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arxiv: 2409.09987 · v4 · submitted 2024-09-16 · 🧮 math.GR

Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups

Milana Golich , Antonio L\'opez Neumann , Mark Pengitore This is my paper

Pith reviewed 2026-05-23 21:14 UTC · model grok-4.3

classification 🧮 math.GR
keywords rational cohomologyZariski dense subgroupssolvable linear algebraic groupsLie algebra cohomologyrestriction mapsQ-defined groupsfinitely generated solvable groups
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The pith

For irreducible solvable Q-defined linear algebraic groups, the rational cohomology rings of the group and its Zariski dense subgroups are isomorphic when the subgroups intersect the Q-split maximal torus discretely.

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

The paper establishes that the cohomology ring of the Q-defined Lie algebra g_Q with coefficients in a finite dimensional rational G-module M is isomorphic to the cohomology ring of certain Zariski dense subgroups Gamma of G(Q). It further shows that the natural restriction map in rational cohomology from the algebraic group G to such a subgroup Gamma is injective. This identification links the cohomology of the algebraic group to that of its discrete subgroups and yields consequences for the cohomology representations of finitely generated solvable groups of finite abelian rank.

Core claim

For an irreducible solvable Q-defined linear algebraic group G there is an isomorphism between the cohomology rings with coefficients in a finite dimensional rational G-module M of the associated Q-defined Lie algebra g_Q and of Zariski dense subgroups Gamma less than or equal to G(Q) that intersect the Q-split maximal torus discretely. The restriction map in rational cohomology from G to Gamma is an injection.

What carries the argument

The restriction map in rational cohomology from G to Gamma, which induces an isomorphism with the Lie algebra cohomology of g_Q.

If this is right

  • The restriction map from G to Gamma is injective in rational cohomology with coefficients in M.
  • The identification produces results on the cohomology representations of finitely generated solvable groups of finite abelian rank.
  • Cohomology computations for such discrete subgroups reduce to computations on the associated Lie algebra.
  • The isomorphism preserves the ring structure of the cohomology.
  • Results extend to several classes of representations on the cohomology of these groups.

Where Pith is reading between the lines

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

  • Lie algebra cohomology methods become available for computing the rational cohomology of certain discrete solvable groups.
  • The result may connect to questions about when cohomology detects Zariski density or discreteness conditions in other algebraic groups.
  • It opens the possibility of transferring finiteness or generation properties from the Lie algebra side to the discrete group side.

Load-bearing premise

The algebraic group G must be irreducible and solvable and defined over Q, while the subgroups Gamma must be Zariski dense in G(Q) and intersect the Q-split maximal torus discretely.

What would settle it

An explicit counterexample of an irreducible solvable Q-defined G, a finite dimensional rational G-module M, and a qualifying Zariski dense Gamma where the cohomology rings of g_Q and Gamma with coefficients in M differ or the restriction map fails to be injective.

read the original abstract

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational $\mathbf{G}$-module $M$ of the associated $\mathbb{Q}$-defined Lie algebra $\mathfrak{g_\mathbb{Q}}$ and Zariski dense subgroups $\Gamma \leq \mathbf{G}(\mathbb{Q})$ that satisfy the condition that they intersect the $\mathbb{Q}$-split maximal torus discretely. We further prove that the restriction map in rational cohomology from $\mathbf{G}$ to a Zariski dense subgroup $\Gamma \leq \mathbf{G}(\mathbb{Q})$ with coefficients in $M$ is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.

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 / 2 minor

Summary. The manuscript claims that for an irreducible solvable ℚ-defined linear algebraic group G, the rational cohomology rings H^*(𝔤_ℚ, M) and H^*(Γ, M) are isomorphic when Γ ≤ G(ℚ) is Zariski dense and intersects the ℚ-split maximal torus discretely, for any finite-dimensional rational G-module M. It further asserts that the restriction map H^*(G(ℚ), M) → H^*(Γ, M) is injective, and derives consequences for the cohomology of finitely generated solvable groups of finite abelian rank.

Significance. If the isomorphism and injectivity hold, the work supplies a concrete bridge between Lie-algebra cohomology and discrete-group cohomology in the solvable setting by reducing via the discrete-torus condition to a virtually unipotent situation where the exponential map identifies the two theories. This could be useful for computing cohomology rings of arithmetic subgroups of solvable groups and for studying their representations.

minor comments (2)
  1. The abstract states the main theorems but does not indicate the key technical steps (e.g., how the discrete-intersection hypothesis is used to invoke the BCH correspondence); a one-sentence outline of the reduction would improve readability.
  2. Notation for the Lie algebra 𝔤_ℚ and the group G(ℚ) is introduced without an explicit reference to the standard definitions of rational points and Q-defined structures; adding a short preliminary paragraph on these conventions would help readers outside algebraic groups.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript, including the recommendation for minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point response at this stage. We will incorporate any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an isomorphism between rational cohomology of the Lie algebra g_Q and that of Zariski-dense subgroups Γ of an irreducible solvable Q-defined algebraic group G, under the explicit hypothesis that Γ intersects the Q-split maximal torus discretely, plus injectivity of the restriction map from G(Q) to Γ. These results are derived from standard external theorems on algebraic groups, Lie algebra cohomology, and solvable group representations; the discrete-intersection condition is stated as an input hypothesis rather than derived internally. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The central claims remain independent of the paper's own inputs and rest on externally verifiable mathematical structures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Claims rest on standard background in algebraic groups, Lie algebras, and rational cohomology; no free parameters, new entities, or ad-hoc constructions appear in the abstract.

