I will relate the subgroup structure of one-relator groups to a measure of complexity for the relator \(w\) introduced by Puder - the * primitivity rank * \(\p (w)\), the smallest rank of a subgroup of \(F\) containing \(w\) as an imprimitive element. A sample application is that every subgroup of \(G\) of rank less than \(\pi(w)\) is free. These results in turn provoke geometric conjectures that suggest a beginning of a geometric theory of one-relator groups.

This is joint work with Larsen Louder.