The Surface Group Conjectures for groups with two generators

Giles Gardam, Dawid Kielak, Alan D. Logan

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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
Pages (from-to)109–123
Number of pages15
JournalMathematical Research Letters
Issue number1
Publication statusPublished - 21 Jun 2023


  • math.GR
  • 20F65 (Primary) 57M10, 20F05, 20F67, 20J05 (Secondary)


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