  A number of years ago, Kumar Murty pointed out to me that the computation of
the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the
computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly
similar. We puzzled over this, in particular over the role of elementary
matrices in both computations. We formulated a very general result on the
fundamental group of a Satake compactification of a locally symmetric space.
This lead to our joint paper [1] with Lizhen Ji and Les Saper on these
fundamental groups. Although the results in it were intriguingly similar to the
corresponding calculations of the congruence subgroup kernel of the underlying
algebraic group in [5], we were not able to demonstrate a direct connection
(cf. [1], ${\S}$7). The purpose of this note is to explain such a connection. A
covering space is constructed from inverse limits of reductive Borel-Serre
compactifications. The congruence subgroup kernel then appears as the group of
deck transformations of this covering. The key to this is the computation of
the fundamental group in [1].
