On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

Itamar Vigdorovich; Tal Cohen

arxiv: 2603.09640 · v3 · pith:35EJ3GQZnew · submitted 2026-03-10 · 🧮 math.GR

On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

Tal Cohen , Itamar Vigdorovich This is my paper

Pith reviewed 2026-05-15 13:14 UTC · model grok-4.3

classification 🧮 math.GR

keywords Lie groupstopological generating setsirredundant setsalgebraic groupsNielsen transformationsfinite simple groupsWiegold conjecture

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The pith

A topologically generating set in a connected compact Lie group must be redundant if its size exceeds a polynomial in the group's rank.

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

The paper shows that any topologically generating set S in a connected compact Lie group G whose size grows faster than a fixed polynomial in the rank of G must contain a proper subset that still topologically generates G. The same conclusion holds for amenable Lie groups and for reductive algebraic groups under the Zariski topology. The size bounds obtained this way are determined by the corresponding polynomial bounds that hold for finite simple groups of Lie type. The work further examines redundancy after Nielsen transformations and proves that several conjectures posed by Gelander are logical consequences of the Wiegold conjecture.

Core claim

We show that a topologically generating set S of a connected compact Lie group G of size larger than a fixed polynomial in the rank of G must be redundant (i.e., some proper subset of S still topologically generates G). Similar results are obtained for amenable Lie groups and for reductive algebraic groups with the Zariski topology. The quantitative bounds produced by our method are controlled by corresponding bounds for finite simple groups of Lie type. We also treat redundancy up to Nielsen transformations, thereby partially answering a few conjectures of Gelander. We show that these conjectures are implied by the Wiegold conjecture.

What carries the argument

Reduction of the topological generation question for Lie groups to polynomial bounds on generating sets of finite simple groups of Lie type, via suitable approximations or quotients.

Load-bearing premise

The method assumes that sufficiently strong polynomial bounds on the size of generating sets already exist or can be proved for finite simple groups of Lie type.

What would settle it

A connected compact Lie group of rank r that admits an irredundant topologically generating set whose size exceeds every polynomial function of r would disprove the main claim.

read the original abstract

We show the following dichotomy for a connected Lie group $G$: If $G$ is amenable, then any topologically generating set $X\subset G$ of size larger than a fixed polynomial in the dimension of $G$ must be redundant (i.e., a proper subset of $X$ still generates $G$). If $G$ is non-amenable, then it admits arbitrarily large topologically generating sets that are irredundant, and remain irredundant even after applying Nielsen transformations. The polynomial bound for amenable groups is obtained by reduction to finite simple groups of Lie type via strong approximation. This partially answers two conjectures by Gelander on generation in compact Lie groups and simple algebraic groups, and moreover shows that these conjectures are implied by the Wiegold conjecture. The construction of large Nielsen irredundant generating sets in non-amenable groups is done by extending Minsky's work to higher rank Lie groups, exhibiting dense representations in the domain of discontinuity of the $\mathrm{Out}(F_{n})$-action on the character variety.

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.

Desk Editor's Note private letter to a colleague

The paper reduces the Lie-group irredundant generation question to finite simple groups of Lie type, with the polynomial bound still conditional on that case.

read the letter

The key point is that this paper reduces bounding the size of irredundant topologically generating sets in connected compact Lie groups to the corresponding problem for finite simple groups of Lie type. They obtain similar polynomial-type bounds for amenable Lie groups and reductive algebraic groups under the Zariski topology. They also address redundancy with respect to Nielsen transformations and show that this implies some of Gelander's conjectures follow from the Wiegold conjecture. The reduction and the Nielsen-equivalence part are the genuinely new elements. The abstract indicates that the method organizes a previously qualitative question by supplying explicit polynomial bounds controlled by the finite case. That reduction step looks like a solid contribution on its own, and linking the conjectures to Wiegold gives a clear path forward even if the finite bounds are not yet settled. The main limitation is the conditional nature of the bound. The quantitative result for Lie groups inherits whatever status the finite simple groups of Lie type bounds have. The abstract does not claim to prove a polynomial bound for those finite groups, so the Lie group statement remains dependent on progress there. The topological details of the approximation or congruence arguments would need close inspection in the manuscript to confirm they go through without extra constants or losses. This work is aimed at researchers in geometric group theory and the study of generating sets in Lie and algebraic groups. Readers following Gelander's conjectures or interested in profinite and topological generation would get the most out of it. The citation pattern seems appropriate, drawing on finite-group literature without overclaiming. I would recommend sending it to peer review. The reduction is worth referee scrutiny, and the partial resolution of the conjectures adds value even with the dependence on the finite case.

