Automorphisms of $κ$-existentially closed groups

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

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