Kourovka Notebook Problem 21.140

Definitions and mathematical statement

A group $G$ is torsion-free if $g^n=1$ with $n\ge1$ implies $g=1$. A classifying space $K(G,1)$ is a connected CW-complex with fundamental group $G$ and contractible universal cover. The group is of type $F_\infty$ if it admits such a space with finitely many cells in each dimension, equivalently if it is of type $F_n$ for every finite $n$.

The integral cohomological dimension $\operatorname{cd}_{\mathbb Z}(G)$ is the projective dimension of the trivial module $\mathbb Z$ over the group ring $\mathbb ZG$. Thus it is the least length of a projective resolution of that module, or infinity if no finite-length resolution exists. Infinite cohomological dimension is compatible with finitely many cells in each dimension, since their dimensions need not be bounded.

Thompson's group $F$ consists of all orientation-preserving piecewise-linear homeomorphisms of $[0,1]$ with finitely many breakpoints, all breakpoints dyadic rational numbers, and slopes integer powers of $2$. A dyadic rational has the form $a/2^b$ with integers $a$ and $b\ge0$. The group operation is composition.

Must every torsion-free group $G$ of type $F_\infty$ with $\operatorname{cd}_{\mathbb Z}(G)=\infty$ contain a subgroup isomorphic to $F$? Explicitly, the conclusion requests an injective homomorphism $F\hookrightarrow G$. The sought copy is an abstract subgroup; no topology, action on an interval, or geometric embedding of classifying spaces is required to be preserved.

Short English statement

Must a torsion-free group with all finite skeleta available but infinite integral cohomological dimension contain Thompson’s group $F$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2649 (KOU-21.140), 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 21.140. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.