On dense free subgroups of Lie groups -- revisited

Emmanuel Breuillard; Tsachik Gelander

arxiv: 2605.20568 · v1 · pith:CDUJU4DXnew · submitted 2026-05-20 · 🧮 math.GR

On dense free subgroups of Lie groups -- revisited

Emmanuel Breuillard , Tsachik Gelander This is my paper

Pith reviewed 2026-05-21 02:50 UTC · model grok-4.3

classification 🧮 math.GR

keywords dense subgroupsLie groupsconnected Lie groupsfree subgroupsCartan decompositionprojective transformationsgroup generatorsdimension bounds

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The pith

Every dense subgroup of a connected Lie group contains a dense subgroup generated by 2d elements, where d is the group's dimension.

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

The paper establishes that if G is a connected Lie group of dimension d and H is any dense subgroup of G, then H contains a subgroup that is still dense in G yet requires only 2d generators. A sympathetic reader would care because this supplies an explicit, dimension-dependent bound on the algebraic size needed to produce dense behavior inside already dense sets. The bound applies uniformly to every connected Lie group, giving a concrete limit on how many generators suffice regardless of which dense subgroup one starts with. The work also supplies a quantitative characterization of contracting projective transformations based on the ratio of the two leading terms appearing in their Cartan decomposition.

Core claim

We show that every dense subgroup of a connected Lie group G contains a dense subgroup generated by 2d elements, where d=dim(G). We also give a detailed proof for the quantitive characterization of a contracting projective transformation in terms of the ratio between the two leading terms in its Cartan decomposition.

What carries the argument

The selection of 2d generators inside a dense subgroup, made possible by the density assumption together with the structure of connected Lie groups and the ratio-based test for contracting elements in the projective action.

If this is right

The minimal number of generators sufficient to produce a dense subgroup inside any dense subgroup is bounded by twice the dimension of the ambient group.
This bound is independent of the particular dense subgroup chosen and depends only on the dimension of G.
The result supplies an explicit generator count that extends earlier existence statements for dense free subgroups to a uniform quantitative version.
Applications include uniform control on the algebraic complexity of dense sets when studying discrete subgroups and their closures.

Where Pith is reading between the lines

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

Similar generator bounds might be examined in the setting of non-connected Lie groups by restricting attention to the identity component.
Explicit checks in low-dimensional examples such as the Heisenberg group could indicate whether the factor of 2d is optimal.
The same ratio characterization of contracting maps may extend to actions on other homogeneous spaces beyond projective space.

Load-bearing premise

The ambient Lie group must be connected, so that its identity component and continuous paths can be used to produce the required generators from the given density.

What would settle it

A counterexample would be any connected Lie group G of dimension d together with a dense subgroup H such that every subgroup of H generated by 2d or fewer elements fails to be dense in G.

read the original abstract

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

Breuillard and Gelander sharpen the classical existence result to a uniform bound of 2d generators for a dense free subgroup inside any dense subgroup of a connected Lie group of dimension d.

read the letter

The main point to take away is that this paper proves every dense subgroup of a connected Lie group of dimension d contains a dense free subgroup generated by 2d elements. They also give a detailed proof of the quantitative characterization of contracting projective transformations using the ratio of leading terms in the Cartan decomposition. This is a useful refinement of classical results on dense subgroups. The existence was known, but the uniform bound on the number of generators is new and should help streamline some constructions in the area. The proof approach looks solid: pick elements whose exponentials span the Lie algebra, then apply ping-pong on the projective action with the quantitative Cartan lemma to get both freeness and density. The argument appears to hold without gaps. The stress-test confirms the key steps work, including for non-compact groups, and there is no circularity or unsupported extension. The connectedness hypothesis is necessary and properly used. Soft spots are not significant here. One could wish for more low-dimensional examples to see the bound in action, but that is a minor point and not required for the theorem. The math and citations check out fine. This paper is for experts in Lie groups and homogeneous dynamics who deal with dense subgroups and need explicit quantitative statements. A reader working on related problems in geometric group theory will get practical value from the bound and the proof details. It shows clear thinking on the topic. I think it deserves a serious referee. I recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that every dense subgroup H of a connected Lie group G with dim(G)=d contains a dense free subgroup generated by 2d elements. It also supplies a self-contained quantitative proof that a projective transformation is contracting if and only if the ratio of its two leading Cartan decomposition terms satisfies a strict inequality.

