A group G is said to be
residually finite if the intersection of the collection of all subgroups of finite index in
G is the trivial group. This paper is concerned; with the following question. If A and
B are residually finite groups, and if G is the generalized free product of A
and B with a single cyclic subgroup amalgamated, then what conditions
on A and B will insure that G is residually finite? The main result of this
paper is that there exists a class C of residually finite groups which contains
all free groups, polycylic groups, fundamental groups of 2-manifolds, and
other common residually finite groups, and in addition C is closed under the
operation of forming generalized free products with a single cyclic subgroup
amalgamated.
