Topological simplicity of the group of automorphisms of the affine plane

J\'er\'Remy Blanc

REVIEW 1 major objections 7 minor 16 references

Topological simplicity of the group of automorphisms of the affine plane

T0 review · 1 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that SAut_k(A^2) is simple as an ind-group over any infinite field, and that closed normal subgroups in higher dimensions contain all tame automorphisms.

desk verdict The main result is likely true and the strategy is mostly sound, but Proposition 4.2 has a genuine unhandled case (affine non-translations) that needs a small repair. read the letter →

arxiv 2411.17143 v1 pith:QIMIQLAR submitted 2024-11-26 math.AG

JérŔemy Blanc This is my paper · ORCID

classification math.AG MSC 14R1014R20

keywords automorphismgroupsofaffinespaceind-groupssimplegrouptameautomorphismsclosednormalsubgroupspolynomialNagatafinitefields

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 proves that the group SAut_k($A^{2}$) of polynomial automorphisms of the affine plane with Jacobian 1 is simple as an infinite-dimensional algebraic group, for every infinite field k: its only closed normal subgroups are the trivial group and the whole group. In higher dimensions, the same argument shows that every nontrivial closed normal subgroup of SAut_k(A^n) contains all tame automorphisms. This resolves a question that had been open since a 1981 proof by Shafarevich was found to rely on a false Lie correspondence. The paper also shows the behavior is completely different over finite fields, where SAut_k($A^{2}$) admits surjective homomorphisms onto the additive group of k[x].

What carries the argument

The load-bearing construction is a family of commutators h = $ρ^{{-1}}$ g(t^a) ρ built from a non-translation f and suitable translations, where replacing t by a nonzero element of k gives elements of the form ρ($τ^{{-1}}$ $f^{{-1}}$ τ f)$ρ^{{-1}}$ that lie in the closed normal subgroup N by normality. Because the map t ↦ h(t) is continuous for the ind-group topology on SAut_k(A^n), the preimage of N is closed; it contains k^*, so by the specialization property it contains 0, and h(0) is a nontrivial translation. This forces N to contain all translations, and Proposition C (due to Lewis) then forces N to contain SL_n(k) and all tame automorphisms. The ind-group structure, in which automorphisms of bounded degree form finite-dimensional affine varieties, is what makes the specialization property available.

What would settle it

Find an infinite field k and a proper nontrivial closed normal subgroup of SAut_k($A^{2}$). Equivalently, find an element f of SAut_{k[t]}(A^n) and a closed normal subgroup N such that the specialized family h(t_0) belongs to N for every t_0 ∈ k^* but h(0) does not; such an example would show the specialization property fails and would block Theorem A.

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Extended reading notes

Core claim

Blanc establishes Theorem A: for any infinite field k and any integer n ≥ 1, if N is a closed normal subgroup of SAut_k(A^n), then either N = {id} or N contains every tame automorphism. For n = 2, since every automorphism is tame, this makes SAut_k($A^{2}$) simple as an ind-group. The proof first shows that any nontrivial closed normal subgroup contains a nontrivial translation, then uses a result of Lewis to lift translations to SL_n(k) and then to all tame automorphisms. The finite-field case is treated separately, yielding surjective homomorphisms to the additive group of k[x] whose kernels contain all translations.

Load-bearing premise

The proof rests on the topological specialization property: if a closed normal subgroup contains h(t) for every nonzero t in k, then it contains the limit h(0) obtained by setting t = 0. If a topology on SAut_k(A^n) fails this property, the nontrivial translation that drives the whole argument could fall outside the subgroup.

Editorial extensions

If this is right

Any nontrivial closed normal subgroup of SAut_k(A^2) is the whole group, over any infinite field k.
For n ≥ 3, every nontrivial closed normal subgroup of SAut_k(A^n) contains all tame automorphisms; in particular, by Example 3.8, it contains the Nagata automorphism.
If N is a closed normal subgroup of Aut_k(A^n) with Jacobian image equal to k^*, then N = Aut_k(A^n) (Proposition E).
Over finite fields, SAut_k(A^2) is not perfect and is not finitely generated, and the normal subgroup generated by affine automorphisms is proper (Proposition D).
The argument isolates a purely topological condition, so the same conclusion holds for any topology on SAut_k(A^n) satisfying the specialization property, such as the two standard ind-topologies or, for local fields, the topology induced by the field.