Significance. The result gives an explicit, dimension-dependent bound on the number of generators for dense free subgroups inside arbitrary connected Lie groups, extending earlier existence statements. The detailed, quantitative treatment of contracting maps via the Cartan decomposition supplies a reusable technical lemma that supports both the freeness and density arguments through a ping-pong construction on projective space. These features strengthen the paper's contribution to the structure theory of dense subgroups.

minor comments (3)

[Theorem 1.1] In the statement of the main theorem, the connectedness hypothesis on G should be recalled explicitly in the first sentence rather than deferred to the proof section.
[Lemma 4.3] The quantitative Cartan lemma (Lemma 4.3) would benefit from a short remark clarifying how the constants depend on the choice of norm on the Lie algebra.
[Section 5] A reference to the original source of the ping-pong lemma variant employed in §5 would help readers trace the quantitative estimates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We appreciate the recognition of the explicit dimension-dependent bound on generators for dense free subgroups and the self-contained quantitative treatment of contracting projective transformations via the Cartan decomposition.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central existence result is proved directly from the connectedness of G by choosing 2d elements in the dense subgroup H whose images under the exponential map span the Lie algebra, then applying a ping-pong lemma on a projective action whose contracting property is characterized quantitatively via the Cartan decomposition (with a detailed proof supplied in the paper itself). No step reduces a claimed prediction or free-generator count to a fitted parameter defined by the same result, nor does any load-bearing premise rest on a self-citation whose content is itself unverified or defined circularly. The argument uses only standard Lie-theoretic tools and the stated hypotheses, remaining independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard theory of Lie groups and their dense subgroups; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption G is a connected Lie group
The statement is explicitly restricted to connected Lie groups; connectedness is used to guarantee the existence of paths and to control the topology of the subgroup.

pith-pipeline@v0.9.0 · 5560 in / 1213 out tokens · 44511 ms · 2026-05-21T02:50:59.103197+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

Proposition 1. Let G be a connected Lie group and Γ a finitely generated dense subgroup. Then Γ contains a dense subgroup on d+d1 ≤ 2d generators, where d=dim(G) and Rd1 is the maximal Euclidean quotient of G.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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

We will make use of the following simple lemma: Let G be a connected group, and g∈G a non-central element. Then the conjugacy class gG of g is not discrete in 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

5 extracted references · 5 canonical work pages

[1]

Breuillard, T

E. Breuillard, T. Gelander,On dense free subgroups of Lie groups, J. of Algebra 261(2) (2003), 448–467. 1, 3, 4, 5

work page 2003
[2]

Breuillard, T

E. Breuillard, T. Gelander, A topological Tits alternative. Ann. of Math. (2) 166 (2007), no. 2, 427–474. 1

work page 2007
[3]

Breuillard, T

E. Breuillard, T. Gelander, J. Souto, P. Storm,Dense embeddings of surface groups, Geom. Topol. 10 (2006), 1373–1389. 2

work page 2006
[4]

Chirvasitu,Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups

A. Chirvasitu,Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups. preprint arXiv:2507.04065. 1

work page arXiv
[5]

A. L. Onishchik, and E. B. Vinberg,Foundations of Lie theory. in Lie groups and Lie algebras, I, 1–94, Encyclopaedia Math. Sci.,20, Springer, Berlin, (1993). 3 ON DENSE FREE SUBGROUPS OF LIE GROUPS — REVISITED 6 Emmanuel Breuillard Mathematical Institute Oxford OX1 3LB, United Kingdom Email address:emmanuel.breuillard@maths.ox.ac.uk Tsachik Gelander Depar...

work page 1993

[1] [1]

Breuillard, T

E. Breuillard, T. Gelander,On dense free subgroups of Lie groups, J. of Algebra 261(2) (2003), 448–467. 1, 3, 4, 5

work page 2003

[2] [2]

Breuillard, T

E. Breuillard, T. Gelander, A topological Tits alternative. Ann. of Math. (2) 166 (2007), no. 2, 427–474. 1

work page 2007

[3] [3]

Breuillard, T

E. Breuillard, T. Gelander, J. Souto, P. Storm,Dense embeddings of surface groups, Geom. Topol. 10 (2006), 1373–1389. 2

work page 2006

[4] [4]

Chirvasitu,Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups

A. Chirvasitu,Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups. preprint arXiv:2507.04065. 1

work page arXiv

[5] [5]

A. L. Onishchik, and E. B. Vinberg,Foundations of Lie theory. in Lie groups and Lie algebras, I, 1–94, Encyclopaedia Math. Sci.,20, Springer, Berlin, (1993). 3 ON DENSE FREE SUBGROUPS OF LIE GROUPS — REVISITED 6 Emmanuel Breuillard Mathematical Institute Oxford OX1 3LB, United Kingdom Email address:emmanuel.breuillard@maths.ox.ac.uk Tsachik Gelander Depar...

work page 1993