The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the C'(1 /6) condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every C'(1/6) positive one-relator group is residually finite. It is shown that positive one-relator groups are generic ally C'(1/6) and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs.