On generic and supertight automorphisms

TL;DR

This paper demonstrates that generic automorphisms in stable groups are inherently supertight, establishing their existence and exploring their relationship with automorphisms of algebraic groups and the principle that fixed points are pseudofinite.

Contribution

It proves the existence of supertight automorphisms in stable groups and clarifies their connection to automorphisms of algebraic groups like PGL2, advancing understanding of automorphism structures.

Findings

Generic automorphisms of stable groups are supertight.

Established a link between supertight automorphisms of PGL2(K) and automorphisms of K.

Provided evidence supporting the 'fixed points are pseudofinite' principle in simple groups.

Abstract

We show that generic automorphisms of stable groups are supertight in a strong sense. In particular, we obtain the existence of supertight automorphisms. We also answer a question concerning the relationship between supertight automorphisms of $\mathrm{PGL}_2(K)$ and generic automorphisms of the underlying field $K$. Moreover, we provide partial evidence-already suggested by Hrushovski-toward the principle that ``fixed points are pseudofinite'' in the setting of generic automorphisms of simple groups of finite Morley rank.