On the Surface Group Conjecture

Abstract

We consider the following conjecture. Suppose that \(G\) is a non-free non-cyclic one- relator group such that each subgroup of finite index is again aone relator group and each subgroup of infinite index is a free group. Must \(G\) be a surface group? We show that if \(G\) is a freely indecomposable fully residually free group and satisfies the property that every subgroup of infinite index is free  then \(G\) is either a cyclically pinched one-relator group or a conjugacy pinched one-relator group. Further such a group \(G\) is either hyperbolic or free abelian of rank 2.