Reading between the lines

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

The limited role of the topology suggests that the same specialization axiom could be a useful test for simplicity of other infinite-dimensional automorphism groups, such as ind-subgroups of the Cremona group.
Because the proof shows that the closure of commutator conjugates of any non-translation contains a translation, similar rigidity may hold for closed normal subgroups of other groups of polynomial automorphisms, for instance those preserving a symplectic form.
Over finite fields, the non-perfectness of SAut_k(A^2) comes from a quotient onto k[x]/V; a natural question is whether the kernels of these quotients are closed in an ind-topology adapted to finite fields, though the usual topology degenerates to discrete.
The method likely extends to automorphism groups of A^n over rings or semi-local bases, as long as the specialization property and the centralizer lemma hold.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main take: this is a serious paper with a real result, but the proof as written has a genuine gap that needs patching. The affine non-translation case in Proposition 4.3 is not covered by Proposition 4.2.

What's new: Theorem A resolves a long-open question. The earlier proof by Shafarevich used a false Lie correspondence, and Furter–Kraft's result gives closed normal subgroups of Aut(A^n) rather than SAut. The paper's one-parameter limit construction (Props 4.2–4.3) and the closedness argument (Prop B) are new and mostly clean. Reproving Lewis's generation results makes the paper self-contained; the finite-field section is a nice bonus with concrete quotient maps to (k[x],+). No free parameters, no circularity, and the citation pattern looks appropriate.

The soft spot: Proposition 4.2 chooses a,b with a/b = max(m/j). If all nonzero q_{i,j,m} have m=0, no such coprime positive integers exist. That case is not exotic—it happens exactly for affine non-translation f in Prop 4.3: e.g. f=(x1+x2,x2), tau=(x1,x2+t) gives g=(x1-t,x2). For such g, h_epsilon(0)=id for all epsilon, so Prop 4.2(2) fails and Prop 4.3 has no valid proof for affine f. The gap is repairable: if f is affine non-translation, take a translation sigma not commuting with f; then sigma^{-1} f^{-1} sigma f is a nontrivial translation in N. But the manuscript doesn't do this, so the proof of Prop B is incomplete as written. The topological specialization property is also stated rather than proved, and some ind-topology facts are used without proof; those are standard and likely fine.

Bottom line: the main theorem is probably correct, and the fix is small. For people in affine algebraic geometry and ind-groups, this is worth reading and deserves a serious referee. I would send it to review and expect acceptance after minor revision.

Referee Report

1 major / 7 minor

Summary. The paper proves Theorem A: for an infinite field k and n≥1, every closed normal subgroup N of SAut_k(A^n) is either trivial or contains all tame automorphisms of SAut_k(A^n); in particular SAut_k(A^2) is simple as an ind-group. The proof strategy is to show (Proposition B) that a nontrivial closed normal subgroup contains all translations, and then to invoke and reproduce Lewis's theorem (Proposition C) that translations generate all tame automorphisms. The translation step is the new ingredient: using the one-parameter family g = τ^{-1} f^{-1} τ f with τ a k[t]-translation, a limit argument produces a non-trivial translation in the closure of N. The paper also treats finite fields (Proposition D) and gives a result for Aut_k(A^n) (Proposition E).

