Questions in Geometric Group Theory — Q 1.7

Definitions and mathematical statement

A finitely generated group $G$ is word-hyperbolic when geodesic triangles in a Cayley graph are uniformly thin. To make the dimension in the question precise, consider torsion-free such groups and use integral cohomological dimension \[ \operatorname{cd}_{\mathbb Z}G =\operatorname{pd}_{\mathbb ZG}\mathbb Z, \] the shortest length of a projective resolution of the trivial group-ring module. A subgroup has infinite index if its left-coset space in $G$ is infinite. A free subgroup may have finite or infinite rank.

For a prescribed integer $n\geq2$, is there a torsion-free word-hyperbolic group $G$ of cohomological dimension $n$ for which every infinite-index subgroup is free? The related weaker existence problem asks, for a prescribed $0\leq k\leq n-2$, whether such a group can have no infinite-index subgroup $H$ with \[ \operatorname{cd}_{\mathbb Z}H\geq n-k. \] One may instead ask this last question only for quasiconvex subgroups: those for which every geodesic between subgroup elements stays a uniformly bounded distance from the subgroup.

The exclusion concerns infinite-index subgroups; otherwise $G$ itself would immediately violate it. Codimension means the difference of cohomological dimensions, not the codimension of an embedded manifold. These are existence questions for each dimension and codimension range, and do not assume that an arbitrary hyperbolic group has the asserted restrictive subgroup structure.

Short English statement

Can higher-dimensional hyperbolic groups have only free infinite-index subgroups, or at least avoid infinite-index subgroups of small cohomological codimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100107 (AMR-010-0107), 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 1.7 (PDF page 3). Original source indexed by AMR-010-0107. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.