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 HNNextension 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 nilpotentbyfinite 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2008
 DOI:
 10.48550/arXiv.0809.3441
 arXiv:
 arXiv:0809.3441
 Bibcode:
 2008arXiv0809.3441O
 Keywords:

 Mathematics  Logic;
 Mathematics  Group Theory;
 03C99;
 20F65;
 20E08
 EPrint:
 2 figures