Characterize finitely presented groups with all infinite-index subgroups free

Determine whether every infinite, finitely presented group G in which every subgroup of infinite index is free must be isomorphic to either a free group or the fundamental group of a closed, aspherical surface (a surface group).

Background

The paper studies when the property that all subgroups of infinite index are free characterizes free and surface groups. Classical results show every subgroup of a free group is free (Nielsen–Schreier) and that every infinite-index subgroup of a surface group is free (Johansson). The author asks if these properties uniquely characterize free and surface groups among finitely presented groups.

The main results provide affirmative answers within important classes (cubulated hyperbolic groups and one-relator groups), but the general question remains open.