Kourovka Notebook Problem 21.36

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 $\mathcal K$ be the class of profinite groups, and work throughout with Hausdorff tdlc groups. Set $\mathbf H_0\mathcal K=\mathcal K$. A successor class $\mathbf H_{\alpha+1}\mathcal K$ consists of groups admitting a continuous admissible cellular action on a finite-dimensional contractible CW-complex, with open cell stabilizers in $\mathbf H_\alpha\mathcal K$. Admissible means that each setwise cell stabilizer fixes that cell pointwise. At a limit ordinal $\lambda$, set $\mathbf H_\lambda\mathcal K=\bigcup_{\alpha<\lambda}\mathbf H_\alpha\mathcal K$. The union over ordinals is $\mathbf H\mathcal K$. For classes $\mathcal A,\mathcal B$, the extension class $\mathcal A\mathcal B$ consists of tdlc groups with a closed normal subgroup in $\mathcal A$ and quotient in $\mathcal B$, with the quotient topology.

Statement recorded in the supplied source.

For every ordinal $\alpha$, is \[ (\mathbf H_\alpha\mathcal K)\mathcal K\subseteq\mathbf H_\alpha\mathcal K? \] Equivalently, if $N$ is a closed normal subgroup of a tdlc group $G$, $N\in\mathbf H_\alpha\mathcal K$, and $G/N$ is profinite, must $G\in\mathbf H_\alpha\mathcal K$? [S2]

Short English statement

Is every stage of the tdlc version of Kropholler’s hierarchy closed under extensions with profinite quotient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2545 (KOU-21.36). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 G. C. Cook, Problem 21.36, 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). See also source definitions in Problems 15.45. https://arxiv.org/abs/1401.0300v48.