  Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there
is a totally disconnected, locally compact (t.d.l.c.) group $\hat{G}_K$ that
contains the profinite completion of $K$ as an open compact subgroup and also
contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover,
given an arbitrary group $G$, then every t.d.l.c. group containing an image of
$G$ as a dense subgroup can be realised as a quotient of $\hat{G}_K$ for some
commensurated subgroup $K$.
