Cyclic subgroup separability of certain graph products of subgroup separable groups. (Q2873011)

A group \(G\) is called \(H\)-separable for the subgroup \(H\) if for each \(x\in G\setminus H\) there exists a normal, finite index subgroup \(N\) of \(G\) such that \(x\not\in HN\). A group \(G\) is called subgroup separable if \(G\) is \(H\)-separable for every finitely generated subgroup \(H\), and is called cyclic subgroup separable (or \(\pi_c\)) if \(G\) is \(H\)-separable for every cyclic subgroup \(H\).NEWLINENEWLINE A graph product of groups is a generalization of the concept of a free product of groups with amalgamated subgroup. (An amalgamated free product \(A*_HB\) is a graph product \(G(Q)\) where \(Q\) is a graph with edge set \(E(Q)=\{e\}\) and where \(A,B\) are the vertex groups assigned to the endpoints of \(e\) and \(H=H_e\) is the edge group.) The intersection property is said to hold for a graph product of groups \(G(Q)\) if for every edge \(uv\in E(Q)\) we have NEWLINE\[NEWLINEH_{uv}\cap\prod_{uw\in E(Q),\;w\neq v}H_{uw}=1.NEWLINE\]NEWLINE The main result is that if \(Q\) is a simple graph that has exactly one cycle (that is not a triangle) and the intersection property holds, then the graph product \(G(Q)\) of subgroup separable groups amalgamating finitely generated normal subgroups is \(\pi_c\). This generalizes a theorem of \textit{R. B. J. T. Allenby} [Bull. Aust. Math. Soc. 54, No. 3, 369-372 (1996; Zbl 0885.20012)], which corresponds to the case when \(Q\) is a cycle. It is noted that the theorem also applies to a graph product \(G(Q)\) of polycyclic-by-finite groups, since such groups are subgroup separable.