The automorphism group of certain HNN extensions (Q1105036)

This paper deals with the automorphism group of a simple HNN-extension \(G=<t,K|\) \(t^{-1}At=B\), \(\phi >\) where the base group K is a finitely generated free abelian group, and generalizes results of \textit{D. Collins} and \textit{F. Levin} [Proc. Lond. Math. Soc., III. Ser. 36, 480- 493 (1978; Zbl 0376.20025) and Arch. Math. 40, 385-400 (1983; Zbl 0498.20021)] who have studied the case where the base group is an infinite cyclic group. The paper is divided into two sections. In the first section the author gives necessary and sufficient conditions under which an endomorphism of the base group K can be extended to an endomorphism of the group G. In the special case where K is mapped into itself the author gives a description of the image of the free generator t. In the second section the author studies the automorphism group of the group G. Under certain conditions it is proved that if \(\theta\in Aut G\), then \(K\theta \leq g^{-1}Kg\) for some \(g\in G\) and in particular if A and B are proper subgroups of K, then for \(\theta\in Aut G\) there exists \(g\in G\) such that \(K\theta =g^{-1}Kg\). In this case a description is given of the group of automorphisms of G as a product of certain subgroups. A description is also given in the case where one of the associated subgroups is the base group. By using the Bass-Serre theory the author also proves that the subgroup H of Aut G which contains the automorphisms \(\theta\) with \(A\theta =A\), \(B\theta =B\) modulo the group of inner automorphisms of G, is also a convenient HNN-extension.