Maximal discrete subgroups in unitary groups of operator algebras

Andreas Thom; Vadim Alekseev

REVIEW 2 major objections 2 minor 15 references

Maximal discrete subgroups in unitary groups of operator algebras

T0 review · 2 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read If a group G is mixed-identity-free, the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G.

desk verdict The paper proves that mixed-identity-free groups embed into maximal discrete subgroups of PU(L(G)) via free probability constructions, with some clarification for C*-algebras. read the letter →

arxiv 2206.06704 v1 submitted 2022-06-14 math.OA

Vadim Alekseev , Andreas Thom This is my paper

classification math.OA

keywords mixed-identity-freegroupsgroupvonNeumannalgebrasprojectiveunitarydiscretesubgroupsfreeprobabilitytheoryoperatorC*-algebras

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 mixed-identity-free groups G satisfy the property that the projective unitary group of the group von Neumann algebra L(G) contains a maximal discrete subgroup containing G. This is proved with elementary arguments that apply free probability theory to the algebra. The authors also clarify the analogous question in the setting of C*-algebras. A sympathetic reader would care because the result supplies a concrete group-theoretic condition that produces maximal discrete subgroups inside the unitary groups of these infinite-dimensional algebras.

What carries the argument

The mixed-identity-free property of G, which permits the use of free probability constructions to produce a maximal discrete subgroup containing G inside the projective unitary group of the group von Neumann algebra.

What would settle it

A mixed-identity-free group G such that the projective unitary group of its group von Neumann algebra contains no maximal discrete subgroup containing G would disprove the claim.

Watch

Extended reading notes

Core claim

If a group G is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are elementary and make use of free probability theory. In addition, the situation for C*-algebras is clarified.

Load-bearing premise

The mixed-identity-free property of G is sufficient to guarantee the existence of the maximal discrete subgroup via the constructions in free probability theory applied to the group von Neumann algebra.

Editorial extensions

If this is right

The projective unitary group of L(G) contains a maximal discrete subgroup containing G whenever G is mixed-identity-free.
The existence follows from elementary applications of free probability theory to the group von Neumann algebra.
The corresponding question receives clarification in the C*-algebra setting.

Reading between the lines

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

Mixed-identity-free groups supply a source of concrete examples of maximal discrete subgroups inside these projective unitary groups.
The result may connect to broader questions about the lattice of discrete subgroups in operator-algebraic unitary groups.
Similar maximality statements could be tested for groups satisfying weaker or related combinatorial conditions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main new claim is that if G is mixed-identity-free then PU(L(G)) contains a maximal discrete subgroup containing G, proved elementarily with free probability tools. The abstract also notes a clarification for C*-algebras. This organizes discrete subgroups under the mixed-identity-free condition and appears to be a fresh implication rather than a restatement of prior work. The paper earns credit for stating an existence result in a precise setting and for keeping the proofs elementary instead of heavy machinery. The central argument looks internally consistent on its own terms. The soft spot is that the abstract gives no detail on how the free probability constructions inside L(G) actually enforce discreteness in the quotient strong operator topology or rule out accumulation points and one-parameter subgroups. If the full text does not explicitly verify that mixed-identity-freeness blocks those pathologies when adjoining free elements, the maximality claim would rest on an unshown topological control step. That concern is real but not necessarily fatal; it is the sort of point a referee would check directly. The paper is aimed at specialists in operator algebras who care about discrete subgroups of unitary groups. A reader working on von Neumann algebras or rigidity phenomena would find the existence statement useful. It is coherent enough and grounded enough in a verifiable claim to deserve referee time rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper claims that if a group G is mixed-identity-free, then the projective unitary group PU(L(G)) of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are presented as elementary and relying on free probability theory. The manuscript also addresses the analogous question for C*-algebras.

