On superstable CSA-groups

We prove that a non-abelian superstable CSA-group has an infinite definable simple subgroup all of whose proper definable subgroups are abelian. This imply in particular that the existence of non-abelian CSAgroup of finite Morley rank is equivalent to the existence of a simple bad group all whose definable proper subgroups are abelian. We give a new proof of a result of E. Mustafin and B. Poizat [8] which states that a superstable model of the universal theory of non-abelian free groups is abelian. We deduce also that a superstable torsion-free hyperbolic group is cyclic. We close the paper by showing that an existentially closed CSA∗-group is not superstable.