Kourovka Notebook Problem 21.46

Definitions and mathematical statement

For a group $G$, its integral group ring $\mathbb ZG$ consists of finite formal integer combinations of group elements. Give $\mathbb Z$ the trivial $\mathbb ZG$-module structure, so every $g\in G$ acts as the identity. The ordinary integral cohomological dimension $\operatorname{cd}_{\mathbb Z}(G)$ is the least length $d$ of a projective resolution \[ 0\longrightarrow P_d\longrightarrow\cdots\longrightarrow P_0 \longrightarrow\mathbb Z\longrightarrow0 \] over $\mathbb ZG$, or $\infty$ if no finite-length resolution exists. Projective means a direct summand of a free module; finite generation of the modules is not built into this definition.

The question asks whether there exists an infinite simple finitely presented group $G$ with \[ 3\le\operatorname{cd}_{\mathbb Z}(G)<\infty. \] Simple means nontrivial with no proper nontrivial normal subgroup. Finitely presented means a quotient of a finitely generated free group by the normal closure of finitely many relators. All three group-theoretic requirements and the cohomological bound apply to the same group.

The dimension sought is an integer strictly greater than two, without a specified upper value. A group of infinite cohomological dimension is not a witness. Neither rational cohomological dimension nor virtual cohomological dimension is substituted for the ordinary integral invariant. A resolution gives an upper bound; to establish a qualifying example one must also exclude dimensions zero, one and two, as well as verify the finite presentation and simplicity.

Short English statement

Does there exist a finitely presented infinite simple group of finite integral cohomological dimension greater than two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2555 (KOU-21.46), 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 21.46. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.