Kourovka Notebook Problem 21.33

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.

A tdlc group is a Hausdorff topological group that is locally compact and totally disconnected. A discrete rational $G$-module is a $\mathbb Q$-vector space with $G$-action such that the stabilizer of each vector is open. The rational discrete cohomological dimension $\operatorname{cd}_{\mathbb Q}^{\rm disc}(G)$ is the projective dimension of the trivial module $\mathbb Q$ in this category. A profinite group is an inverse limit of finite groups with their discrete topologies. A graph of profinite groups has profinite vertex and edge groups and continuous injective edge-to-vertex maps with open image. Its topological fundamental group is the Bass--Serre fundamental group with the topology induced by its open vertex groups; no finiteness or cocompactness of the underlying graph is assumed.

Statement recorded in the supplied source.

If $G$ is a tdlc group with $\operatorname{cd}_{\mathbb Q}^{\rm disc}(G)\leq1$, must $G$ be topologically isomorphic to the fundamental group of a graph of profinite groups? In tree-action language, must $G$ admit a continuous action, without inversions, on a tree with compact open vertex stabilizers? [S2]

Short English statement

Must a totally disconnected locally compact group of rational discrete cohomological dimension at most one arise from a graph of profinite groups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2542 (KOU-21.33). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 I. Castellano, Problem 21.33, 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.