It's a conjecture that surface groups are characterized by being the only 1-relator groups such that every finite-index subgroup is also 1-relator and every infinite index subgroup is free. 

---

**Addendum July 2024:** This conjecture has been proved for 2-generator groups:

* Giles Gardam, Dawid Kielak, Alan D. Logan, _The Surface Group Conjectures for groups with two generators_, Math. Res. Lett. **30**  Number 1 (2023) pp 109–123, doi:[10.4310/MRL.2023.v30.n1.a5](https://doi.org/10.4310/MRL.2023.v30.n1.a5), [arXiv:2202.11093][1].

**Further addendum July 2024:** Recently, this conjecture has been proved in full generality. More precisely, every one-relator group such that every subgroup of infinite index is free is either a free group or a surface group. (The related conjecture, that every residually finite one-relator group with every finite-index subgroup one-relator must be either free, a surface group or a Baumslag--Solitar group, remains open.)

* Henry Wilton, _Surface groups among cubulated hyperbolic and one-relator groups_, [arXiv:2406.02121][2].


  [1]: https://arxiv.org/abs/2202.11093
  [2]: https://arxiv.org/abs/2406.02121