\(3\)-manifolds and generalized Baumslag-Solitar groups (Q724442)

A \textit{Baumslag-Solitar group} is a group with two generators \(x\) and \(t\) and one relation \(t^{-1}x^mt = x^n\). \textit{W. H. Heil} [Proc. Am. Math. Soc. 53, 497--500 (1975; Zbl 0332.57001)] showed that such a group is isomorphic to the fundamental group of a 3-manifold if and only if \(|m| = |n|\), and by a result of \textit{P. B. Shalen} [Topology Appl. 110, No. 1, 113--118 (2001; Zbl 0976.57002)], if two elements \(x\) and \(t\) in the fundamental group of a compact 3-manifold satisfy such a Baumslag-Solitar relation then either \(x\) has finite order or \(|m| = |n|\). A \textit{generalized Baumslag-Solitar group} is defined as the fundamental group of a graph of groups whose vertex and edge groups are infinite cyclic, hence it is obtained as an iterated free product with amalgation and HNN-extension of infinite cyclic groups. In the present paper the generalized Baumslag-Solitar groups are classified which are fundamental groups of compact orientable 3-manifolds. ``More generally, we show that many generalized Baumslag-Solitar groups which are not 3-manifold groups are special types of quotients of generalized Baumslag-Solitar groups'' (which are obtained by ``pinching'').