorem 3.3. The paper introduces no new theoretical entities such as particles or forces; ball-preserving bijections are a definition, not an invention.

assumptions (5)

ad hoc to paper The Rasiowa-Sikorski lemma is applicable to the partial order P = K_G, which requires P to be countable (or to have a countable cofinal subfamily).
Invoked in Theorem 3.3; the paper does not establish countability of K_G, and for structures with function symbols it can fail. This is an unstated assumption introduced to make the proof work.
domain assumption The classes K_N and K_N^prec of finite ordered ultrametric spaces with bp-injections are Fraisse classes.
Stated as 'We leave it to the reader to verify' in Section 5; the amalgamation proofs rely on this.
domain assumption Krawczyk-Kubis theory of weak Fraisse limits and games (Theorem 5.1 and the game BM_p from [11]) is correct.
The paper builds the main game arguments on [11] without reproving the foundational limit existence.
domain assumption Ahlman's characterization of homogenizable structures (Theorem 1.1 of [1]) is correct.
Used directly in Theorem 4.3.
standard math Rasiowa-Sikorski lemma (standard set theory).
Used in Theorem 3.3; standard result.

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Pith. "Pith review of Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups." pith.science (2026). https://pith.science/paper/YA6DNOWV

@misc{pith2026190809494,
  author       = {Pith},
  title        = {Pith review of: Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YA6DNOWV}},
  note         = {Machine review of arXiv:1908.09494}
}

abstract

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fra\"{i}ss\'{e} classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ahlman, Homogenizable structures and model completeness , Arch

O. Ahlman, Homogenizable structures and model completeness , Arch. Math. Logic 55 (2016) 977– 999

work page 2016

[2] [2]

Atserias, S

A. Atserias, S. Torunczyk, Non-homogenizable classes of ﬁnite structures

work page

[3] [3]

Covington, Homogenizable relational structures , Illinos J

J. Covington, Homogenizable relational structures , Illinos J. Math. 34 (1990), 731–743

work page 1990

[4] [4]

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

W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, 42. Cambridge Un i- versity Press, Cambridge, 1993

work page 1993

[5] [5]

Ivanov, Generic expansions of ω -categorical structures and semantics of generalized quan tiﬁes, J

A. Ivanov, Generic expansions of ω -categorical structures and semantics of generalized quan tiﬁes, J. Symbolic Logic, 64 (1999), 775–789

work page 1999

[6] [6]

Jasinski, Ramsey degrees of boron tree structures , Combinatorica 33 (2013), 23-44

J. Jasinski, Ramsey degrees of boron tree structures , Combinatorica 33 (2013), 23-44

work page 2013

[7] [7]

Rosendal, Turbulence, amalgamation, and generic automorphisms of ho mogeneous structures, Proc

A.Kechris, C. Rosendal, Turbulence, amalgamation, and generic automorphisms of ho mogeneous structures, Proc. Lond. Math. Soc. 94 (2007), 302–350

work page 2007

[8] [8]

Kwiatkowska, Universal minimal ﬂows of generalized Waewski dendrites , J

A. Kwiatkowska, Universal minimal ﬂows of generalized Waewski dendrites , J. of Symbolic Logic 83 (2018), 1618–1632

work page 2018