Questions in Geometric Group Theory — Q 1.11

Definitions and mathematical statement

A finitely generated group $G$ is word-hyperbolic when the geodesic triangles of a Cayley graph are uniformly thin. It is one-ended when removing any finite vertex set from that graph leaves exactly one infinite component. A group is virtually a closed hyperbolic surface group if it has a finite-index subgroup isomorphic to $\pi_1(\Sigma_g)$ for some closed orientable surface of genus $g\geq2$; using nonorientable hyperbolic surfaces gives the same virtual class.

Does there exist a one-ended word-hyperbolic group $G$ which is not virtually a closed hyperbolic surface group and nevertheless satisfies \[ \forall H\leq G\quad [G:H]=\infty \Longrightarrow H\text{ is a free group}? \] Here a free group may have a finite or infinite basis, and the trivial subgroup is free of rank zero. The quantifier covers all subgroups, not only finitely generated or quasiconvex ones.

Equivalently, a negative answer would say that every one-ended hyperbolic group outside the virtual surface class has some nonfree infinite-index subgroup. The witnessing subgroup need not have finite presentation or finite rank. The one-ended hypothesis excludes ordinary free groups, while the virtual surface exclusion removes the motivating surface examples. No restriction on the dimension of a classifying space is included. In particular, demonstrating that many finitely generated subgroups are free would not establish the universal subgroup property requested of an example.

Short English statement

Can a one-ended hyperbolic group outside the virtual surface class have only free subgroups of infinite index?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100111 (AMR-010-0111), 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 1.11 (PDF page 3). Original source indexed by AMR-010-0111. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.