Automorphisms of $\kappa$-existentially closed groups

TL;DR

This paper studies the automorphism groups of certain large, highly homogeneous groups called $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{kappa}}}}}$-existentially closed groups, revealing their structure and cardinality under specific set-theoretic conditions.

Contribution

It characterizes the automorphism groups of $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{kappa}}}}}$-existentially closed groups, showing their union structure and exact size when $oldsymbol{oldsymbol{ ext{kappa}}}$ is inaccessible.

Findings

$Aut(G)$ is the union of level preserving automorphisms.

$|Aut(G)|=2^{oldsymbol{ ext{kappa}}}$ for inaccessible $oldsymbol{ ext{kappa}}$.

For certain groups, $|Aut(G)|=eth_{oldsymbol{ ext{kappa}}+1}$.

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.