Significance. If the proof is completed, Theorem A resolves a long-standing open problem on the simplicity of SAut_k(A^2) as an ind-group and provides the first general statement for higher-dimensional special automorphism groups. The paper is largely self-contained: it reproduces the relevant results of Lewis, gives explicit computations in Sections 3 and 4, and cleanly isolates the topological specialization property needed for the limit argument. The finite-field results and the Aut_k(A^n) proposition are valuable complements. However, the proof of Proposition B currently has a gap (see Major Comment 1), so the central claim is not yet established as written.

major comments (1)

[§4.1, Proposition 4.2] Proposition 4.2 is false as stated. If all non-zero q_{i,j,m} have m=0, the maximum of m/j is 0, so no coprime integers a,b ≥ 1 exist with a/b = max{m/j}; moreover, for any a,b ≥ 1, the specialization h_ε(0) is the identity for every ε. The example g = (x_1 - t, x_2, ..., x_n) illustrates this: h_ε = (x_1 - t^a, x_2, ..., x_n), so h_ε(0) = id. This case occurs in Proposition 4.3 when f is affine but not a translation, since then g = τ^{-1} f^{-1} τ f is a k[t]-translation; for f = (x_1+x_2, x_2) and τ = (x_1, x_2+t), g = (x_1 - t, x_2). Consequently the proof of Proposition 4.3, and hence of Proposition B, does not cover affine non-translation f. The gap is repairable: if f is affine and does not commute with a translation σ, then σ^{-1} f^{-1} σ f is a non-trivial translation in N. The manuscript should either prove Proposition 4.2 under a hypothesis that excludes this case (and then handle the affine case separately) or adjust the statement.

minor comments (7)

[§2] The word 'inclustion' should be 'inclusion'.
[Introduction, Proposition D] The phrase 'There exists moreover a a surjective group homomorphism' contains a duplicated article 'a'.
[§3.3, proof of Proposition 3.6] In the definition of h_{q,ǫ}, the expression 'q(x2, . . . , qn)' should read 'q(x2, . . . , xn)'.
[§3.3, proof of Proposition 3.6] In the characteristic-2 case for n=2, the translation is written as '(x1 + θµν(θ^2 + µ^2), x2, )' with a stray comma; please clarify the intended composition.
[§4.1, Proposition 4.2] The notation 'g(ta)' should be 'g(t^a)' with a superscript, to avoid confusion with the product 't a'.
[§4.2, Lemma 4.1] In the induction step, 'the homogeneous part of degree i of ∑...' should be 'degree j', not 'degree i'.
[Introduction and §4.1, Proposition B] The topological specialization property (if h(t0) ∈ N for all t0 ∈ k*, then h(0) ∈ N for a closed normal N) is used essentially in Proposition B but is asserted rather than proved or referenced for the two ind-topologies; please add a proof or a precise reference showing that k* is dense in the relevant topology on A^1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A rests on explicit constructions and reproduced external normal-subgroup results, not on its own conclusion.

full rationale

The derivation chain is self-contained. Proposition 4.2 explicitly constructs h from g and epsilon; the assertion that h(0) is a translation is computed from the summands with m/j = a/b, not from the desired conclusion. Proposition 4.3 produces a family by conjugation, and Proposition B uses the stated topological specialization property (an assumption about the ind-topology, not a disguised version of Theorem A) to force h(0) into N. Proposition C, the main external input, is reproduced in Section 3.3 with full proofs (Propositions 3.6 and 3.7), so the citation to Lewis is not an unverified load-bearing self-citation. The references to [Bla16] and [FK18] are contextual, and the paper explicitly rejects the Shafarevich Lie-algebra route, so no uniqueness theorem is imported from the author's prior work. There are no fitted parameters, no renamed known result, and no quantity defined in terms of the target claim. A separate possible gap for affine non-translation f in Proposition 4.2 is a case-analysis defect, not a circular reduction, and does not affect the circularity verdict.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof introduces no free parameters and no new entities. The claims rest on the infinite-field hypothesis, classical structure theorems for Aut_k(A^2) (Jung-van der Kulk), the topological specialization property of the ind-group topology, and elementary linear algebra (SL_n transitivity). Lemma 4.1 and other auxiliary lemmas are proved within the paper.

