Automorphism groups of non-Archimedean groups

Abstract

Let $\Aut(G)$ denote the group of (bi-)continuous automorphisms %and $\Out(G)$ the outer automorphism group of a non-Archimedean Polish group~$G$. We show that for any such $G$ with an invariant countable basis of open subgroups, the group $\Aut(G)$ 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 (Coarse groups, and the isomorphism problem for oligomorphic groups, Journal of Mathematical Logic, 2021). This provides an alternative description of the topology of $\Aut(G)$. 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 $\Out(G)$ of an oligomorphic group $G$ is locally compact, a result due to Paolini and the first author (arXiv:2410.02248).