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Non-strong ergodicity of canonical actions of the Thompson groups

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Pith's one-line read This paper proves that the canonical actions of the Thompson group and its generalizations are never strongly ergodic, yielding non-full type III crossed-product factors and a non-embedding theorem.

desk verdict False as stated but probably fixable; the real Thompson-group results look right. read the letter →

arxiv 2510.13309 v2 pith:VHTXKSZQ submitted 2025-10-15 math.DS math.GRmath.OA

Ryoya Arimoto This is my paper · ORCID
classification math.DSmath.GRmath.OA MSC 37A4020F3820L05
keywords StrongergodicityThompsongroupsTopologicalfullAmplegroupoidsAmenableCrossedproductvonNeumannalgebrasTypeIIIfactorsCantorset
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 establishes that the canonical action of the Thompson group V on the Cantor set — and, much more generally, of the topological full group [[G]] of any topologically principal, amenable, ample groupoid G on its unit space — is not strongly ergodic for any quasi-invariant probability measure. The key step is showing that the orbit equivalence relation of such an amenable groupoid is Borel amenable, which yields a non-trivial almost invariant sequence for [[G]]. Because full crossed-product von Neumann algebras force strong ergodicity, this implies the associated group-measure-space algebras cannot be full. For the Higman–Thompson groups V_{d,k}, the paper combines this with a non-amenability argument and a type computation to conclude that L∞(X_{d,k}) ⋊ V_{d,k} is a non-amenable, non-full factor of type III_{1/d}; for the classical Thompson group V this is a non-amenable, non-full type III_{1/2} factor. It also yields the non-embedding of strongly ergodic actions, in particular free-group Bernoulli shifts, into these canonical actions.

What carries the argument

The central object is the canonical action of the topological full group [[G]] — the group of compact open full bisections of an ample groupoid G — on the unit space G^(0). The load-bearing mechanism is the transfer of amenability from the groupoid G to its associated equivalence relation R_G: a net of probability measures on the source fibers, asymptotically invariant under G, is summed over each equivalence class to produce a net of probability measures asymptotically invariant under R_G, proving R_G is Borel amenable. Then the standard almost-invariant-sequence property of amenable equivalence relations yields non-trivial sets A_n with μ(A_n △ sA_n)→0 for all s in [[G]]. For the Higman–Th

What would settle it

Search for a topologically principal, amenable, ample groupoid G and a quasi-invariant probability measure μ on G^(0) such that the canonical action of [[G]] is strongly ergodic; the theorem forbids any such example. Concretely, one can test the Higman–Thompson group V_{3,3} on the Cantor set: prove that some almost invariant sequence of measurable sets is non-trivial (as the theorem requires) or find a contradiction showing every such sequence is trivial.

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Extended reading notes

Core claim

The central claim is that the canonical action [[G]]↷G^(0) of the topological full group of a topologically principal, amenable, ample groupoid is never strongly ergodic, regardless of the quasi-invariant probability measure on the unit space. The proof's engine is the transfer of amenability from the etale groupoid G to the associated orbit equivalence relation R_G = {(r(g), s(g)) : g∈G}: the fiber-wise probability measures witnessing amenability are pushed forward to equivalence classes, making R_G Borel amenable. Since [[G]] is contained in the full group of R_G, a non-trivial almost invariant sequence for R_G — obtained from the amenability of equivalence relations — is also almost invar

Load-bearing premise

The proof assumes that for every amenable etale groupoid the associated orbit equivalence relation is Borel amenable; if that transfer step (Lemma 3.1) were to fail for some topologically principal, ample groupoid, the existence of the non-trivial almost invariant sequence would no longer be guaranteed and the main theorem would collapse.

Editorial extensions

If this is right

  • The canonical actions of the Thompson group V, the Higman–Thompson groups V_{d,k}, and the Brin–Thompson groups are all non-strongly-ergodic, so their group-measure-space von Neumann algebras are not full factors.
  • For any topologically principal, amenable, ample groupoid G, no subgroup of [[G]] can realize a strongly ergodic action on G^(0); in particular, no free-group Bernoulli shift can be embedded as a restricted action.
  • The crossed products L∞(X_{d,k}) ⋊ V_{d,k} are non-amenable, non-full factors of type III_{1/d}; for the classical Thompson group V, the algebra L∞(X_2) ⋊ V is a non-amenable, non-full type III_{1/2} factor.
  • Since strong ergodicity is impossible, fullness of these crossed products cannot be obtained from central-sequence arguments based on the action; any fullness, if it exists, must come from a different mechanism.

