Kourovka Notebook Problem 21.119

Definitions and mathematical statement

A classifying space $K(H,1)$ for a group $H$ is a connected CW-complex with fundamental group isomorphic to $H$ and contractible universal cover. Equivalently, its higher homotopy groups vanish. A group has type $F_n$ if it admits such a space with finite $n$-skeleton, meaning finitely many cells in dimensions at most $n$. In particular type $F_1$ is finite generation and type $F_2$ is finite presentability.

Does there exist one group $G$ with the following simultaneous property? For every positive integer $n$, there are a finite-index subgroup $G_n\le G$ and a homomorphism to the additive infinite cyclic group, \[ f_n:G_n\longrightarrow\mathbb Z, \qquad K_n=\ker f_n, \] such that $K_n$ is of type $F_n$ but is not of type $F_{n+1}$.

The negated condition means that no classifying space for $K_n$ has a finite $(n+1)$-skeleton, even though some classifying space has a finite $n$-skeleton. All the subgroups $G_n$ must belong to the same ambient group $G$; independently constructing a different ambient group for each $n$ does not meet this requirement. The subgroups need not form a nested family and need not be normal. No relation between the homomorphisms for different indices is stipulated, and the source does not require them to be surjective.

Short English statement

Can one group have finite-index subgroups with integer-valued homomorphisms whose kernels realize every successive gap between types $F_n$ and $F_{n+1}$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2628 (KOU-21.119), 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, version 47 (30 September 2026), new problems, Problem 21.119. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.