Homomorphisms into totally disconnected, locally compact groups with dense image

Abstract

Let $ϕ: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $ϕ$ 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 $(G,ϕ^{-1}(L))$, where $L$ is a compact open subgroup of $H$. These results generalize the well-known properties of profinite completions to the locally compact setting.