On Generic Automorphisms

Abstract

In this article we investigate generic automorphisms of countable models. Hodges et al. 1993 introduces the notion of SI (small index) generic automorphisms. They used the existence of small index generics to show the small index property of the model. Truss 1989 defines the notion of Truss generic automorphisms. An automorphism f ofM is called Truss generic if its conjugacy class is comeagre in the automorphism group ofM. We study the relationship between these two types of generic automorphisms. We show that either the countable random graph or a countable arithmetically saturated model of True Arithmetic have both SI generic and Truss generic automorphisms. We prove that the dense linear order has the small index property and Truss-generic automorphisms but it does not have SI generic automorphisms. We also construct an example of a countable structure which has SI generics but it does not have Truss generics.