Kourovka Notebook Problem 21.138

Definitions and mathematical statement

A group is finitely presented if it has a presentation $\langle S\mid R\rangle$ with both the generating set $S$ and the list of defining relators $R$ finite. For a subgroup $H\le G$, its index $[G:H]$ is the cardinality of the set of left cosets. Infinite index means that this coset set is infinite.

A free group on a set $B$ is a group in which arbitrary assignments of $B$ into any group extend uniquely to homomorphisms. Free subgroups in the hypothesis may have any rank, including zero, finite rank, or infinite rank. A surface group here means the fundamental group of a closed connected aspherical surface, orientable or nonorientable. Aspherical means its universal cover is contractible; thus this convention includes the torus and Klein bottle and excludes the sphere and projective plane.

Suppose $G$ is infinite and finitely presented, and every subgroup $H\le G$ of infinite index is a free group. Must $G$ itself be either a free group or a surface group?

The hypothesis is imposed on all infinite-index subgroups, not just finitely generated ones. Finite-index subgroups are not required to be free. The desired conclusion is an isomorphism of $G$ itself with one of the two indicated kinds of groups, rather than just existence of a finite-index subgroup of such a kind. No one-relator, residual-finiteness, or geometric-action assumption is present.

Short English statement

Are free groups and closed aspherical surface groups the only infinite finitely presented groups whose every infinite-index subgroup is free?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2647 (KOU-21.138), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.138. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 H. Wilton, Surface groups among cubulated hyperbolic and one-relator groups, Question 0.1; specifies closed aspherical surfaces and distinguishes known restricted cases. https://arxiv.org/html/2406.02121v3.