An unusual example of a universal automorphism group

TL;DR

This paper presents an example of a Fraïssé structure with a universal automorphism group that does not have a group-extensible $ ext{omega}$-age, illustrating the non-equivalence of these properties.

Contribution

It provides the first known example demonstrating that having a universal automorphism group does not imply group-extensible $ ext{omega}$-age.

Findings

Constructed a Fraïssé structure with a universal automorphism group

Showed the $ ext{omega}$-age of this structure is not group-extensible

Established the non-equivalence of the two properties

Abstract

Let $M$ be a Fra\"{i}ss\'{e} structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $\omega$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $\omega$-age of $M$, there is an embedding $\textrm{Aut}(A) \to \textrm{Aut}(M)$, and we say that $M$ has group-extensible $\omega$-age if, for each $A$ in the $\omega$-age of $M$, there is an embedding $A \to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $\omega$-age, then $M$ has a universal automorphism group. We give an example of a Fra\"{i}ss\'{e} structure with a universal automorphism group whose $\omega$-age is not group-extensible, showing that the above two properties are not equivalent.