GENERALIZED FREE-PRODUCTS OF PI-C GROUPS

A group G is said to be pi(c) if and only if for every pair of elements g1 and g2 of G either g1 = g2k for some integer k or there exists a normal subgroup N of finite index in G such that g1 is not identically equal to g2z mod N for all integers z. In this note we prove that a certain generalized free product amalgamating a cyclic subgroup is pi(c) and then apply the result to show some one-relator groups are pi(c).