The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in [FKMRR]. First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if $G$ is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then $G$ is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results in [FKMRR] on Property IF with a theorem of H. Wilton [W] and results of Stallings [St] and Kharlampovich and Myasnikov [KhM4] we show that Surface Group Conjecture C proposed in [FKMRR] is true, namely: If $G$ is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then $G$ is a surface group.