Mathematical Research Letters

Volume 30 (2023)

Number 1

The Surface Group Conjectures for groups with two generators

Pages: 109 – 123



Giles Gardam (Mathematisches Institut, Universität Münster, Germany)

Dawid Kielak (Mathematical Institute, University of Oxford, United Kingdom)

Alan D. Logan (School of Engineering and Physical Sciences, Heriot–Watt University, Edinburgh, Scotland, United Kingdom)


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.

Received 3 May 2022

Received revised 21 July 2022

Accepted 4 August 2022

Published 21 June 2023