The Surface Group Conjectures for groups with two generators

Giles Gardam, Dawid Kielak, Alan D. Logan

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Number of pages14
JournalMathematical Research Letters
Publication statusAccepted/In press - 4 Aug 2022


  • One-relator group
  • Surface group


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