Decomposing one-relator products of cyclic groups into free products with amalgamation

Abstract

This paper deals with the problem of decomposing one-relator products of cyclics into non-trivial free product with amalgamation. Two theorems have been proved, we state the following one. Let $G=<a,b | a^{2n}=R^m(a,b)=1>$, where m>1, R(a,b) is a cyclically reduced word in the free group on a and b which involves b. Then G is a non-trivial free product with amalgamation. As a corollary of this theorem we obtain the proof of the conjecture of Fine, Levin and Rosenberger which says that any two generator one-relator group with torsion is a non-trivial free product with amalgamation.