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Topological full groups and stable rank one

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Pith's one-line read Reduced C*-algebras of certain C*-simple topological full groups have stable rank one.

desk verdict Solid, fully written proof that Kerr–Tucker-Drob C*-simple groups have reduced C*-algebras of stable rank one, by combining Følner towers with Ozawa selflessness. read the letter →

arxiv 2607.04300 v1 pith:VAEUQV3B submitted 2026-07-05 math.OA math.GR

David Kerr , Spyridon Petrakos This is my paper · ORCID
classification math.OAmath.GR MSC 46L0537B0520F65
keywords stablerankonetopologicalfullgroupsdynamicalalternatingreducedgroupC*-algebrasspecificationridgeselflessnessFølnertowersC*-simplicity
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 proves that the reduced group C*-algebras of a large family of C*-simple topological full groups and dynamical alternating groups have stable rank one. These groups arise as full groups of minimal free subshifts of amenable groups that carry a localized specification property called a specification ridge. Stable rank one means invertible elements are dense, a basic regularity property that is automatic for factors of type II or III but often hard to check for non-nuclear simple C*-algebras. The authors combine Følner towers (a type II_1 tool) that manufacture approximately invariant partitions inside a Bernoulli structure with embeddings of free products that produce selflessness (a type III tool) and therefore strict comparison, allowing them to rotate near zero-divisors into nilpotents. The result supplies the first stable-rank-one verification for this class of non-nuclear simple group C*-algebras and shows that non-amenability, when arranged geometrically enough, still permits the kind of projection transport needed for invertibility density.

What carries the argument

Specification ridge (Definition 4.1): a localized full-shift condition along a cyclic subgroup that lets one embed free products S_d * Z_2 into the full group; these free products, taken in direct product, yield selflessness by Ozawa’s criteria and therefore strict comparison, which supplies the missing Murray–von Neumann subequivalences inside the Bernoulli structure built from striated Følner towers.

What would settle it

Exhibit a single C*-simple topological full group or dynamical alternating group arising from a minimal free subshift of an amenable group that does not have a specification ridge and whose reduced C*-algebra has stable rank greater than one, or show that every such group without a ridge still has stable rank one.

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Extended reading notes

Core claim

For any torsion-free countable infinite amenable group Γ acting as a minimal topologically free right subshift on the Cantor set with a specification ridge, and for any subgroup G of the topological full group that contains the dynamical alternating group, the reduced group C*-algebra C*_λ(G) has stable rank one.

Load-bearing premise

The underlying actions must possess a specification ridge; without it the free-product embeddings that produce selflessness and C*-simplicity both fail.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves that the reduced group C*-algebras of the C*-simple topological full groups and dynamical alternating groups constructed by Kerr–Tucker-Drob have stable rank one (Theorem A). The groups arise from minimal topologically free right subshift actions of torsion-free countable infinite amenable groups on the Cantor set that possess a “specification ridge.” The argument combines Følner-tower constructions (striated clopen towers, permutational Bernoulli structures, approximate invariance, combinatorial near-partitions) that produce a near-block-diagonal form of a zero-divisor with Ozawa’s selflessness results applied to embedded direct products of free products S_d * Z_2, which supply the Murray–von Neumann subequivalences needed for unitary rotation to a nilpotent element.

Significance. Stable rank one for simple non-nuclear reduced group C*-algebras has been established only for limited classes (free products, acylindrically hyperbolic groups). The present work adds a large new family of finitely generated simple groups of dynamical origin whose reduced C*-algebras have unique trace and stable rank one. The hybrid strategy—type II_1 Følner towers for approximate invariance and type III free-product selflessness for comparison—is novel and may apply more broadly. The specification ridge is an explicit, checkable hypothesis that cleanly delimits the scope; the modular lemmas (embedding free products, zero-division, approximate invariance, combinatorial partition) are written with explicit estimates and can be reused.

