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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.

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.

It's a conjecture that surface groups are characterized (among infinite groups, with the exception of $\mathbb{Z}/2\mathbb{Z}$) 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.

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.

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.

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.

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.

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: This conjecture has been proved for 2-generator groups.

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.