Analogs of Magnus' theorem for groups with small contraction, and the fact that they cannot be strengthened (Q1077537)

Let r, s be elements of the free group F with basis X. It is a consequence of Magnus' Freiheitssatz that the identity map on F induces an isomorphism between the one-relator groups \(<X| r>\) and \(<X| s>\) if and only if s is a conjugate of r or of \(r^{-1}\) in F. There is an analogue of this result, due to \textit{M. D. Greendlinger} [Izv. Akad. Nauk SSSR, Ser. Mat. 29, 245-268 (1965; Zbl 0166.014)] for presentations satisfying the \(<metric>\) small cancellation condition \(C(<1/10):\) the identity on F induces an isomorphism from \(<X| R>\) to \(<X| S>\) if and only if R and S have the same symmetrized closure. In this paper, Greendlinger's result is extended to groups satisfying either \(C(<1/5)\) or \(C(<1/3) \& T_ 3\). Examples are given to show that \(C(<\lambda)\) cannot be replaced by the weaker condition C(\(\leq \lambda)\) in these results.