Superstable groups acting on trees

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 $Λ=\mathbb Z$ or $Λ=\mathbb R$, and if $G$ is either $α$-connected and $Λ=\mathbb Z$, 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 $Λ=\mathbb Z$ or $Λ=\mathbb R$. 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 $Λ$-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.