Reading between the lines

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

  • Because the proof is non-constructive, no explicit almost invariant sequence is given even for the Thompson group V; constructing one on the tail equivalence relation might reveal whether the sequence can be chosen to be clopen, giving a combinatorial obstruction visible in the Cantor set.
  • The non-embedding corollary suggests a general rigidity principle: topological full groups of amenable groupoids are poorly suited to hosting strongly ergodic actions of their subgroups; one could test whether this persists for actions that are only asymptotically strongly ergodic in a weaker sense.
  • The non-amenable, non-full type III_{1/d} factors exhibited here could serve as concrete test cases for questions about central sequences and property Gamma in type III settings, where few explicit examples are known.
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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 proves a general theorem: for a topologically principal, amenable, ample groupoid G, the canonical action of its topological full group [[G]] on the unit space is claimed to be not strongly ergodic with respect to any quasi-invariant probability measure. The proof combines Lemma 3.1 (groupoid amenability implies amenability of the associated orbit equivalence relation R_G) with Schmidt's Rokhlin lemma to produce a non-trivial almost invariant sequence. As applications, the paper shows that the canonical actions of the Thompson, Higman–Thompson, and Brin–Thompson groups are not strongly ergodic, their associated crossed products are not full, and derives a non-embedding corollary for strongly ergodic actions. It then specialises to the Higman–Thompson group V_{d,k} and computes the crossed product to be a non-amenable, non-full factor of type III_{1/d}.

Significance. If the universal statement were correct, the paper would give a clean structural result: no topological full group of an amenable ample groupoid can admit a strongly ergodic quasi-invariant action on its unit space, and the accompanying non-embedding and non-fullness corollaries are natural and useful. The proof strategy is transparent and mostly standard: Lemma 3.1 gives a direct transfer from groupoid amenability to Borel amenability of R_G, and the subsequent ergodicity and type III computations are classical modulo minor typos. The paper is honest about the non-constructive nature of the almost invariant sequence. However, the central theorem as printed is false: a finite two-point groupoid is a counterexample. The fix is likely local and the Thompson-group applications survive, but the overstatement is load-bearing in the current version.

major comments (2)
  1. [Section 3, proof of Theorem 3.2] Theorem 3.2 is false as stated. Let G be the finite groupoid with unit space {a,b}, identity arrows id_a,id_b, and one arrow α:b→a with inverse α^{-1}. This groupoid is locally compact, Hausdorff, étale, ample, and topologically principal. It is amenable: for example, take m^a=δ_α and m^b=δ_{id_b}; then α·m^b=m^a and α^{-1}·m^a=m^b, so the amenability net in §2.1 is exactly invariant. The topological full group [[G]] is {id,σ}, where σ swaps a and b, and the uniform probability μ on {a,b} is quasi-invariant. The action σ↷({a,b},μ) is strongly ergodic: any almost invariant sequence is eventually σ-invariant, hence trivial. This directly contradicts the theorem. The missing hypothesis is an aperiodicity assumption (e.g., G^(0) has no isolated points, or every G-orbit is infinite, or μ gives zero mass to finite G-orbits). Please add such a hypothesis; the Thompson/Higman–Thompson groupoids
  2. [Section 3, proof of Theorem 3.2] The proof invokes [19, Proposition 2.2] to obtain, from μ-amenability of R_G, a non-trivial almost invariant sequence for the full group [R_G]. This is the exact step that fails for the finite groupoid counterexample described above. The citation's hypotheses must be checked explicitly: an amenable equivalence relation with finite classes does not, in general, admit such a sequence. Please either add the missing aperiodicity hypothesis to Theorem 3.2 and verify it before applying the Rokhlin lemma, or prove the existence of the almost invariant sequence directly under the stated hypotheses. Without this, the universal statement is unsupported even apart from the explicit counterexample.
minor comments (3)
  1. [Lemma 3.1] In the final display of the proof, the norm should be ∥g m_i^{s(g)} − m_i^{r(g)}∥_1, not ∥g m_i^{s(g)} − m_i^{s(g)}∥_1 as printed. The preceding line makes the intended estimate clear, but it is a typo in a load-bearing inequality.
  2. [Section 2.2] The notation for R_x and R^x is confusing and likely misprinted: the text defines both R_x and R^y as {(y,x)∈R}. In Lemma 3.1, p_i^x is used as a measure on the range fibre, so the definition of R^x should be clarified (presumably {(x,y)∈R}). This would improve readability.
  3. [Section 3, type III computation] The sentence 'and (L∞(X_{d,k}) ⋊ V_{d,k})^{σ^φ} is a factor' is stated without proof. Since this fact is needed to apply [6, Corollary 3.2.7] for the S-invariant computation, a brief justification (or a reference to a standard argument) would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is non-circular and anchored in external results.