axioms (2)
  • standard math Standard definitions and functoriality properties of rational cohomology for algebraic groups and their Lie algebras.
    The isomorphism and restriction statements presuppose the usual cohomology theory in this category.
  • domain assumption Zariski density and discrete intersection with the Q-split maximal torus are well-defined and compatible with the module action.
    These conditions are invoked to obtain the isomorphism.

pith-pipeline@v0.9.0 · 5682 in / 1206 out tokens · 30045 ms · 2026-05-23T21:14:15.305910+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups.Topology, 43(4):903–924, 2004

    Oliver Baues. Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups.Topology, 43(4):903–924, 2004

    work page 2004

  2. [2]

    Automorphism groups of polycyclic-by-finite groups and arithmetic groups.Publ

    Oliver Baues and Fritz Grunewald. Automorphism groups of polycyclic-by-finite groups and arithmetic groups.Publ. Math. Inst. Hautes ´Etudes Sci., (104):213–268, 2006

    work page 2006

  3. [3]

    Springer-Verlag, New York, second edition, 1991

    Armand Borel.Linear algebraic groups, volume 126 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991

    work page 1991

  4. [4]

    Commensurability of Baumslag-Solitar groups.Indiana Univ

    Montserrat Casals-Ruiz, Ilya Kazachkov, and Alexander Zakharov. Commensurability of Baumslag-Solitar groups.Indiana Univ. Math. J., 70(6):2527–2555, 2021

    work page 2021

  5. [5]

    Automorphism groups of solvable groups of finite abelian ranks.arXiv preprint arXiv:2506.07991, 2025

    Jonas Dere and Mark Pengitore. Automorphism groups of solvable groups of finite abelian ranks.arXiv preprint arXiv:2506.07991, 2025

    work page arXiv 2025

  6. [6]

    Cambridge University Press, Cambridge, 2002

    Allen Hatcher.Algebraic topology. Cambridge University Press, Cambridge, 2002

    work page 2002

  7. [7]

    Hochschild

    G. Hochschild. Cohomology of algebraic linear groups.Illinois J. Math., 5:492–519, 1961

    work page 1961

  8. [8]

    Hochschild and J.-P

    G. Hochschild and J.-P. Serre. Cohomology of Lie algebras.Ann. of Math. (2), 57:591–603, 1953

    work page 1953

  9. [9]

    Central theorems for cohomologies of certain solvable groups.Transactions of the American Mathematical Society, 369(4):2879–2896, 2017

    Hisashi Kasuya. Central theorems for cohomologies of certain solvable groups.Transactions of the American Mathematical Society, 369(4):2879–2896, 2017

    work page 2017

  10. [10]

    Extended simplicial rational Nomizu’s theorem and Sullivan’s minimal models for non-nilpotent groups.Geom

    Hisashi Kasuya. Extended simplicial rational Nomizu’s theorem and Sullivan’s minimal models for non-nilpotent groups.Geom. Dedicata, 216(3):Paper No. 30, 9, 2022

    work page 2022

  11. [11]

    Paul J. Kunkel. Rational cohomology of algebraic solvable groups. InProceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), volume 44, pages 251–268, 1987

    work page 1985

  12. [12]

    On everywhere dense imbedding of free groups in Lie groups.Nagoya Math

    Masatake Kuranishi. On everywhere dense imbedding of free groups in Lie groups.Nagoya Math. J., 2:63–71, 1951

    work page 1951

  13. [13]

    Lambe and Stewart B

    Larry A. Lambe and Stewart B. Priddy. Cohomology of nilmanifolds and torsion-free, nilpotent groups.Trans. Amer. Math. Soc., 273(1):39–55, 1982

    work page 1982

  14. [14]

    Lennox and Derek J

    John C. Lennox and Derek J. S. Robinson.The theory of infinite soluble groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004

    work page 2004

  15. [15]

    G. D. Mostow. Cohomology of topological groups and solvmanifolds.Ann. of Math. (2), 73:20–48, 1961

    work page 1961

  16. [16]

    M. S. Raghunathan. Discrete subgroups of Lie groups.Math. Student, (Special Centenary V olume):59–70 (2008), 2007

    work page 2008

  17. [17]

    Cambridge University Press, Cambridge, 1983

    Daniel Segal.Polycyclic groups, volume 82 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1983

    work page 1983

  18. [18]

    T. A. Springer.Linear algebraic groups. Modern Birkh ¨auser Classics. Birkh¨auser Boston, Inc., Boston, MA, second edition, 2009

    work page 2009

  19. [19]

    Weibel.An introduction to homological algebra, volume 38 ofCambridge Studies in Advanced Mathematics

    Charles A. Weibel.An introduction to homological algebra, volume 38 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994. RATIONAL COHOMOLOGY AND ZARISKI DENSE SUBGROUPS OF SOLV ABLE LINEAR ALGEBRAIC GROUPS 21 DEPARTMENT OFMATHEMATICS, PURDUEUNIVERSITY, WESTLAFAYETTE, IN 47907 Email address:mgolich@purdue.edu DEPARTMENT...

    work page 1994