Subgroup separability of certain HNN extensions of finitely generated Abelian groups
A group \(G\) is called subgroup separable if, for each finitely generated subgroup \(M\) and for each \(x\not\in M\), there exists a normal subgroup \(N\) of finite index in \(G\) such that \(x\not\in MN\). In this paper, the author gives a characterization of certain HNN-extensions to be subgroup separable. More precisely, he proves the following theorem: Let \(G=\langle t,K;\;t^{-1}At=B,\;\phi\rangle\) be an HNN-extension, where \(K\) is a finitely generated Abelian group and \(A\) and \(B\) have finite index in \(K\). Then the following are equivalent: (i) \(G\) is subgroup separable; (ii) Either \(K=A=B\) or there exists a subgroup \(H\) of finite index in \(K\) and \(H\) is normal in \(G\); (iii) There exists a finitely generated Abelian group \(X\) such that \(K\) is a subgroup of finite index in \(X\) and an automorphism \(\overline\phi\in\Aut X\) with \(\overline\phi|_A=\phi\). For the proof, the author uses also three lemmas from a paper of the reviewer and \textit{E. Raptis} and \textit{D. Varsos} [Arch. Math. 53, No. 2, 121-125 (1989; Zbl 0651.20060)]. Note that Lemma 7 is Theorem 2 of the above mentioned paper, phrased slightly differently and with essentially the same proof. (Also submitted to MR).
scientific article; zbMATH DE number 1051541