Generalised Hrushovski Constructions and Generic Automorphisms

Hrushovski's amalgamation method is a powerful tool to construct (finite rank) structures with interesting combinatorial geometries. Ehud Hrushovski first used it in [2]Ê to construct a counter-example to Zilber's Trichotomy Conjecture, then to fuse two strongly minimal theories into a single one. The method later served to construct various unexpected expansions of algebraically closed fields, most recently a bad field in characteristic 0 [1]. It is useful to divide the method into two steps: the free amalgamation, where a theory (usually of infinite rank) is obtained; then the collapse of onto the desired finite rank theory.

We will focus on generalisations of the free amalgamation method, in particular the free fusion of two simple rank 1 theories over a common -categorical reduct. In interesting cases, the resulting theory is supersimple [3]. The situation is similar to the expansion of some (stable) theory by a generic automorphism.

In important algebraic contexts, e.g. for algebraically closed fields, the class of models together with a generic automorphism is axiomatisable. We show that the same is true for some of the infinite rank structures obtained by Hrushovski's free amalgamation method, in particular for Poizat's "bad field of infinite rank".