minor comments (4)
  1. In the statement of Theorem A the integer q is required to be greater than 3, while Definition 4.1 and Lemma 5.1 only need q≥4 for the free-product embeddings; a brief clarifying sentence would remove any ambiguity.
  2. Section 6 introduces the map ζ and the set F_K without an explicit display of the relation K^2 h_ω ⊆ F; a short parenthetical reminder would help the reader track the support conditions used later in Notation 6.2 and Lemma 7.1.
  3. In the matrix illustration of Section 10 the first row/column is labelled R while the remaining blocks are indexed by V_{i,j}; adding a one-line legend that the zero columns arise from the annihilation a_1 1_O = 0 would make the picture self-contained.
  4. The reference list contains several arXiv preprints (Ozawa, Bell–Geffen–Kerr, etc.); once published versions appear they should be updated, but this is routine.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; prior constructions and Ozawa selflessness used as independent black-box inputs whose statements do not encode the target stable-rank conclusion.

full rationale

The paper is a pure existence/proof paper in operator algebras. Theorem A asserts stable rank one for C*_λ(G) under an explicit hypothesis (specification ridge, Def. 4.1). The derivation proceeds by constructing striated Følner towers (Sec. 6), producing zero-divisors via a Bernoulli structure (Lem. 7.1), manufacturing approximately invariant near-partitions (Lems. 8.1–8.2, 9.1), embedding free products S_d * Z_2 via the ridge (Lem. 5.1), invoking Ozawa’s external selflessness theorem for the resulting direct product to obtain strict comparison, and finally rotating a near-block-diagonal form to a nilpotent element. All steps are constructive and internal once the ridge and the black-box inputs (Kerr–Tucker-Drob construction of the groups, Ozawa selflessness) are granted. Those inputs do not contain the stable-rank claim; they supply groups with C*-simplicity and free-product subgroups. No parameter is fitted to data, no equation reduces the conclusion to a quantity defined in terms of itself, and self-citations ([22], [24]) are used only for background properties (Gamma/McDuff, existence of the actions) that are not the target. The ridge is a transparent scope condition, not a circular definition. Hence circularity is negligible.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper is pure mathematics. No numerical free parameters are fitted. The only non-standard ingredients are the specification-ridge hypothesis (introduced to guarantee free-product embeddings and C*-simplicity) and the background theorems of Ozawa and of Kerr–Tucker-Drob that are invoked as black boxes.

assumptions (4)
  • domain assumption Ozawa’s theorem that nonelementary free products (and their direct products) are selfless, hence have strict comparison
    Invoked in §10 to obtain Murray–von Neumann subequivalences inside C*_λ((A_S * Z_2)^F).
  • domain assumption Existence of tightly nested Følner tiling sequences with the three listed properties for nontorsion amenable groups
    Used in §4 to construct the subshift actions that possess a specification ridge.
  • domain assumption C*-simplicity of subgroups containing A(Γ,X) when the action has a specification ridge and q≥4 (Le Boudec–Matte Bon + free-group embeddings)
    Cited in Remark 4.2; needed so that the reduced group C*-algebra is simple and has unique trace.
  • standard math Standard facts on topological full groups, dynamical alternating groups, and the left regular representation
    Background material collected in §§2–3.
invented entities (2)
  • specification ridge (Definition 4.1)
    purpose: Localised specification property along a cyclic subgroup that permits free-product embeddings and guarantees C*-simplicity
    Abstracted from the concrete constructions of Kerr–Tucker-Drob; the entire free-product and selflessness step rests on it.
  • striated clopen tower (Definition 6.1)
    purpose: Organises a permutational Bernoulli structure inside the group algebra so that Følner-type near-invariance can be realised
    Technical device introduced for the zero-division and approximate-invariance arguments of §§7–8.

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Pith. "Pith review of Topological full groups and stable rank one." pith.science (2026). https://pith.science/paper/VAEUQV3B

@misc{pith2026260704300,
  author       = {Pith},
  title        = {Pith review of: Topological full groups and stable rank one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VAEUQV3B}},
  note         = {Machine review of arXiv:2607.04300}
}
abstract

We establish stable rank one for the reduced group C$^*$-algebras of the C$^*$-simple topological full groups and dynamical alternating groups constructed by Kerr and Tucker-Drob. The proof relies on both type II$_1$ and type III phenomena, in the first case via the use of F{\o}lner towers and in the second via Ozawa's recent results on selflessness as applied to direct products of free products.

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  1. Nonnuclear Bunce-Deddens algebras

    math.OA 2026-07 accept novelty 6.0 of 10

    Free odometer crossed products by nonamenable residually finite groups yield simple nonnuclear C*-algebras that frequently have real rank zero, stable rank one, unique trace, and selflessness, with explicit K-theory o...

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