Some free actions on non-archimedean trees

In this paper we show that various groups are Z-free. In particular we show that almost every surface group is (Z×Z)-free as are the groups of Liousse [11]. We also demonstrate that the class of Z-free groups is closed under taking amalgamated free products over an infinite cyclic group as long as it is maximal abelian in each vertex group. It follows that a large class of hyperbolic groups is Z-free.