Referee Report

2 major / 1 minor

Summary. The paper establishes that any topologically generating set S of a connected compact Lie group G whose cardinality exceeds a fixed polynomial in the rank of G must be redundant (some proper subset still topologically generates G). Analogous statements are proved for amenable Lie groups and for reductive algebraic groups equipped with the Zariski topology. The quantitative bounds obtained are controlled by corresponding (currently unknown) polynomial bounds on the maximal size of irredundant generating sets in finite simple groups of Lie type. The authors also study redundancy up to Nielsen transformations and show that several conjectures of Gelander are implied by the Wiegold conjecture.

Significance. If the reduction to the finite case is valid, the work supplies the first explicit polynomial control on the maximal size of irredundant topological generating sets in compact Lie groups, thereby giving a uniform quantitative answer to questions about generation in infinite groups by reducing them to finite-group problems. The link to the Wiegold conjecture and the partial resolution of Gelander’s conjectures are additional contributions. The result is conditional on the existence of polynomial bounds for finite simple groups of Lie type; until those bounds are established or cited, the quantitative statement remains formally open.

major comments (2)

[reduction to finite groups / main theorem] The central quantitative claim (a fixed polynomial bound in the rank) is stated to be controlled by polynomial bounds for finite simple groups of Lie type, yet the manuscript supplies neither a proof nor a reference establishing that such polynomial bounds exist for those finite groups. This renders the explicit polynomial degree for the Lie-group statement conditional on an unresolved question (see the reduction argument and the paragraph following the statement of the main theorem).
[Nielsen transformations section] The argument that Gelander’s conjectures on Nielsen redundancy follow from the Wiegold conjecture is presented as a corollary, but the reduction step does not explicitly verify that the topological-generation property is preserved under the relevant Nielsen moves in the infinite setting; a short additional paragraph confirming this preservation is needed.

minor comments (1)

[Introduction] The notation for the rank function and for the polynomial degree is introduced without a dedicated symbol table; adding a short notation paragraph would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and describe the planned revisions.

read point-by-point responses

Referee: [reduction to finite groups / main theorem] The central quantitative claim (a fixed polynomial bound in the rank) is stated to be controlled by polynomial bounds for finite simple groups of Lie type, yet the manuscript supplies neither a proof nor a reference establishing that such polynomial bounds exist for those finite groups. This renders the explicit polynomial degree for the Lie-group statement conditional on an unresolved question (see the reduction argument and the paragraph following the statement of the main theorem).

Authors: The central contribution is the reduction from the Lie-group setting to the finite simple groups of Lie type; the quantitative bound for compact Lie groups is explicitly described as being controlled by the (presently unknown) corresponding bound in the finite case. We do not claim an unconditional explicit degree. To make this conditional character fully transparent, we will revise the statement of the main theorem and the subsequent paragraph to state explicitly that the polynomial degree depends on the finite-group bounds. revision: yes
Referee: [Nielsen transformations section] The argument that Gelander’s conjectures on Nielsen redundancy follow from the Wiegold conjecture is presented as a corollary, but the reduction step does not explicitly verify that the topological-generation property is preserved under the relevant Nielsen moves in the infinite setting; a short additional paragraph confirming this preservation is needed.

Authors: We agree that an explicit verification improves clarity. We will add a short paragraph in the Nielsen transformations section confirming that topological generation is preserved under the relevant Nielsen moves in the infinite (topological) setting, using the continuity of the group operations and the fact that the closed subgroup generated by the set is unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds explicitly conditional on external finite-group results

full rationale

The manuscript states that its quantitative polynomial bounds for Lie groups are 'controlled by corresponding bounds for finite simple groups of Lie type' and that certain conjectures are implied by the (external) Wiegold conjecture. No equation or step in the derivation reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation whose content is itself unverified within the paper. The reduction to finite groups is presented as a transfer of the quantitative question rather than a resolution of it, leaving the central claim dependent on independent external progress. This is a standard conditional reduction and satisfies the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts about topological generation in Lie groups and on the existence of polynomial generating-set bounds for finite simple groups of Lie type; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Standard properties of topological generation and connectedness in compact Lie groups
Invoked to equate topological generation with generation of the identity component.
domain assumption Existence of polynomial bounds on the size of irredundant generating sets for finite simple groups of Lie type
The paper states that its quantitative bounds are controlled by these finite-group bounds.

pith-pipeline@v0.9.0 · 5403 in / 1385 out tokens · 50143 ms · 2026-05-15T13:14:25.768622+00:00 · methodology

discussion (0)

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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Theorem 1.3: m(G) ≤ a·dim(G)^b for amenable Lie G; m_C(G) ≤ a·rank(G)^b for reductive algebraic G. Bounds controlled by m(G_p(F_p)) for finite simple groups of Lie type.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Reduction via congruence images and strong approximation (Prop 2.1, Lemma 2.4) to finite simple groups.

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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.