Submission date: 30 November 2008.

Abstract:

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 ω-stable. 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=G_1*_AG_2, with the index of A in G_1 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 H_1 \lhd H_2 \lhd G, H_2 is of finite index in G, such that if H_1 is not nilpotent-by-finite then any action of H_1 on a Λ-tree is trivial, and H_2/H_1 is either soluble or simple and acts nontrivially on a Λ-tree. We are interested particularly in the case where H_2/H_1 is simple and we show that H_2/H_1 has some properties similar to those of bad groups.

Mathematics Subject Classification: 03C99;20F65;20E08

Keywords and phrases: Groups, Superstable groups, actions on trees, free product with amalgamation, HNN-extension.

Full text arXiv: pdf, ps.