Automorphism groups of non-Archimedean groups

TL;DR

This paper investigates the automorphism groups of non-Archimedean Polish groups, establishing a unique Polish topology, exploring dualities with groupoids, and proving local compactness of outer automorphism groups in oligomorphic cases.

Contribution

It introduces a unique Polish topology on automorphism groups, develops a functorial duality with groupoids, and provides a model-theoretic proof of local compactness for outer automorphism groups.

Findings

Automorphism groups have a unique Polish topology compatible with their action.

A functorial duality links automorphism groups to countable groupoids with meet operations.

Outer automorphism groups of oligomorphic groups are locally compact.

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).