Automorphisms of $\kappa$-existentially closed groups

Abstract

We investigate the automorphisms of some $\kappa$- existentially closed groups. In particular, we prove that $Aut(G)$ is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^{\kappa }$ whenever $\kappa$ is inaccessible and $G$ is the unique $\kappa$-existentially closed group of cardinality $\kappa$. Indeed, the latter result is a byproduct of an argument showing that, for any uncountable $\kappa$ and any group $G$ that is the limit of regular representation of length $\kappa$ with countable base, we have $|Aut(G)|={\beth }_{\kappa +1}$, where $\beth$ is the beth function. Such groups are also $\kappa$-existentially closed if $\kappa$ is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.