Simplicity of Special Automorphism Groups of Affine Space

Definitions and mathematical statement

Let $k$ denote the base field intended by the source, and let $\mathbb A_k^n$ be affine $n$-space over $k$. The supplied record does not specify $k$, its characteristic, or whether it is algebraically closed; these hypotheses remain unresolved. Write $\operatorname{Aut}_k(\mathbb A_k^n)$ for the group of polynomial automorphisms over $k$. For $f=(f_1,\ldots,f_n)$, its Jacobian determinant is a nonzero constant, giving a homomorphism \[ \operatorname{jac}\colon \operatorname{Aut}_k(\mathbb A_k^n)\longrightarrow k^\times, \qquad f\longmapsto \det\!\left(\frac{\partial f_i}{\partial x_j}\right)_{1\leq i,j\leq n}. \] In the source's notation, define \[ \operatorname{SAut}(\mathbb A_k^n):=\ker(\operatorname{jac}). \] A group $G$ is simple as an abstract group if $G\neq\{1\}$ and every normal subgroup $N\trianglelefteq G$ satisfies $N=\{1\}$ or $N=G$, without any restriction that $N$ be closed in a topology.

The supplied question is: for each integer $n\geq3$, is $\operatorname{SAut}(\mathbb A_k^n)$ simple as an abstract group? This formulation leaves the intended base-field hypotheses unresolved and does not assert that the question is meant uniformly over all fields.

The supplied excerpt attributes the failure of abstract simplicity in dimension $2$ to Danilov in 1974, with reference label [Dan74]. The supplied literature assessment also attributes topological simplicity in all dimensions to Blanc and describes the higher-dimensional abstract simplicity question as unresolved. These are supplied contextual claims, not independently verified conclusions; the topology and hypotheses of the reported topological result are not specified in the record.

Short English statement

For each $n\geq3$, ask whether the group of polynomial automorphisms of affine $n$-space having Jacobian determinant $1$ has any nontrivial proper normal subgroup. The intended base field and its hypotheses are missing. Abstract simplicity is the target; the supplied topological simplicity claim concerns a different condition.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006254$, source problem code catalogue entry, title "Simplicity of Special Automorphism Groups of Affine Space." The record assigns proposed year 2025 and names no proposer.
• Supplied report citation: Algebraic Groups (2025), Oberwolfach Reports, pp. 941--1010, DOI 10.4171/owr/2025/20: https://doi.org/10.4171/owr/2025/20. No problem-specific page is supplied.
• Supplied literature source, labeled "Blanc result as reported": https://doi.org/10.4171/OWR/2025/20. No separate Blanc publication or theorem citation is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem "partially solved," reports that $\operatorname{SAut}(\mathbb A^2)$ is not abstractly simple, attributes topological simplicity in all dimensions to Blanc, and describes higher-dimensional abstract simplicity as unresolved. This is supplied dated curation derived from the Oberwolfach report, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The original excerpt attributes the dimension-two result to Danilov (1974), using [Dan74], without providing its bibliographic entry. Interleaved surrounding exposition also contains [LRU23] and [RvS25], without complete bibliographic entries; these are not supplied as references for the simplicity question.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated, and notation repaired. This is a supplied source-verification claim, not verification performed for this draft.