  In this paper we initiate a systematic study of the abstract commensurators
of profinite groups. The abstract commensurator of a profinite group $G$ is a
group $Comm(G)$ which depends only on the commensurability class of $G$. We
study various properties of $Comm(G)$; in particular, we find two natural ways
to turn it into a topological group. We also use $Comm(G)$ to study topological
groups which contain $G$ as an open subgroup (all such groups are totally
disconnected and locally compact). For instance, we construct a topologically
simple group which contains the pro-2 completion of the Grigorchuk group as an
open subgroup. On the other hand, we show that some profinite groups cannot be
embedded as open subgroups of compactly generated topologically simple groups.
Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong
rigidity theorem for simple algebraic groups defined over local fields and the
Neukirch-Uchida theorem, can be reformulated as structure theorems for the
commensurators of certain profinite groups.
