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arxiv: 2507.18821 · v2 · pith:AJ7TAJA7new · submitted 2025-07-24 · 🧮 math.OA · math.GR

On Similarity Structure Groups and their W^* and C^*-Algebras

Eli Bashwinger , Patrick DeBonis This is my paper

Pith reviewed 2026-05-25 08:18 UTC · model grok-4.3

classification 🧮 math.OA math.GR
keywords CSS* groupssimilarity structure groupsgroup von Neumann algebrasC*-simplicityThompson groupsvon Neumann algebra primenessC*-algebrastopological full groups
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The pith

CSS* groups produce prime group von Neumann algebras and satisfy a C*-simplicity dichotomy with their commutator subgroups.

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

CSS* groups form a subclass of generalized Thompson groups that includes Higman-Thompson groups V_{d,r}, Röver-Nekrashevych groups, and topological full groups of subshifts of finite type. The paper proves that many of these groups generate prime group von Neumann algebras, extending earlier results on this property. It also shows that CSS* groups are either C*-simple with a simple commutator subgroup or lack both of these features. Many CSS* groups are non-inner amenable and properly proximal, and none are acylindrically hyperbolic.

Core claim

CSS* groups give rise to prime group von Neumann algebras, expanding the class of groups known to satisfy this property. In the process many CSS* groups are shown to be non-inner amenable and properly proximal. CSS* groups are either C*-simple with a simple commutator subgroup, or lack both properties. CSS* groups are not acylindrically hyperbolic.

What carries the argument

The CSS* subclass of countable similarity structure groups, which preserves the dynamical and algebraic features used to establish von Neumann algebra primeness and the C*-simplicity dichotomy.

If this is right

  • Many CSS* groups give rise to prime group von Neumann algebras.
  • Many CSS* groups are non-inner amenable and properly proximal.
  • CSS* groups are either C*-simple with a simple commutator subgroup or lack both properties.
  • CSS* groups are not acylindrically hyperbolic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The results add new examples of groups with prime von Neumann algebras beyond those previously identified.
  • The dichotomy may help distinguish algebraic properties across broader families of generalized Thompson groups.
  • Proof techniques independent of acylindrical hyperbolicity are needed to establish rigidity features for these groups.

Load-bearing premise

The specific definition of the CSS* subclass must guarantee that the dynamical and algebraic features needed for the primeness and dichotomy proofs are preserved across the included families.

What would settle it

A CSS* group whose group von Neumann algebra is not prime, or a CSS* group that is C*-simple but has a non-simple commutator subgroup, would falsify the claims.

Figures

Figures reproduced from arXiv: 2507.18821 by Eli Bashwinger, Patrick DeBonis.

Figure 1
Figure 1. Figure 1: The Fibonacci tree corresponds to the Golden Mean sub￾shift of finite type in Example 4.8. smaller and actually agrees with the local order preserving similarity structure that leads to Vd,r being an FSS groups from Example 4.3. Remark 4.9. In [Mat15], Matui studied the groupoids of subshifts of finite type, and it turns out the topological full group is exactly the FSS∗ groups defined in this section. Not… view at source ↗
Figure 2
Figure 2. Figure 2: The tree on whose end space the Houghton group H2 acts on, from Example 4.14. The compact ultrametric space for which the Houghton group H2 is the correspond￾ing FSS group is given in [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The tree on whose end space QAut(T2,c) acts on, from Ex￾ample 4.15. all share the same label x, so the second FSS∗ condition is always satisfied without even passing to a subball. Given that every vertex labeled by x is followed by two labeled by x and one labeled by a, we take the balls corresponding to the two subballs labeled by x to see every infinite ball contains at least two disjoint infinite balls.… view at source ↗
read the original abstract

Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups essentially introduced by Hughes. In this paper, we study CSS$^*$ groups, a subclass that includes the Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of subshifts of finite type of Matui. We prove that many CSS$^*$ groups give rise to prime group von Neumann algebras, greatly expanding the class of groups satisfying the result of the second named author, de Santiago, and Khan. In the process, we also prove that many CSS$^*$ groups are non-inner amenable and properly proximal. We then prove CSS$^*$ groups are either $C^*$-simple with a simple commutator subgroup, or lack both properties. This extends $C^*$-simplicity results of Le Boudec and Matte Bon and recovers the simple commutator subgroup results of Bleak, Elliott, and Hyde. Lastly, we observe that CSS$^*$ groups are not acylindrically hyperbolic, motivating the need to prove many of these results by other methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies Countable Similarity Structure (CSS) groups and their subclass CSS*, which includes the Higman-Thompson groups V_{d,r}, the countable Röver-Nekrashevych groups V_d(G), and the topological full groups of subshifts of finite type. It proves that many CSS* groups yield prime group von Neumann algebras (expanding results of the second author, de Santiago, and Khan), are non-inner amenable and properly proximal, and satisfy a dichotomy: they are either C*-simple with simple commutator subgroup or lack both properties (extending Le Boudec-Matte Bon and Bleak-Elliott-Hyde). It further shows that CSS* groups are not acylindrically hyperbolic.

Significance. If the claims hold, the work substantially enlarges the class of groups known to produce prime von Neumann algebras and supplies a clean dichotomy for C*-simplicity and commutator simplicity within an explicitly delineated family of generalized Thompson groups. The adaptation of dynamical and algebraic techniques to the CSS* setting, together with the explicit inclusion of concrete families such as V_{d,r} and Matui's full groups, constitutes a concrete advance.

minor comments (3)
  1. [§1] §1: The precise definition of the CSS* subclass (as a restriction of Hughes' CSS groups) is stated only informally in the introduction; a numbered definition with explicit axioms would improve readability before the main theorems.
  2. [§3] The statement that 'many CSS* groups' satisfy the primeness and dichotomy results is repeated in the abstract and introduction without a clear enumeration of which concrete families are covered; a table or explicit list in §3 or §4 would clarify the scope.
  3. [Preliminaries] Notation for the group von Neumann algebra L(G) and the reduced C*-algebra C*_r(G) is used without a preliminary reminder of the standard conventions; a short paragraph in the preliminaries would prevent ambiguity for readers outside operator algebras.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the results substantially enlarge the class of groups known to produce prime von Neumann algebras and provide a clean dichotomy for C*-simplicity within CSS* groups. We appreciate the recommendation for minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point at this stage. We will incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

Minor self-citation to co-author's prior result; not load-bearing

full rationale

The paper introduces no fitted parameters or self-definitional loops. CSS* is defined by reference to Hughes and restricted to listed families (V_{d,r}, etc.); proofs adapt dynamical arguments from external citations (Le Boudec-Matte Bon, Bleak-Elliott-Hyde). The single self-citation to the second author's earlier joint work on primeness is used only to state the expansion, not to derive the new claims by construction. No equation reduces a claimed result to a quantity defined from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms appear in the abstract; the work relies on standard definitions and background results from group theory and operator algebras.

axioms (1)
  • standard math Standard axioms and definitions of countable groups, von Neumann algebras, C*-algebras, inner amenability, proper proximality, C*-simplicity, and acylindrical hyperbolicity.
    The paper invokes these established mathematical structures without introducing new ones.

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Reference graph

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