Compressible subgroups and simplicity

Alejandra Garrido; Colin D. Reid

REVIEW 1 major objections 4 minor 2 cited by

Compressible subgroups and simplicity

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Pith's one-line read Joinable compression families yield simple derived subgroups

desk verdict A solid unification of simplicity criteria via compression families, with a real but localized gap in the openness theorem. read the letter →

arxiv 2412.18891 v1 pith:L5KEZAQ6 submitted 2024-12-25 math.GR

Alejandra Garrido , Colin D. Reid This is my paper · ORCID

classification math.GR MSC 20E3222F5022D0520E0837B05

keywords almostsimplegroupderivedcommutatortotallydisconnectedlocallycompactactiononCantorspacedynamicsmicro-supportedcompressionfamily

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

This paper gives sufficient conditions, expressed through a family of subgroups called a compression family, for a group to be almost simple. The main theorem shows that when the family satisfies a closure property called joinability and generates a subgroup $H$, the derived subgroup $D(H)$ is the monolith of $G$; with an extra transitivity condition $D(H)$ is simple, and with a shift condition $D(H)=H$ is simple. A second theorem shows that in a totally disconnected locally compact group with a faithful, minimal, locally decomposable action, any normal subgroup—closed or not—whose action is compressible must be open. These results unify and extend earlier simplicity criteria for tree automorphism groups and topological full groups, and imply that many previously studied monolithic t.d.l.c. groups are simple-by-discrete.

What carries the argument

The central object is a joinable compression family: a conjugation-invariant family $\mathcal{C}$ of non-abelian subgroups satisfying (A) each member is non-abelian, (B) every nontrivial element of the ambient group has a member $A$ with $[A,gAg^{-1}]=\{1\}$, (C) semitransitivity, meaning any member can be conjugated inside any other by $H$, and (D) joinability: whenever $A$ commutes with $C$ and $C$ is not contained in $B$, some $D\in\mathcal{C}$ contains $\langle A,B\rangle$. Joinability is the load-bearing closure condition that lets the proof push commutators into the monolith; shiftability (E) adds an infinite-product shift endomorphism that turns every element of $H$ into a commutator, giving perfectness.

What would settle it

Exhibit a group $G$ and a joinable compression family $\mathcal{C}$ for $H=\langle\mathcal{C}\rangle$ such that $D(H)$ is not the monolith of $G$, or such that $D(H)$ has a proper nontrivial normal subgroup while $H$ is semi-transitive on $\mathcal{C}$. Concretely, one could search for a compressible micro-supported action satisfying the union closure of Corollary 2.7 whose derived subgroup is not simple; the paper leaves this open in Question 1, so no such example is currently known.

Watch

Extended reading notes

Core claim

Under the hypotheses of Theorem 1.6, the derived group of the subgroup generated by a joinable compression family is exactly the intersection of all nontrivial normal subgroups, and it is simple whenever the family is semi-transitive; when the family is also shiftable, the generated subgroup is perfect and simple. The proof is commutator-theoretic: condition (D) lets any commutator $[a,b]$ with $a,b$ in members of $\mathcal{C}$ be expressed as a product of commutators falling inside the monolith, forcing $D(H)$ into every normal subgroup, while Lemma 2.8 supplies the reverse containment. Theorem 1.13 adds a topological conclusion: in a faithful minimal locally decomposable action of a t.d.l.c. group, a normal subgroup with compressible action is open, because compressibility spreads rigid stabilisers whose contraction groups lie inside the subgroup.

Load-bearing premise

Everything rests on condition (D), joinability: whenever one member of the family commutes with another that is not contained in a third, a fourth member must contain the subgroup generated by the first and third; this closure under taking unions is not a formal consequence of compressibility and is verified in the paper only when the underlying subsets are closed under unions up to density.

Editorial extensions

If this is right

For any compressible micro-supported action whose rigid stabilisers satisfy the union condition of Corollary 1.7, the ambient group is almost simple with monolith $D(H)$, and the action of $D(H)$ is fully compressible.
Tree automorphism groups acting geometrically densely with nontrivial $G^{++}$ are almost simple with monolith $D(G^{++})$; under local compactness, $G^{++}$ is abstractly simple.
Compressible piecewise full groups on zero-dimensional spaces are almost simple with monolith $D(H)$; if the space is compact and the action minimal, $D(G)$ is simple and its action fully compressible.
In a t.d.l.c. group with a faithful minimal locally decomposable action, every normal subgroup with compressible action is open, hence inherits local decomposability and minimality.
Robustly monolithic t.d.l.c. groups with an open direct-product subgroup have an open, abstractly simple monolith, making them simple-by-discrete.

