Kourovka Notebook Problem 21.63

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $F$ be a field of characteristic $p>0$, $n\geq1$, and $\mathfrak m=(x_1,\ldots,x_n)\subset F[x_1,\ldots,x_n]$. Let \[ \Gamma=\{\gamma\in\operatorname{Aut}_F(F[x_1,\ldots,x_n]):\gamma(x_i)-x_i\in\mathfrak m^2\ (1\leq i\leq n)\}, \] where automorphisms fix $F$ pointwise. The source notes that $\Gamma$ is residually $p$: its finite $p$-group quotients separate its elements. Its pro-$p$ completion $\widehat\Gamma_p$ is the inverse limit of all finite $p$-group quotients. A pro-$p$ identity is a nonidentity element $w$ of a finitely generated free pro-$p$ group such that every continuous substitution homomorphism into $\widehat\Gamma_p$ sends $w$ to $1$. A free pro-$p$ group is the pro-$p$ completion of a free group, with its universal property for maps into pro-$p$ groups.

Statement recorded in the supplied source.

For every such $F$ and $n$, does $\Gamma$ satisfy a pro-$p$ identity? Explicitly, is there some $d\geq1$ and $1\ne w\in\widehat{F_d}_p$ such that $w(g_1,\ldots,g_d)=1$ in $\widehat\Gamma_p$ for every $g_1,\ldots,g_d\in\widehat\Gamma_p$? The word is allowed to depend on $F$ and $n$. [S2]

Short English statement

Do polynomial automorphisms tangent to the identity over a positive-characteristic field satisfy a nontrivial identity in their pro-p completion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2572 (KOU-21.63). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. Zelmanov and E. I. Khukhro, Problem 21.63, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). https://arxiv.org/abs/1401.0300v48.