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Locally compact piecewise full groups of homeomorphisms

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Pith's one-line read The paper proves that alternating full groups of locally decomposable piecewise full actions are compactly generated simple open subgroups realizing every local isomorphism class of simple locally decomposable t.d.l.c. groups.

desk verdict Major, likely-correct reorganization of simple t.d.l.c. group theory via piecewise full groups, but the main theorems lean on two unproved companion results and one definitional slip. read the letter →

arxiv 2501.00908 v1 pith:3TVVXSUC submitted 2025-01-01 math.GR

Alejandra Garrido , Colin D. Reid This is my paper · ORCID
classification math.GR MSC 22D9922D0522F5020E3222A1537C8520E0837B05
keywords totallydisconnectedlocallycompactgroupspiecewisefullalternatingBooleaninversemonoidsdecomposableactionsgenerationfullycompressiblelocalisomorphism
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 sets out to show that piecewise full groups of homeomorphisms of the Cantor space—groups built by gluing together restrictions of a smaller group—can carry a canonical non-discrete locally compact topology, and that the alternating full subgroup inside them is the clean object to study. Working through inverse monoids of partial homeomorphisms, it proves a compact-generation criterion: if the Boolean completion of the monoid is compactly generated and every orbit has at least five points, then the alternating full group is compactly generated. Under minimality and local decomposability, that implies the alternating full group is abstractly simple, open, of countable index in the full group, and equal to its commutator subgroup. The payoff is a classification statement: every compactly generated simple t.d.l.c. group with a faithful locally decomposable action occurs as an open subgroup of such an alternating full group, and every robustly monolithic locally decomposable group is locally isomorphic to one.

What carries the argument

The central object is the Boolean inverse monoid BI(M) of clopen partial homeomorphisms of the Cantor space, whose group of units is the piecewise full group; it is equipped with a locally decomposable topology in which restriction to clopen sets and compatible joins are continuous. Inside it lives the alternating full group A(M), generated by the '3-cycles' of the action—homeomorphisms supported on three disjoint clopen pieces cyclically permuted. The load-bearing mechanism is the equivalence between compact generation of BI(M) and the existence of an expansive, piecewise dense, compactly generated open subgroup of the acting group; this converts the expansive dynamics of the action into a compact generating set for A(M), and then compressibility forces A(M) to be simple and open.

What would settle it

Exhibit a faithful minimal locally decomposable t.d.l.c. action on the Cantor space whose alternating full group is compactly generated but is not open in the full group, or whose quotient by the alternating full group is not discrete abelian. A more targeted check of Theorem 9.1: construct a compactly generated group in the robustly monolithic locally decomposable class whose group of germs has no compactly generated open subgroup that is expansive.

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Extended reading notes

Core claim

The central claim is that local decomposability of the action is exactly the condition that lets the topology of a t.d.l.c. group extend to its piecewise full group, and that once the alternating full group A(G) is compactly generated, structure follows almost for free: A(G) is simple, open in F(G), of countable index, equal to D(F(G)), and acts fully compressibly on the Cantor space. The universality theorem then asserts that the class A of groups between D(F) and F built from such actions is the natural ambient space for the local isomorphism classes in SLD and RLD: every group in SLD is an open subgroup of a group in A∩SLD, and every group in RLD is locally isomorphic to a group in SLD. If true, this means that alternating full groups are the canonical representatives of the local isomorphism classes of all simple t.d.l.c. groups admitting faithful locally decomposable actions.

Load-bearing premise

The structural conclusions of Sections 5–9 depend on two theorems imported from the companion paper: a normal subgroup with compressible action is open, and a compressible piecewise full action has simple monolith (the intersection of all nontrivial normal subgroups) equal to its derived subgroup; if either theorem is false, or its hypotheses are not met in a particular action, the openness and universality claims collapse.

