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. 

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**Addendum July 2024:** This conjecture has been proved for [2-generator groups](https://groupprops.subwiki.org/wiki/2-generated_group):

* 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].


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