The Surface Group Conjectures for groups with two generators - Gardam, G; Kielak, D; Logan, AD (International Press, 2023)
The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.