On the residual finiteness of generalized free products of finite rank groups. (Q359352)

Let \(P\) be a free product of residually finite groups \(A\) and \(B\) with amalgamated subgroups \(H\) and \(K\) different from \(A\) and \(B\), respectively. The author proves that if \(A\) and \(B\) are finitely generated groups of finite rank, \(H\) and \(K\) are normal subgroups then \(P\) is residually finite if and only if \(H\) and \(K\) are residually separable in \(A\) and \(B\), respectively. He obtains the analogous result for a free product \(P\) of residually finite groups \(A\) and \(B\) with cyclic amalgamated subgroups \(H\) and \(K\). Also the author finds a criterion of the residual finiteness of \(P\) in the case when \(H\) and \(K\) have finite indices in \(A\) and \(B\), respectively.