assumptions (5)

domain assumption The field k is infinite (in Theorem A, Propositions B, 3.6, 3.7, and E).
The centralizer lemma (Lemma 3.2) and the diagonal-scaling argument in Proposition 3.7 fail for finite fields; Proposition D provides explicit quotients over finite fields showing the main statement is false there.
standard math Jung-van der Kulk theorem: Tame_k(A^2) = Aut_k(A^2) for every field k.
Used in the introduction and in the n=2 conclusion of Theorem A, where 'contains all tame automorphisms' becomes 'equals SAut_k(A^2)'. Cited as [Jun42].
domain assumption The ind-group topology satisfies the specialization property: for f ∈ SAut_{k[t]}(A^n) and closed N, f(t0) ∈ N for all t0 ∈ k^* implies f(0) ∈ N.
Stated after Theorem A in the introduction and used in the proof of Proposition B: the preimage of N under the continuous map t0 ↦ h(t0) is closed and contains A^1 \ {0}, hence contains 0. This is the load-bearing topological premise.
standard math SL_n(k) acts transitively on nonzero vectors, hence on nonzero translations by conjugation, for n ≥ 2.
Used in Proposition B to conclude that one nontrivial translation inside N implies all translations are inside N.
standard math SAut_k(A^2) is the amalgamated product of its triangular subgroup and its affine subgroup over their intersection.
Used in Section 3.4 (proof of Proposition 3.10) to define the finite-field quotient homomorphisms ρ_T and ρ_A and check agreement on the intersection. This is the Jung-van der Kulk decomposition.

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Pith. "Pith review of Topological simplicity of the group of automorphisms of the affine plane." pith.science (2026). https://pith.science/paper/QIMIQLAR

@misc{pith2026241117143,
  author       = {Pith},
  title        = {Pith review of: Topological simplicity of the group of automorphisms of the affine plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIMIQLAR}},
  note         = {Machine review of arXiv:2411.17143}
}

abstract

We prove that the group $\mathrm{SAut}_{\mathrm{k}}(\mathbb{A}^2)$ is simple as an algebraic group of infinite dimension, over any infinite field $\mathrm{k}$, by proving that any closed normal subgroup is either trivial or the whole group. In higher dimension, we show that closed normal subgroups contain all tame automorphisms. The case of finite fields, very different, is also discussed.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Conjugacy classes of special automorphisms of the affine spaces

J \'e r \'e my Blanc. Conjugacy classes of special automorphisms of the affine spaces. Algebra Number Theory , 10(5):939--967, 2016

work page 2016

[2] [2]

V. I. Danilov. Nonsimplicity of the group of unimodular automorphisms of the affine plane. Math. Notes , 15:165--167, 1974

work page 1974

[3] [3]

Closed subgroups of the polynomial automorphism group containing the affine subgroup

Eric Edo. Closed subgroups of the polynomial automorphism group containing the affine subgroup. Transform. Groups , 23(1):71--74, 2018

work page 2018

[4] [4]

On the geometry of the automorphism groups of affine varieties, 2018

Jean-Philippe Furter and Hanspeter Kraft. On the geometry of the automorphism groups of affine varieties, 2018

work page 2018

[5] [5]

Normal subgroup generated by a plane polynomial automorphism

Jean-Philippe Furter and St \'e phane Lamy. Normal subgroup generated by a plane polynomial automorphism. Transform. Groups , 15(3):577--610, 2010

work page 2010

[6] [6]

H. E. W. Jung. \"U ber ganze birationale Transformationen der Ebene . J. Reine Angew. Math. , 184:161--174, 1942

work page 1942

[7] [7]

Kambayashi

T. Kambayashi. Pro-affine algebras, ind-affine groups and the Jacobian problem. Appendix : Linear compactness for rings and modules. J. Algebra , 185(2):481--498, 1996

work page 1996

[8] [8]

Kac- Moody groups, their flag varieties and representation theory , volume 204 of Prog

Shrawan Kumar. Kac- Moody groups, their flag varieties and representation theory , volume 204 of Prog. Math. Boston: Birkh \"a user, 2002

work page 2002