Reading between the lines

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

Going beyond the paper, the same commutator strategy suggests that any minimal compressible action generated by elements of compressible support may have simple derived subgroup; Question 1 marks this as the natural sharpening if joinability can be weakened.
The openness conclusion of Theorem 1.13 does not require the normal subgroup to be closed, so it may apply to dense subgroups such as derived subgroups, potentially turning abstract simplicity into topological simplicity in many t.d.l.c. piecewise full groups.
The property (NGT) argument in Theorem 3.3 is stated for groups with a joinable compression family, so the obstruction to general-type hyperbolic actions should transfer to any such family satisfying the transitivity hypotheses, not only to piecewise full groups.
A testable extension suggested by condition (E) is to study the quotient $H/D(H)$ when only (A)–(D) hold; the paper notes this quotient is often mysterious, and the shiftability hypothesis may be exactly what forces it to vanish.

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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 does something genuinely useful: it distills the common core of Tits, Matui, Le Boudec, and Möller-Vonk into compression families, and proves a clean monolith/simplicity theorem. The proof of Theorem 1.6 is careful and I found no error in the commutator arguments. The corollaries recovering known results are correct, and the new consequences, like abstract simplicity of G++ for locally compact tree automorphism groups, are worthwhile. Section 3's adaptation of BFFG's NGT criterion is also clever and checks out.

The main soft spot is exactly where the reader flagged it. In the proof of Theorem 1.13, the paper invokes 'locally weakly decomposable' from [CRW17b] without defining the term or proving that local decomposability (as defined in Section 1.4) implies it. This is not a cosmetic omission: [CRW17b, Proposition 6.14] is the only input that forces con_G(g) ∩ L to be open in L, and that openness is what makes the argument go through. Corollary 1.15 inherits the gap. I agree with the reader's conditional verdict.

Two lesser observations. First, condition (D) in Definition 1.5 is strong, and the paper itself acknowledges via Question 1 that it is not a formal consequence of compressibility. That is honest, not a flaw. Second, the paper leans on several previous results from the same research group; those citations look accurate and are not circular, so I do not hold that against it.

Who is this for? People working on simplicity of groups acting on trees, Cantor spaces, and t.d.l.c. groups. The paper deserves a serious referee, but the referee should insist that Theorem 1.13 be fixed: either add the missing definition, prove the implication, or adjust the hypothesis. The rest of the paper is solid and would survive that revision intact.

Referee Report

1 major / 4 minor

Summary. The paper develops a general commutator framework for proving almost simplicity and simplicity of derived subgroups. It defines compression families of subgroups (nomadic, joinable, shiftable) and proves Theorem 1.6: for a joinable compression family C with H=<C>, D(H) is the monolith of G; under semi-transitivity D(H) is simple; under shift-joinability D(H)=H is a simple monolith. Applications include compressed micro-supported actions (Corollary 1.7), tree automorphism groups (Corollaries 1.8–1.9), and piecewise full groups (Corollary 1.10, Theorem 1.11). The paper also proves a criterion for non-closed normal subgroups of t.d.l.c. groups to be open (Theorem 1.13), and derives consequences for robustly monolithic groups (Corollary 1.15). Section 3 adapts arguments of Balasubramanya–Fournier-Facio–Genevois to give property (NGT) for subgroups G0 with D(G) ≤ G0 ≤ G (Theorem 3.3 and Corollary 3.5).

Significance. The compression-family framework is a clean and potentially useful unification of several existing simplicity criteria, including work of Tits, Matui, Le Boudec, Möller–Vonk, and others. The applications to tree actions and piecewise full groups are natural and the derivations are mostly self-contained. The paper is honest about the limitations of condition (D) (Question 1) and provides weaker statements when it fails (Lemma 2.8, Proposition 2.9). The central gap is that Theorem 1.13 depends on the undefined term 'locally weakly decomposable'; until that term is defined and the implication from 'locally decomposable' is supplied, the openness theorem and its corollary are not established. This is a load-bearing but likely fixable issue.

major comments (1)

[Section 4.1, proof of Theorem 1.13] The proof invokes 'locally weakly decomposable in the terminology of [CRW17b]' without defining the term or proving that the standing hypothesis of local decomposability (defined in Section 1.4) implies it. The sentence 'since the action is locally weakly decomposable ... we see by [CRW17b, Proposition 6.14] that the intersection L* = con_G(g) ∩ L is open in L' is the only step that produces an open contraction-group intersection, and that openness is then used to conclude that A is open in G. With the undefined term left as is, Theorem 1.13 is unproved. The same gap propagates to Corollary 1.15, which is derived directly from Theorem 1.13.

