Infinite conjugacy classes in groups acting on trees.
The author considers an amalgam \(\Gamma=A*_HB\) acting on its Bass-Serre tree which is a tree with two given vertices \(\alpha,\beta\) with stabilizers \(A,B\) linked by an oriented edge \(\varepsilon\) with stabilizer \(H\). In case \(\Gamma=\text{HNN}(K,H,\theta)\) its Bass-Serre tree is a tree having one vertex \(\alpha\) with stabilizer \(K\) and one oriented edge from \(\alpha\) to \(\theta(\alpha)\) with stabilizer \(H\). All free groups considered are non-Abelian. The main aim of the paper is to characterize \(\Gamma\) both in case it is an amalgam and when it is a Higman-Neumann-Neumann extension. This is done in terms of infinite conjugacy classes. By this, the author answers a question posed by \textit{P. de la Harpe} [Bull. Lond. Math. Soc. 39, No. 1, 1-26 (2007; Zbl 1123.22004), Problems 27, 28].
scientific article; zbMATH DE number 5550904