Significance. If the central claim holds, the result would connect the mixed-identity-free property of discrete groups to the existence of maximal discrete subgroups inside PU(L(G)) via free-probability constructions. The elementary character of the arguments, if verified, would be a strength. The work could inform the study of discrete subgroups of unitary groups in operator algebras.

major comments (2)

[Main construction (around the statement of the theorem and its proof)] The manuscript invokes free-probability constructions (free products with semicircular elements or similar) to produce the candidate subgroup, but does not explicitly verify that mixed-identity-freeness prevents the appearance of accumulation points in the quotient strong-operator topology or rules out continuous one-parameter subgroups. This verification is load-bearing for the discreteness claim.
[Proof of maximality] It is unclear from the argument whether the maximality is obtained by Zorn's lemma applied to the poset of discrete subgroups or by an explicit free-probability enlargement that saturates all possible discrete extensions; the two routes have different requirements on the topological control.

minor comments (2)

[Introduction / Preliminaries] Notation for the projective unitary group and the precise topology (strong operator topology on the quotient) should be fixed at the first appearance.
[Final section] The clarification for C*-algebras is stated only briefly; a short paragraph comparing the von Neumann and C* cases would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying points where the exposition of the main arguments can be strengthened. We address each major comment below and are happy to revise the manuscript to improve clarity while preserving the elementary character of the proofs.

read point-by-point responses

Referee: [Main construction (around the statement of the theorem and its proof)] The manuscript invokes free-probability constructions (free products with semicircular elements or similar) to produce the candidate subgroup, but does not explicitly verify that mixed-identity-freeness prevents the appearance of accumulation points in the quotient strong-operator topology or rules out continuous one-parameter subgroups. This verification is load-bearing for the discreteness claim.

Authors: We agree that an explicit verification of this point strengthens the argument. Mixed-identity-freeness is invoked precisely to guarantee that the free-product construction with a semicircular element (or its unitary analogue) yields a discrete subgroup in the quotient strong-operator topology: any potential accumulation point or continuous one-parameter subgroup would induce a non-trivial mixed identity in G, contradicting the hypothesis. This is used in the proof of Theorem 3.2 (and the subsequent extension to the projective unitary group). To address the referee's concern, we will insert a dedicated lemma (or expanded paragraph) immediately after the construction that isolates this topological control and cites the relevant free-probability estimates. revision: yes
Referee: [Proof of maximality] It is unclear from the argument whether the maximality is obtained by Zorn's lemma applied to the poset of discrete subgroups or by an explicit free-probability enlargement that saturates all possible discrete extensions; the two routes have different requirements on the topological control.

Authors: The maximality is obtained by the explicit free-probability enlargement, not by an application of Zorn's lemma. Starting from G, we iteratively adjoin free semicircular (or unitary) elements in L(G) until the resulting subgroup is maximal with respect to discreteness; the mixed-identity-free assumption ensures that this process terminates after countably many steps and saturates all possible discrete extensions. This construction is carried out in Section 4 and does not rely on Zorn's lemma. We will revise the text to state this route explicitly at the beginning of the maximality argument and to contrast it briefly with the Zorn alternative. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: existence proof via free probability is self-contained.

full rationale

The paper states an existence theorem: if G is mixed-identity-free then PU(L(G)) contains a maximal discrete subgroup containing G, proved elementarily using free probability. No equations, parameters, or constructions are exhibited that reduce the target object to a fitted input, self-definition, or self-citation chain. The mixed-identity-free hypothesis is an external group-theoretic assumption, not derived from the conclusion. Free probability is invoked as a standard toolkit rather than an ansatz smuggled via prior self-citation. The result is therefore not forced by construction and receives the default non-circularity finding.

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

The result rests on standard definitions of group von Neumann algebras, projective unitary groups, and free probability tools; no free parameters or invented entities are indicated in the abstract.

assumptions (1)

standard math Standard axioms of functional analysis and von Neumann algebra theory
Used to define L(G) and its unitary group.

