Two-generated groups acting on trees

Abstract.

We study 2-generated subgroups of groups that act on simplicial trees. We show that any generating pair $\{{g},h\}$ of such a subgroup is Nielsen-equivalent to a pair $\{f,s\}$ where either powers of f and s or powers of f and $sfs^{-1}$ have a common fixed point if the subgroup $\langle {g},h\rangle$ is freely indecomposable. Analogous results are obtained for generating pairs of fundamental groups of graphs of groups. Some simple applications are given.