Editorial extensions

If this is right

  • The classical almost automorphism groups of regular trees and earlier tree-based constructions become special cases of a uniform criterion: any locally decomposable t.d.l.c. group with a compactly generated expansive piecewise dense open subgroup has a compactly generated simple alternating full group.
  • Groups in A act freely on their Furstenberg boundary, so every conjugacy class of closed relatively amenable subgroups accumulates at the trivial subgroup in the space of closed subgroups.
  • Groups in A admit no general-type actions on hyperbolic spaces; their perfect alternating full subgroups have no loxodromic elements at all and are one-ended.
  • Every group in SLD is an open subgroup of a group in A∩SLD, and every group in RLD is locally isomorphic to a group in SLD, so a local isomorphism class is controlled by one compactly generated expansive open subgroup together with a faithful minimal micro-supported action.
  • For profinite branch groups, being locally isomorphic to a group in SLD is equivalent to some compactly generated open subgroup of the group of germs being expansive; otherwise, in the just-infinite case, all compactly generated subgroups have arbitrarily small invariant identity neighbourhoods.

Reading between the lines

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

  • Editorial inference: The five-point orbit condition in the compact-generation theorem looks like a real boundary, not a convenience: because the proof requires the alternating group on five letters to be perfect, actions whose orbits have only three points—the range of the minimal homeomorphism constructions—may fail compact generation even when the Boolean inverse monoid is compactly generated.
  • Editorial inference: Theorem 9.1 turns the problem of constructing new simple locally compact groups into a search for compactly generated expansive t.d.l.c. groups with a faithful minimal micro-supported action; any such group can be completed to a group in A and then stripped down to a simple alternating full group, which should be a productive source of new examples.
  • Editorial inference: The paper's open question about adapting a homology theory to non-discrete full groups suggests a testable programme: if such homology is built, the quotient F(G)/A(G) could be computed by an index map that detects the modular function, giving new invariants to distinguish groups within the same local isomorphism class.
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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

2 major / 3 minor

Summary. The paper develops a theory of locally compact piecewise full groups of homeomorphisms of the Cantor space, working partly through topological Boolean inverse monoids. The main results are: local decomposability is necessary and sufficient for extending a group topology to the piecewise full group; a compact generation criterion for Nekrashevych's alternating full group A(G); and a structural theorem (Theorem 1.7) saying that under faithful, minimal, locally decomposable action with A(G) compactly generated, A(G) is open, abstractly simple, equal to D(F(G)), and has fully compressible action. The paper then introduces the class A of t.d.l.c. groups containing such alternating full groups, proves Neretin-type properties for A (free action on the Furstenberg boundary, absence of general-type hyperbolic actions), and derives a universality result for A within the local structure theory of simple t.d.l.c. groups with locally decomposable actions. The final section lists open questions.

Significance. If the main results are correct, the paper gives a substantial unifying framework for non-discrete piecewise full groups, placing Neretin, Röver and Lederle constructions in a common setting, and it yields a structural characterisation of local isomorphism classes of a large family of simple t.d.l.c. groups. The inverse-monoid topology extension result (Theorem 1.3, Corollary 3.34) is a genuinely useful technical contribution, and the applications to Furstenberg boundaries and hyperbolic actions give strong new constraints on groups in A. The paper is carefully written and the proofs inspected are detailed; the authors also clearly separate their own open questions.

