Non-universality of automorphism groups of uncountable ultrahomogeneous structures

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.