Conjugacy separability and outer automorphism groups of certain HNN extensions.
A group \(G\) is conjugacy separable if for every \(x\in G\) its conjugacy class \(\{x\}^G\) is closed with respect the profinite topology in \(G\). Evidently conjugacy separability implies residual finiteness. It is well known that the automorphism group of a finitely generated residually finite group is residually finite. However [see \textit{D. T. Wise}, Bull. Lond. Math. Soc. 35, No. 1, 23-29 (2003; Zbl 1027.20014)], there exist finitely generated residually finite groups where the outer automorphism group is not residually finite. On the other hand \textit{E. K. Grossman} [in J. Lond. Math. Soc., II. Ser. 9, 160-164 (1974; Zbl 0292.20032)] proved that if a finitely generated group is conjugacy separable and every conjugating automorphism is also an inner automorphism then its outer automorphism group is residually finite. In this paper, on one hand, the authors study the conjugacy separability of certain HNN-extensions with associated subgroups in the center of the base group and obtain: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the polycyclic-by-finite group \(A\) and proper associated subgroups \(H\) and \(K\) in the center of \(A\). The group \(G\) is conjugacy separable if and only if it is residually finite. This theorem is inspired by an analogous theorem of \textit{E. Raptis, O. Talelli} and \textit{D. Varsos} [J. Algebra 199, No. 1, 327-336 (1998; Zbl 0891.20023)] where the base group is a finitely generated Abelian group and generalizes the results there. On the other hand they prove: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the group \(A\) and proper associated subgroups \(H\) and \(K\) in the center of \(A\). Then every conjugating endomorphism of \(G\) is an inner automorphism. Combing this result with the Grossman's result it is obtained: Theorem: Let \(G=\langle t,A\mid t^{-1}Ht=K,\;\varphi\rangle\) be the HNN-extension of the polycyclic-by-finite group \(A\) and proper associated subgroups \(H\) and \(K\) in the center of \(A\). If the group \(G\) is residually finite then \(\text{Out}(G)\) is residually finite.
scientific article; zbMATH DE number 5990145
scientific article