Two-generated groups acting on trees (Q1125388)

Let \(G\) be a group acting on a simplicial tree \(T\) without inversions, and let the edge stabilizers be non-trivial. If \(g,h\in G\) generate \(G\) (\(\langle g,h\rangle=G\)) or if \(\langle g,h\rangle\) is neither cyclic nor a free product of cyclic groups, then it is proved that the pair \(\{g,h\}\) is Nielsen equivalent to \(\{f,s\}\) and some non-trivial powers of either \(f\) and \(s\) or \(f\) and \(sfs^{-1}\) have a common fixedpoint. An analogous result is obtained for a Bass-Serre tree \(T\) associated to the presentation of the 2-generated group \(G\) as the fundamental group of a graph of groups. The proof is strictly geometric. The authors obtain stronger versions of several results of Karrass and Solitar, S. Bleiler and A. Jones and S. Pride on subgroups of amalgamated products.