full rationale

The derivation chain is non-circular. Lemma 3.1 takes the defining almost-invariant probability measures on the amenable étale groupoid G and explicitly constructs measures on the associated equivalence relation R_G, checking invariance directly from the groupoid amenability estimate; this is a translation of the external definition, not an assumption of the target result. Theorem 3.2 then applies Schmidt's Rokhlin lemma ([19, Proposition 2.2]) to the amenable equivalence relation to obtain a non-trivial asymptotically invariant sequence, and uses the inclusion [[G]]<[R_G] to transfer it to the topological full group action. The non-fullness conclusion follows by the standard centralizing-sequence argument, while non-amenability and the type III classification are obtained from separate external criteria ([4], [9], [20], [6]); there are no fitted parameters, no self-citations, and no step in which an equation is used as its own input. The paper's remark that the proof is non-constructive is a transparency note, not a circularity admission. Whether Theorem 3.2 requires an aperiodicity hypothesis to avoid the finite-groupoid counterexample is a correctness question about the applicability of an external lemma, not evidence that the conclusion is presupposed.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests entirely on standard background results and theorems from the literature: groupoid amenability, Rokhlin's lemma for amenable equivalence relations, the Douglas–Nowak/Vaes–Wahl amenability criterion, Kesten's characterization of non-amenable groups, and Connes' S-invariant theory. No free parameters are introduced and no new entities are postulated.

assumptions (6)
  • standard math Amenable étale groupoid ⇒ associated equivalence relation R_G is Borel amenable (Lemma 3.1)
    Proved in the paper; depends on the definition of groupoid amenability (Anantharaman-Delaroche–Renault) and the uniform-on-compact convergence of the net (m_i^x).
  • standard math Rokhlin's lemma for µ-amenable equivalence relations (Schmidt [19, Prop 2.2])
    Produces a non-trivial asymptotically invariant sequence for the full group [R_G]; this is the engine of Theorem 3.2.
  • standard math Amenable étale groupoid iff reduced groupoid C*-algebra is nuclear (Brown–Ozawa [4, Thm 5.6.18])
    Used to verify amenability of G_{d,k} and G_2^m via nuclearity of M_k(C)⊗O_d and O_2^{⊗m}.
  • standard math Douglas–Nowak / Vaes–Wahl criterion (Theorem 3.4): if ∃ finite F with Σ_{s∈F}∫√ω dµ > ∥Σ_{s∈F}λ_s∥, then the action is non-amenable in Zimmer's sense
    Used in §3 to prove non-amenability of V_{d,k}↷X_{d,k}; cited to [9, Thm 7] and [20, Prop 5.3].
  • standard math Higman–Thompson group V_{d,d} is non-amenable (contains non-abelian free subgroups), hence ∃ finite F with ∥Σ_{s∈F}λ_s∥ < |F| (Kesten criterion)
    Needed to choose n in the non-amenability argument.
  • standard math Connes' S-invariant computation for irreducible inclusions into the centralizer (Connes [6, Cor 3.2.7])
    Yields type III_{1/d} for L∞(X_{d,k})⋊V_{d,k}.

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Pith. "Pith review of Non-strong ergodicity of canonical actions of the Thompson groups." pith.science (2026). https://pith.science/paper/VHTXKSZQ

@misc{pith2026251013309,
  author       = {Pith},
  title        = {Pith review of: Non-strong ergodicity of canonical actions of the Thompson groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHTXKSZQ}},
  note         = {Machine review of arXiv:2510.13309}
}
read the original abstract

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.

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