Automorphism groups of non-Archimedean groups
Abstract
Let $\operatorname{Aut}(G)$ denote the group of (bi-)continuous automorphisms of a non-Archimedean Polish group $G$. We show that for any such $G$ with an invariant countable basis of open subgroups, $\operatorname{Aut}(G)$ carries a unique Polish topology that makes its natural action on $G$ continuous. Furthermore, for any class of groups allowing a Borel assignment of such bases, there is a functorial duality to a class of countable groupoids with a meet operation, extending work of the authors with Tent [J. Math. Logic, 2022]. This provides an alternative description of the topology of $\operatorname{Aut}(G)$. The results hold for instance for the class of locally Roelcke precompact non-Archimedean groups, which contains most classes studied previously. We further provide a model-theoretic proof that the outer automorphism group $\operatorname{Out}(G)$ of an oligomorphic group $G$ is locally compact, a result originally due to Paolini and the first author [arXiv:2410.02248].