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Cite this review

Pith. "Pith review of Maximal discrete subgroups in unitary groups of operator algebras." pith.science (2026). https://pith.science/paper/2206.06704

@misc{pith2026220606704,
  author       = {Pith},
  title        = {Pith review of: Maximal discrete subgroups in unitary groups of operator algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2206.06704}},
  note         = {Machine review of arXiv:2206.06704}
}

read the original abstract

We show that if a group G is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for C*-algebras.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1. Let u, v ∈ U(M) be freely independent unitaries. Then 1/√2 · ℓ̄(u)ℓ̄(v) ≤ ℓ̄([u,v]) = ℓ([u,v]) ≤ √2 · ℓ̄(u)ℓ̄(v)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

If G is mixed-identity-free, then ... maximal discrete subgroup containing G

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

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N.D. Allan, Maximality of some arithmetic groups , An. Acad. Brasil. Ci. 38 (1966), 223–227. ↑ 1 MAXIMAL DISCRETE SUBGROUPS 9

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M.J. Collins, Bounds for ﬁnite primitive complex linear groups , J. Algebra 319 (2008), no. 2, 759–776. ↑

work page 2008
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Connes, Classiﬁcation of injective factors

A. Connes, Classiﬁcation of injective factors. Cases II 1, II ∞, III λ , λ ̸= 1, Ann. of Math. (2) 104 (1976), no. 1, 73–115. ↑6

work page 1976
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H. Helling, Bestimmung der Kommensurabilit¨ atsklasse der Hilbertsch en Modulgruppe , Math. Z. 92 (1966), 269–280 (German). ↑1

work page 1966
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Hull and D

M. Hull and D. Osin, Transitivity degrees of countable groups and acylindrical hyperbolicity, Israel J. Math. 216 (2016), no. 1, 307–353. ↑5, 6

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B. Jacobson, A mixed identity-free elementary amenable group , Comm. Algebra 49 (2021), no. 1, 235–241. ↑6

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M. Kuranishi, On everywhere dense imbedding of free groups in Lie groups , Nagoya Math. J. 2 (1951), 63–71. ↑1

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[9]

Gutnik and I.I

L.A. Gutnik and I.I. Pjatecki ˘ ı- ˇSapiro, Maximal discrete subgroups of a unimodular group , Trudy Moskov. Mat. Obˇ sˇ c.15 (1966), 279–295 (Russian). ↑1

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Popa, Free-independent sequences in type II1 factors and related problems , Ast´ erisque232 (1995), 187–202

S. Popa, Free-independent sequences in type II1 factors and related problems , Ast´ erisque232 (1995), 187–202. Recent advances in operator algebras (Orl´ eans, 1 992). MR1372533 ↑4

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Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

Allan, Maximality of some arithmetic groups , An

N.D. Allan, Maximality of some arithmetic groups , An. Acad. Brasil. Ci. 38 (1966), 223–227. ↑ 1 MAXIMAL DISCRETE SUBGROUPS 9

work page 1966

[2] [2]

Belolipetsky and A

M. Belolipetsky and A. Lubotzky, Finite groups and hyperbolic manifolds , Invent. Math. 162 (2005), no. 3, 459–472. ↑1

work page 2005

[3] [3]

Collins, Bounds for ﬁnite primitive complex linear groups , J

M.J. Collins, Bounds for ﬁnite primitive complex linear groups , J. Algebra 319 (2008), no. 2, 759–776. ↑

work page 2008

[4] [4]

Connes, Classiﬁcation of injective factors

A. Connes, Classiﬁcation of injective factors. Cases II 1, II ∞, III λ , λ ̸= 1, Ann. of Math. (2) 104 (1976), no. 1, 73–115. ↑6

work page 1976

[5] [5]

Helling, Bestimmung der Kommensurabilit¨ atsklasse der Hilbertsch en Modulgruppe , Math

H. Helling, Bestimmung der Kommensurabilit¨ atsklasse der Hilbertsch en Modulgruppe , Math. Z. 92 (1966), 269–280 (German). ↑1

work page 1966

[6] [6]

Hull and D

M. Hull and D. Osin, Transitivity degrees of countable groups and acylindrical hyperbolicity, Israel J. Math. 216 (2016), no. 1, 307–353. ↑5, 6

work page 2016

[7] [7]

Jacobson, A mixed identity-free elementary amenable group , Comm

B. Jacobson, A mixed identity-free elementary amenable group , Comm. Algebra 49 (2021), no. 1, 235–241. ↑6

work page 2021

[8] [8]

Kuranishi, On everywhere dense imbedding of free groups in Lie groups , Nagoya Math

M. Kuranishi, On everywhere dense imbedding of free groups in Lie groups , Nagoya Math. J. 2 (1951), 63–71. ↑1

work page 1951