DECOMPOSING ONE-RELATOR PRODUCTS OF CYCLIC GROUPS INTO FREE-PRODUCTS WITH AMALGAMATION

The problem of the decomposition of one-relator products of cyclics in to non-trivial free products with amalgamation is considered. Two theo rems are proved, one of which is as follows. Let G = [a, b \ a(2n) = R -m(a, b) = 1], where n greater than or equal to 0, m greater than or e qual to 2, and R(a, b) is a cyclically reduced word containing b in th e free group on a and b. Then G is a n:on-trivial free product with am algamation. One consequence of this theorem is a proof of the conjectu re of Fine, Levin, and Rosenberger that each two-generator one-relator group with torsion is a non-trivial free product with amalgamation.