On superstable CSA-groups

We show that an existentially closed CSA -group is not superstable. We prove that a non-abelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. We also show that a model of the universal theory of non-abelian free groups is not simple. This result combined with the above gives a new proof of a result of E. Mustafin and B. Poizat [12] which states that a superstable model of the universal theory of non-abelian free groups is abelian. We end by showing that a superstable torsion-free hyperbolic group is cyclic.

• Type: Journal
• Year: 2008
• Where: APAL
• Authors: Abderezak Ould Houcine
• Keywords: APAL 2008, Definable Simple Subgroup, Non-abelian Free Groups, Non-abelian Superstable Csa-group