The surface group conjectures for groups with two generators (Q6100648)

A Mel'nikov group is a non-free infinite one-relator group with every subgroup of finite index also a one-relator group. Residually finite Mel'nikov groups are not necessarily surface groups, with the Baumslag-Solitar groups \(\mathrm{BS}(1, n)\) forming a family of counterexamples. The main results of the paper under review are summarized in Theorem 1.5: Let \(G\) be a two-generated group. \begin{itemize} \item[(1)] If \(G\) is a residually finite Mel'nikov group, then \(G\) is a surface group or \(\mathrm{BS}(1, n)\) for some nonzero integer \(n\). \item[(2)] If \(G\) is a Mel'nikov group with every subgroup of infinite index free, then \(G\) is a surface group. \item[(3)] If \(G\) is an infinite (non-free) one-relator group with every subgroup of infinite index free, then \(G\) is a surface group. \end{itemize}