Virtual Properties of cyclically pinched One-Relator Groups

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general the amalgamated product of free groups with cyclic amalgamations is free-by-(torsionfree nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-byabelian. Analagous results are obtained for certain one-relator HNN-extensions.