Questions in Geometric Group Theory — Q 6.2

Definitions and mathematical statement

For a torsion-free group $H$, its integral cohomological dimension $\operatorname{cd}_{\mathbb Z}H$ is the projective dimension of the trivial $\mathbb ZH$-module $\mathbb Z$. Equivalently, it is the least $d$ such that $H^i(H;M)=0$ for every module $M$ and all $i>d$. A virtually torsion-free group $G$ has virtual cohomological dimension \[ \operatorname{vcd}(G)=\operatorname{cd}_{\mathbb Z}H \] for a torsion-free subgroup $H$ of finite index; the value is independent of that choice when finite.

Does there exist such a group $G$ with $d=\operatorname{vcd}(G)<\infty$ for which there is no $d$-dimensional acyclic $G$-complex with finite stabilizers? Here a $G$-complex is a CW complex with cellular group action, using the usual convention that a cell stabilizer fixes that cell pointwise. Acyclic means nonempty with $\widetilde H_i(X;\mathbb Z)=0$ in every degree. Every point stabilizer must be finite.

The complex need not be contractible, locally finite, or cocompact under the action. In particular, the question is weaker than demanding a classifying space for proper actions, where all finite-subgroup fixed-point sets must be contractible. A counterexample to that stronger requirement does not settle this question. Asking for dimension at most $d$ gives the same existence problem, since equivariant harmless cells can increase the dimension if necessary. Integral acyclicity, rather than rational acyclicity, is the convention throughout.

Short English statement

Can virtual cohomological dimension be too small for any acyclic complex on which the group acts with finite stabilizers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100602 (AMR-010-0602), 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 6.2 (PDF page 14). Original source indexed by AMR-010-0602. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.