minor comments (4)

[Section 3, proof of Theorem 3.3, Claim 1] The text 'there is k ∈ A ∩ gC' appears to be a typo: in the preceding lines the relevant group element is h, not g. The intended statement should be 'there is k ∈ A ∩ h C' (or an explicit redefinition), and the derivation should be spelled out.
[Section 3, Theorem 3.3, property (L)] The statement of property (L) is hard to parse: the quantifiers over g_i and the condition 'g_j ∈ A_j for at least one j' are not fully clear. Please restate the property with explicit variables and explain how it is used to produce the element k in Claim 1.
[Section 4.1, proof of Theorem 1.13, final paragraph] The assertion that ⟨a_i L* a_i^{-1} | 1 ≤ i ≤ n⟩ is open in G after using the finite cover X = ⋃ a_i(Z \ gZ) is not immediate. The proof should explicitly say that one passes to a finite clopen partition refining the cover and applies local decomposability to that partition, so that each a_i L* a_i^{-1} is open in the corresponding rigid stabiliser.
[Section 2.1, Lemma 2.4(iv)] The identity [a,b] = [[a,g],[b,g^2]] is used without proof. Since the paper's commutator convention is [x,y] = x y x^{-1} y^{-1}, a short verification would help the reader, especially because the identity relies on the pairwise commutation of distinct conjugates from part (ii).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 1.6 is derived from Definition 1.5 rather than from its conclusion; the flagged Theorem 1.13 step is a missing-support/correctness gap, not a circular reduction.

full rationale

The central simplicity theorem is not circular. Theorem 1.6 is proved in Section 2.3 from the stated conditions (A)-(D) of Definition 1.5: Lemma 2.8 and Proposition 2.9 derive the monolith statements directly from the compression family, and condition (D) (joinability) is used to push commutators [a,b] into the monolith. No fitted parameter, no equation of the form conclusion = input, and no renaming is used. The tree and piecewise-full applications (Corollaries 1.8, 1.10, 2.12) instantiate the same framework, and the recovery of Matui's theorem is a comparison with an external result, not an input. Section 3's NGT results are adaptations of the independent [BFFG24]/[Gen19] arguments. The one genuinely concerning passage is in the proof of Theorem 1.13 (Section 4.1, p. 18): 'since the action is locally weakly decomposable in the terminology of [CRW17b], we see by [CRW17b, Proposition 6.14] that the intersection L* = con_G(g) ∩ L is open in L.' The theorem's hypothesis is only 'locally decomposable', and the paper neither defines 'locally weakly decomposable' nor proves that the former implies the latter; [CRW17b] is prior work by the same research group (Reid is an author), and its Proposition 6.14 is the sole source of the openness that makes A open. This is a load-bearing missing-support/correctness gap, but it is not circularity under the hard rule: one cannot exhibit L* being open as an equation equal to the hypotheses by construction, and the missing implication might be true. It therefore does not raise the circularity score above 2, though it should be fixed in revision.

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

No free parameters are fitted; the theory is qualitative. The main theorems rest on unproved structural assumptions (A)-(E) and on several external results, including prior work by the same research group. These external facts are standard within the t.d.l.c. literature and are not assumptions of the target conclusions, but a fully self-contained verification would require reproducing them.

assumptions (9)

standard math Standard group theory and commutator identities in ZFC
Used throughout Section 2, e.g. Lemma 2.2, without proof beyond the short derivations given.
domain assumption C is a joinable compression family for G satisfying conditions (A)-(D) of Definition 1.5
This is the main structural hypothesis of Theorem 1.6; it is not derived from more basic principles.
domain assumption For Theorem 1.6(iii), the extra shift condition (E) holds
Condition (E) requires an infinite product embedding inside H, a strong hypothesis that the paper notes is not available for Neretin-type examples.
domain assumption Local decomposability and compressibility of the action in Theorem 1.13
These are stated hypotheses; the proof also silently assumes the 'locally weakly decomposable' notion from [CRW17b].
standard math Tits' lemmas on geometrically dense tree actions (Lemma 2.14, citing [Tit70], [MV12], [RS])
Used to verify compression-family conditions for tree actions and to prove Corollaries 1.8 and 1.9.
standard math Contraction groups are contained in normal subgroups (Lemma 4.2, citing [CRW14, Prop. 5.1])
Used in Theorem 1.13 and Corollary 1.9 to identify rigid stabilisers inside the derived group.
standard math Facts on locally normal subgroups and robustly monolithic groups from [CRW17a], [CRW17b], [CRW21] (Lemma 4.4)
Used for Corollary 1.15 to produce a compressible locally decomposable action.
standard math Genevois's criterion [Gen19, Theorem 1.1] and BFFG24 machinery for property (NGT)
These are imported in Theorem 3.3 and Corollary 3.5; the paper adapts rather than reproves them.
standard math Ends theory results (Houghton [Hou74]; Caprace-Marquis-Reid [CMR24, Prop. 3.6])
Used in Corollary 3.4 to convert absence of general-type tree actions into an at-most-one-end conclusion.