major comments (2)
  1. [Section 5, Theorem 1.7 and Corollary 5.14] The proof of Theorem 1.7 and the subsequent structural results (Corollary 5.14, Theorem 8.3, Theorem 9.1) rest on two results imported from the companion article [20] without proof: Theorem 5.12 ([20, Theorem 1.13]) and Proposition 5.13 ([20, Corollary 1.10]). These results carry the claims that A(G) is open in F(G) and equals D(F(G)), and that compressible piecewise full actions have simple monolith M(G)=D(H). Because the hypotheses of these results must be verified in the present setting (notably compressibility of the A(G)-action on X), and because no proof or proof sketch is supplied, the central claims cannot be fully checked from this manuscript alone. Please state the full hypotheses of these results and either reproduce their proofs in an appendix or indicate clearly where they can be found if [20] is publicly available.
  2. [Theorem 9.1, proof of (iii)⇒(vi)] The text asserts that, by Lemma 5.11, a compactly generated open subgroup H in RLD is 'an expansive t.d.l.c. group'. Lemma 5.11(i) only gives regional expansivity, i.e. the existence of a compactly generated subgroup (not necessarily open) that is expansive. Containing such a subgroup does not imply that H itself is expansive, and conclusion (vi) explicitly requires an open expansive subgroup A of L. This step is therefore not justified as written unless the definition of 'regionally expansive' (Definition 5.8) is strengthened to require an open witness, or an additional argument is supplied.
minor comments (3)
  1. [Theorem 1.7] The statement contains the typo 'acts mimimally' for 'acts minimally'; the same typo appears in the introduction where Theorem 1.7 is stated.
  2. [Definition 5.8] The definition of 'regionally expansive' says 'it has a compactly generated subgroup H that is expansive' without specifying whether H must be open. Since Theorem 9.1(vi) needs an open expansive subgroup, please clarify this point or adjust the proof accordingly.
  3. [Definition 6.5] The phrase 'faithul continuous action' should read 'faithful continuous action'.

Circularity Check

0 steps flagged · score 0.0 of 10

No demonstrated circularity: the main results are proved from stated hypotheses using self-contained Sections 2–4 plus quoted external theorems; heavy self-citation in Sections 5 and 9 is a verification risk but not a circular reduction.

full rationale

No step in the derivation chain reduces a target claim to its own input by construction. Sections 2 and 3 contain direct proofs of the topology-extension mechanism (Proposition 2.12, Theorem 3.33, Corollary 3.34), and Section 4 proves compact generation of A(M) from compact generation of the Boolean inverse monoid (Theorem 4.14) by an explicit generating-set argument, not by assuming the conclusion. Theorem 1.7 and the later universal-structure results invoke Theorem 5.12 and Proposition 5.13 as black boxes from the companion article [20], and compressibility/decomposition-lattice facts from [11]–[13]; these are external imported results whose hypotheses do not include the theorem being proved, so their use is dependence rather than circularity. The preprint is fragile if the companion results are not independently verified, and the proof of Theorem 9.1 appears to overstate Lemma 5.11 by passing from regional expansivity to expansivity of a compactly generated open subgroup; this is a logical-support gap, not a circular reduction. No fitted parameter is relabelled as a prediction, no quantity is defined in terms of the claimed conclusion, and no equation is reused as its own output. Consequently no circular step can be quoted and the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

Pure mathematics, so there are no fitted free parameters. The axioms are explicit or cited premises: (a) imported theorems from the companion article [20] and the authors' earlier work [11], [12], [13], which supply the decomposition lattice, the classes R and S, and the compressibility lemmas; (b) Nekrashevych's theorems on alternating full groups [39]; (c) standard background (Stone duality, Wagner-Preston, Pettis, Baire, Van Dantzig); (d) explicit domain hypotheses of faithfulness, minimality, local decomposability, and at least 5 points per orbit in Theorem 4.14. The only invented object is the class A and the notion of A-action, which are definitions, not hidden postulates: Theorem 9.1(vi) characterizes local isomorphism classes containing A-groups by conditions that do not mention A, giving the class an external anchor.

