Superstable groups acting on trees

Abstract

We study superstable groups acting on trees. We prove that an action of an $\omega$-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not $\omega$-stable. It is also shown that if $G$ is a superstable group acting nontrivially on a $\Lambda$-tree, where $\Lambda=\mathbb Z$ or $\Lambda=\mathbb R$, and if $G$ is either $\alpha$-connected and $\Lambda=\mathbb Z$, or if the action is irreducible, then $G$ interprets a simple group having a nontrivial action on a $\Lambda$-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 $\omega$-stable group. We will deal with "minimal" superstable groups of finite Lascar rank acting nontrivially on $\Lambda$-trees, where $\Lambda=\mathbb Z$ or $\Lambda=\mathbb R$. 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 $\Lambda$-tree is trivial, and $H_2/H_1$ is either soluble or simple and acts nontrivially on a $\Lambda$-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.