Superstable groups acting on trees

We study superstable groups acting on trees. We prove that an action of an ω-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not ωstable. It is also shown that if G is a superstable group acting nontrivially on a Λ-tree, where Λ = Z or Λ = R, and if G is either α-connected and Λ = Z, or if the action is irreducible, then G interprets a simple group having a nontrivial action on a Λ-tree. In particular if G is superstable and splits as G = G1 ∗A G2, with the index of A in G1 different from 2, then G interprets a simple superstable non ω-stable group. We will deal with ”minimal” superstable groups of finite Lascar rank acting nontrivially on Λ-trees, where Λ = Z or Λ = R. We show that such groups G have definable subgroups H1 ⊳H2 ⊳G, H2 is of finite index in G, such that if H1 is not nilpotent-by-finite then any action of H1 on a Λ-tree is trivial, and H2/H1 is either soluble or simple and acts nontrivially on a Λ-tree. We are interested particularly in the case where H2/H1 is simple and we show that H2/H1 has some properties similar to those of bad groups.