Questions in Geometric Group Theory — Q 3.3

Definitions and mathematical statement

Let $G$ be a finitely generated group and let $\mathbb ZG$ be its integral group ring. Regard $\mathbb Z$ as the trivial module, with every group element acting as the identity. Its integral homological dimension is the flat dimension of this module: \[ \operatorname{hd}_{\mathbb Z}G =\min\{d:\mathbb Z\text{ has a flat }\mathbb ZG \text{-resolution of length }d\}. \] A module is flat when tensoring with it preserves exact sequences. Equivalently, $\operatorname{hd}_{\mathbb Z}G\leq d$ means that $\operatorname{Tor}^{\mathbb ZG}_i(M,\mathbb Z)=0$ for every right $\mathbb ZG$-module $M$ and every $i>d$. These Tor groups are group homology with arbitrary module coefficients.

If $\operatorname{hd}_{\mathbb Z}G=1$, must $G$ be a free group? A free group is one having a basis subject to no relations beyond the group axioms; finite generation would make that basis finite.

The coefficient ring is the integers, not the rationals or a finite field. Homological dimension uses flat resolutions and must not be replaced by cohomological dimension, which uses projective resolutions. The group is assumed finitely generated, not finitely presented; adding finite presentation would remove the main finiteness difficulty. The equality to one excludes the dimension-zero case. The problem requests freeness of the group itself, not merely that all its finite subsets lie in free quotients.

Short English statement

Is every finitely generated group of integral homological dimension one free?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100303 (AMR-010-0303), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 3.3 (PDF page 10). Original source indexed by AMR-010-0303. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.