assumptions (5)
  • domain assumption The results quoted from the companion article [20] are correct and apply here: Theorem 5.12 ([20, Theorem 1.13]) and Proposition 5.13 ([20, Corollary 1.10]), plus Lemma 8.7 ([20, Corollary 3.5]).
    Theorem 5.12 and Proposition 5.13 are invoked verbatim to prove Theorem 1.7, that A(G) is open, simple and equal to D(F(G)), which anchors the class A and Theorem 9.1. The companion article is by the same authors and its proofs are not reproduced here, so the present results inherit its correctness risk.
  • domain assumption Nekrashevych's theorems on alternating full groups hold: [39, Theorem 4.1] (simplicity and monolith property) and [39, Theorem 5.10] (finite generation).
    Invoked at Theorem 4.11 and used again in Corollary 5.14 and Lemma 4.13; the entire alternating-group machinery assumes these external results are correct and transfer to the topological setting developed here.
  • standard math Standard background facts: Stone duality, the Wagner-Preston theorem, Pettis's theorem, the Baire category theorem, Van Dantzig's theorem, and uniqueness of Polish group topologies from Borel separating families.
    Used in Sections 2 and 3 (Corollary 2.5, Lemma 2.6, Proposition 2.15, Lemma 3.18, Proposition 3.36). These are standard results in the field, unproblematic for the target audience.
  • domain assumption The prior theory of the authors and collaborators is correct: decomposition lattices LD(G), the classes R and S, local decomposability of actions, and compressibility (from [11], [12], [13], [20]).
    Sections 5 and 9 rest on [12, Theorem 4.5 and 5.18], [11, Proposition 5.1.2 and Theorem 7.3.3] and [13]; these supply the decomposition lattice, the robustly monolithic class R, and the structure theory of S that Theorem 9.1 extends.
  • domain assumption The main theorems assume faithful, minimal, locally decomposable actions on compact zero-dimensional spaces, and Theorem 4.14 requires every orbit to have at least 5 points.
    These are explicit hypotheses, not hidden ones, but they delimit the results: minimality and non-degeneracy appear in Corollary 4.22 and Theorem 1.7, while the bound n >= 5 enters Lemma 4.16 through perfectness of Alt(5); the 3-point Juschenko-Monod style setting is excluded from the compact-generation conclusion.
invented entities (1)
  • Class A of t.d.l.c. groups and A-actions independent evidence
    purpose: Organizes the structural results: groups in A are robustly monolithic, act freely on the Furstenberg boundary, have no general-type hyperbolic actions, and every SLD group embeds as an open subgroup of a group in A; A-actions are essentially unique when they exist (Theorem 8.3, definitions in Section 8.1).
    Definitional rather than hypothesized: Theorem 9.1(vi) characterizes the local isomorphism classes containing A-groups by conditions mentioning expansivity and micro-supported actions but not A itself, and Proposition 8.2 gives an equivalent definition via compactly generated expansive subgroups. This gives the class external anchors. No fitted or free entity is introduced.

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Pith. "Pith review of Locally compact piecewise full groups of homeomorphisms." pith.science (2026). https://pith.science/paper/3TVVXSUC

@misc{pith2026250100908,
  author       = {Pith},
  title        = {Pith review of: Locally compact piecewise full groups of homeomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3TVVXSUC}},
  note         = {Machine review of arXiv:2501.00908}
}
abstract

We study when a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor space $X$ can be given a non-discrete totally disconnected locally compact (t.d.l.c.) topology and give a criterion for the alternating full group (in the sense of Nekrashevych's group A(G) to be compactly generated. As a result, starting from qualitative criteria, we obtain a large class of t.d.l.c. groups such that the derived group is non-discrete, compactly generated, open and simple, putting previous constructions of Neretin, Roever and Lederle in a more systematic context. We also show some notable properties of Neretin's groups apply to this class in general. General consequences are derived for the theory of simple t.d.l.c. groups, prime among them the universal role that alternating full groups play in the class of simple t.d.l.c. groups that are non-discrete, compactly generated and locally decomposable. Some of the theory is developed in the setting of topological inverse monoids of partial homeomorphisms of $X$. In particular, we obtain a sufficient condition to extend the topology to a monoid equipped with all restrictions with respect to compact open subsets of $X$ and all joins of compatible pairs of elements. The compact generation criterion is also naturally expressed in this context.

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  1. The regular representation of Neretin groups is factorial

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    The regular representation of every Neretin group is factorial, yielding the first non-discrete simple group whose von Neumann algebra is a factor.

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