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Pith. "Pith review of Compressible subgroups and simplicity." pith.science (2026). https://pith.science/paper/L5KEZAQ6

@misc{pith2026241218891,
  author       = {Pith},
  title        = {Pith review of: Compressible subgroups and simplicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5KEZAQ6}},
  note         = {Machine review of arXiv:2412.18891}
}

abstract

In this article we give sufficient conditions for a group to have simple derived subgroup; the argument is based on generalising properties observed for extremely proximal micro-supported actions on the Cantor space, and generalises previous results of Matui, Le Boudec and others in this direction. We give a sufficient condition for a non-trivial normal subgroup (not assumed closed) of a locally compact group $G$ to be open, also based on the theory of micro-supported actions. This shows in particular that many of the class of robustly monolithic groups introduced by Caprace--Reid--Wesolek are simple-by-discrete.

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Forward citations

Cited by 2 Pith papers

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Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
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Lim-free dynamical criterion for faithful boundary actions on hyperbolic spaces is equivalent to the algebraic property of being mixed identity free, with transitivity degree bounded by 3 for non-lim-free groups.
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Local decomposability characterises when a piecewise full group admits a unique non-discrete t.d.l.c. topology, and the resulting alternating full group is an open, compactly generated, simple derived subgroup that is...

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages · cited by 2 Pith papers

[1]

Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, J

Christopher Banks, Murray Elder, and George A. Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, J. Group Theory 18 (2015), no. 2, 235--261 (English)

work page 2015
[2]

Sahana Balasubramanya, Francesco Fournier-Facio, and Anthony Genevois, Property ( NL ) for group actions on hyperbolic spaces (with an appendix by alessandro sisto , Groups Geom. Dyn. (2024), published on-line first DOI 10.4171/GGD/806

doi:10.4171/ggd/806 2024
[3]

Pierre-Emmanuel Caprace, Automorphism groups of right-angled buildings: simplicity and local splittings., Fundam. Math. 224 (2014), no. 1, 17--51 (English)

work page 2014
[4]

Reid, Growing trees from compact subgroups, Groups, Geometry, and Dynamics 18 (2024), no

Pierre-Emmanuel Caprace, Timoth\' e e Marquis, and Colin D. Reid, Growing trees from compact subgroups, Groups, Geometry, and Dynamics 18 (2024), no. 1, 327--352

work page 2024
[5]

Reid, and George A

Pierre-Emmanuel Caprace, Colin D. Reid, and George A. Willis, Limits of contraction groups and the T its core , Journal of Lie Theory 24 (2014), 957--967

work page 2014
[6]

P art I : G eneral theory , Forum Math

, Locally normal subgroups of totally disconnected groups. P art I : G eneral theory , Forum Math. Sigma 5 (2017), e11

work page 2017
[7]

P art I I : C ompactly generated simple groups , Forum Math

, Locally normal subgroups of totally disconnected groups. P art I I : C ompactly generated simple groups , Forum Math. Sigma 5 (2017), e12

work page 2017
[8]

Caprace, Colin D

P.-E. Caprace, Colin D. Reid, and Phillip R. Wesolek, Approximating simple locally compact groups by their dense locally compact subgroups, Int. Math. Res. Not. (2021), no. 7, 5037--5110

work page 2021

Show all 24 references

[9]

Silva, and Koen Struyve, Universal groups for right-angled buildings, Groups Geom

Tom De Medts, Ana C. Silva, and Koen Struyve, Universal groups for right-angled buildings, Groups Geom. Dyn. 12 (2018), no. 1, 231--287

2018
[10]

Group Theory 22 (2019), no

Anthony Genevois, Hyperbolic and cubical rigidities of T hompson's group V , J. Group Theory 22 (2019), no. 2, 313--345

2019
[11]

Garrido and C.D

A. Garrido and C.D. Reid, Locally compact piecewise full groups of homeomorphisms, in preparation, 2024

2024
[12]

C. H. Houghton, Ends of locally compact groups and their coset spaces, J. Aust. Math. Soc. 17 (1974), no. 3, 274--284

1974
[13]

181--248 (French)

Fr \'e d \'e ric Haglund and Fr \'e d \'e ric Paulin, Simplicity of automorphism groups of spaces with negative curvature, The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday, Warwick: University of Warwick, Institute of Mathematics, 19...

