Realizing Non-Archimedean Polish Groups as Outer Automorphism Groups

Abstract

We prove that every non-Archimedean Polish group is topologically isomorphic to the outer automorphism group of a countable discrete group. Furthermore, our construction is Borel. This implies that determining whether two countable discrete groups have topologically isomorphic outer automorphism groups is at least as complicated as classifying non-Archimedean Polish groups up to topological isomorphism, and so in particular not classifiable by countable structures. Our construction is based on the theory of right-angled Coxeter groups. Along the way, we prove results of independent interest on the topological group $\mathrm{Aut}(W)$, for $W$ a countable right-angled Coxeter group. This has purely group-theoretic applications; in particular, we give a graph-theoretic characterization of when the group $\mathrm{Spe}(W)$ of special automorphisms of $W$ coincides with $\mathrm{Inn}(W)$. Combined with previous work, this gives a characterization of when $\mathrm{Aut}(W)$ is inner-by-graph, for $W$ countable.