Non-universality of automorphism groups of uncountable ultrahomogeneous structures

TL;DR

This paper demonstrates that automorphism groups of various uncountable ultrahomogeneous structures are generally not universal topological groups, contrasting with the countable case, and offers a shorter proof of a related known result.

Contribution

It extends non-universality results to a broader class of uncountable ultrahomogeneous structures and provides a more direct proof of a key existing theorem.

Findings

Automorphism groups of uncountable ultrahomogeneous structures are rarely universal.

Contrasts with the universality of automorphism groups in countable structures.

Provides a shorter proof of Mbombo-Pestov's non-universality result.

Abstract

In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $\kappa$, for uncountable $\kappa$ such that $\kappa^{<\kappa}=\kappa$, is not a universal topological group of weight $\kappa$. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of countable (separable) ultrahomogeneous structures. We also provide a more direct and shorter proof of the Mbombo-Pestov's result.