Questions in Geometric Group Theory — Q 9.3

Definitions and mathematical statement

A finite two-dimensional CW complex $X$ is built from finitely many vertices, edges, and two-cells. Require $X$ to be connected. Its Euler characteristic is \[ \chi(X)=c_0-c_1+c_2, \] where $c_i$ is the number of $i$-cells. The value does not depend on the chosen finite cell decomposition. The complex is aspherical if its universal cover is contractible; equivalently, all its homotopy groups in degrees at least two vanish.

A group is coherent if every finitely generated subgroup is finitely presented. This is a condition on all finitely generated subgroups, not only finite-index ones. A finite presentation consists of finitely many generators and finitely many defining relations.

Does there exist such an aspherical finite two-complex $X$ with coherent fundamental group and \[ \chi(X)\geq2? \] The dimension bound and asphericity are both essential parts of the target. Attaching spheres could increase Euler characteristic but would destroy asphericity, while adding cells in higher dimensions would leave the prescribed class.

One may equivalently start with a coherent group admitting a finite two-dimensional classifying space and ask whether its Euler characteristic can exceed one. The group need not be hyperbolic, torsion-free by a separate assumption, or equipped with a particular geometric action; torsion-freeness follows from the finite-dimensional classifying-space condition. The question asks for a single example in the full class, not a family with arbitrarily large Euler characteristic.

Short English statement

Can a coherent group have a finite aspherical two-dimensional classifying space with Euler characteristic at least two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100903 (AMR-010-0903), 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 9.3 (PDF page 18). Original source indexed by AMR-010-0903. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.