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, arXiv:2202.11093.
