Homomorphisms into totally disconnected, locally compact groups with dense image

TL;DR

This paper characterizes homomorphisms from a group into totally disconnected, locally compact groups with dense image, showing they are essentially completions with specific uniformities, extending properties known from profinite completions.

Contribution

It generalizes the concept of profinite completions to the locally compact setting, providing a structural description of such homomorphisms and the groups involved.

Findings

Homomorphisms are completions with respect to specific uniformities.

The target group is determined up to a compact normal subgroup by the preimage of a compact open subgroup.

Results extend known properties of profinite completions to locally compact groups.

Abstract

Let $\phi: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $\phi$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to uniformities of a particular kind. Moreover, $H$ is determined up to a compact normal subgroup by the pair $(G,\phi^{-1}(L))$, where $L$ is a compact open subgroup of $H$. These results generalize the well-known properties of profinite completions to the locally compact setting.