1998
[14]

Christophe Kapoudjian, Simplicity of Neretin 's group of spheromorphisms. , Ann. Inst. Fourier 49 (1999), no. 4, 1225--1240

1999
[15]

Nir Lazarovich, On regular CAT (0) cube complexes and the simplicity of automorphism groups of rank-one CAT (0) cube complexes , Comment. Math. Helv. 93 (2018), no. 1, 33--54

2018
[16]

2, 253--293

Adrien Le Boudec, Groups acting on trees with almost prescribed local action, Commentarii Mathematici Helvetici 91 (2016), no. 2, 253--293

2016
[17]

5, 1464--1501

, Amenable uniformly recurrent subgroups and lattice embeddings, Ergodic Theory and Dynamical Systems 41 (2021), no. 5, 1464--1501

2021
[18]

Waltraud Lederle, Coloured N eretin groups , Groups Geom. Dyn. 13 (2019), no. 2, 467--510

2019
[19]

Reine Angew

Hiroki Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math. 705 (2015), 35--84

2015
[20]

R. G. M\"oller and J. Vonk, Normal subgroups of groups acting on trees and automorphism groups of graphs, J. Group Theory 15 (2012), no. 6, 831--850

2012
[21]

3, 707--732

Volodymyr Nekrashevych, Simple groups of dynamical origin, Ergodic Theory and Dynamical Systems 29 (2019), no. 3, 707--732

2019
[22]

Reid and Simon M

Colin D. Reid and Simon M. Smith, Groups acting on trees with T its' independence property ( P ) , arXiv:2002.11766v2

2002 arXiv
[23]

188--211

Jacques Tits, Sur le groupe des automorphismes d'un arbre, Essays on topology and related topics ( M \'emoires d\'edi\'es \`a G eorges de R ham), Springer, New York, 1970, pp. 188--211

1970
[24]

Katrin Tent and Martin Ziegler, On the isometry group of the Urysohn space , J. Lond. Math. Soc., II. Ser. 87 (2013), no. 1, 289--303

2013

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, J

Christopher Banks, Murray Elder, and George A. Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, J. Group Theory 18 (2015), no. 2, 235--261 (English)

work page 2015

[2] [2]

Sahana Balasubramanya, Francesco Fournier-Facio, and Anthony Genevois, Property ( NL ) for group actions on hyperbolic spaces (with an appendix by alessandro sisto , Groups Geom. Dyn. (2024), published on-line first DOI 10.4171/GGD/806

doi:10.4171/ggd/806 2024

[3] [3]

Pierre-Emmanuel Caprace, Automorphism groups of right-angled buildings: simplicity and local splittings., Fundam. Math. 224 (2014), no. 1, 17--51 (English)

work page 2014

[4] [4]

Reid, Growing trees from compact subgroups, Groups, Geometry, and Dynamics 18 (2024), no

Pierre-Emmanuel Caprace, Timoth\' e e Marquis, and Colin D. Reid, Growing trees from compact subgroups, Groups, Geometry, and Dynamics 18 (2024), no. 1, 327--352

work page 2024

[5] [5]

Reid, and George A

Pierre-Emmanuel Caprace, Colin D. Reid, and George A. Willis, Limits of contraction groups and the T its core , Journal of Lie Theory 24 (2014), 957--967

work page 2014

[6] [6]

P art I : G eneral theory , Forum Math

, Locally normal subgroups of totally disconnected groups. P art I : G eneral theory , Forum Math. Sigma 5 (2017), e11

work page 2017

[7] [7]

P art I I : C ompactly generated simple groups , Forum Math

, Locally normal subgroups of totally disconnected groups. P art I I : C ompactly generated simple groups , Forum Math. Sigma 5 (2017), e12

work page 2017

[8] [8]

Caprace, Colin D

P.-E. Caprace, Colin D. Reid, and Phillip R. Wesolek, Approximating simple locally compact groups by their dense locally compact subgroups, Int. Math. Res. Not. (2021), no. 7, 5037--5110

work page 2021