Kourovka Notebook Problem 21.9

Definitions and mathematical statement

Fix a prime $p$. A pro-$p$ group is a topological group that is an inverse limit of finite $p$-groups, with its compact inverse-limit topology. Let $F$ be a free pro-$p$ group of finite rank $r\ge2$: it is topologically generated by $r$ free generators, with the universal mapping property for continuous maps of those generators into pro-$p$ groups. The rank restriction makes $F$ nonabelian.

Use the pro-$p$ convention that subgroups under consideration are closed and automorphisms are continuous. A subgroup $U\le F$ is open when it is open in this topology; it then has finite index. A subgroup $C\le U$ is characteristic in $U$ if every continuous automorphism of $U$ maps $C$ onto itself.

Does there exist a finite list of open subgroups $U_1,\ldots,U_m$ of $F$, including $F$ itself, such that \[ \left[C\le\bigcap_{i=1}^m U_i\quad\text{and}\quad C\text{ is characteristic in every }U_i\right] \quad\Longrightarrow\quad C=1 \] for every closed subgroup $C$ of $F$?

The subgroup $C$ is the same subgroup in all conditions; one is not intersecting arbitrarily selected characteristic subgroups of different $U_i$. The chosen list may depend on $F$ and $p$, but must be finite. The open subgroups need not themselves be characteristic or normal in $F$. The member $U_i=F$ ensures that a candidate $C$ is also characteristic in the full free pro-$p$ group.

Short English statement

Can finitely many open subgroups of a finite-rank nonabelian free pro-$p$ group have no common nontrivial subgroup characteristic in each of them?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2518 (KOU-21.9), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.9. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.