Completeness in Zariski Groups

Zariski groups are @0-stable groups with an axiomatically given Zariski topology and thus abstract generalizations of algebraic groups. A large part of algebraic geometry can be developed for Zariski groups. As a main result, any simple smooth Zariski group interprets an algebraically closed field, hence is almost an algebraic group over an algebraically closed field. Introduction Model theory came naturally across the notion of @ 0 -stable groups of finite Morley rank. These are groups where a finite dimension is assigned to all definable sets that behaves much as dimension in algebraic geometry. In fact, these groups share many properties with algebraic groups. It was even conjectured that they were essentially algebraic groups. Removing trivial obstacles, Cherlin's conjecture states that a simple @ 0 -stable group (of finite rank) is an algebraic group over an algebraically closed field. The theory now is far developed: see [BN] for an algebraic and [P2] for a more model theoretic ..