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Math-1993 · mathematics · Group theory

Questions in Geometric Group Theory — Q 2.3

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Statement

Definitions and mathematical statement

For a group $G$, let $\mathbb ZG$ be its integral group ring, and regard $\mathbb Z$ as the trivial module. The integral cohomological dimension $\operatorname{cd}_{\mathbb Z}G$ is the minimum length of a projective resolution \[ 0\longrightarrow P_n\longrightarrow\cdots\longrightarrow P_0 \longrightarrow\mathbb Z\longrightarrow0 \] over $\mathbb ZG$, or infinity if there is no finite-length resolution. The modules are not required to be finitely generated. A classifying space $K(G,1)$ is a connected CW complex with fundamental group $G$ and contractible universal cover. The geometric dimension $\operatorname{gd}G$ is the least dimension of such a space.

Is there a group satisfying \[ \operatorname{cd}_{\mathbb Z}G=2\quad\hbox{and}\quad \operatorname{gd}G=3? \] Equivalently, must every group of integral cohomological dimension two admit a two-dimensional classifying space? The question places no finite-generation, finite-presentation or finite-cell assumption on $G$ or on the desired classifying space. Integral coefficients are essential: replacing them by rational coefficients changes the problem and allows torsion phenomena excluded here.

A proposed example must show both that a length-two projective resolution exists and that no two-dimensional $K(G,1)$ exists. Exhibiting a particular three-dimensional classifying space alone does not establish minimality. This is the exceptional dimension in the comparison of algebraic cohomological dimension with the smallest dimension of an aspherical CW model.

Short English statement

Can a group have algebraic cohomological dimension two while every classifying space has dimension at least three?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1993-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1993

The main theorem provides exactly the requested existence example; finite generation and residual finiteness are additional properties. Within these assumptions, this is a claimed affirmative answer to the stated question.

Questions in Geometric Group Theory — Q 2.3

OpenAI's claim

OpenAI presents a proof in the selected manuscript. A finitely generated residually finite group is claimed to have integral cohomological dimension two but no two-dimensional classifying space.

Mathematical statement of the claimed result

Result 1.
Write $\operatorname{cd}$ and $\operatorname{gd}$ for integral cohomological dimension and the minimum dimension of a classifying space. There is a group $G$, residually finite and finitely generated, for which \[ \operatorname{cd}G=2,\qquad\operatorname{gd}G=3. \] It takes a height kernel in a right-angled Artin group built from an acyclic flag triangulation of the presentation complex \[ \langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle. \] Its nonexistence conclusion applies to every two-dimensional $K(G,1)$, including those with infinitely many cells.

Outline of the argument

Degree estimates on conjugation-invariant parts of word fibers obstruct arbitrary aspherical presentations. Exact boundary-chain counts and quadratic bounds compare any candidate low-dimensional model with the constructed group, forcing geometric dimension three despite the shorter cohomological resolution.

Formal verification

For family 249, the documented example is finitely generated and residually finite, has integral cohomological dimension two, and has geometric dimension three. The listed Comparator files are EilenbergGanea.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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