Ciobanu–Fine–Rosenberger's Surface Group Conjecture A

Let $G$ be a Mel'nikov group, meaning a non-free infinite one-relator group such that every subgroup of finite index is also a one-relator group. A group is residually finite if the intersection of its finite-index subgroups is trivial. A surface group is the fundamental group of a closed surface of non-positive Euler characteristic, and $\operatorname{BS}(1,n)$ denotes the Baumslag–Solitar group with non-zero integer parameter $n$.

Surface Group Conjecture A. Let $G$ be a residually finite Mel'nikov group. Then $G$ is a surface group or $\operatorname{BS}(1,n)$ for some non-zero integer $n$.

The conjecture seeks to characterize residually finite Mel'nikov groups, prompted by the Baumslag–Solitar groups as known non-surface examples. Its resolution status is not specified in the source.