Fundamental group of locally symmetric varieties

We investigate the fundamental group of a compactified locally symmetric variety after resolution of singularities. We show that it can be any finite group and give examples to show that even for Siegel modular threefolds it can be complicated. We also calculate some cases of special interest, including the fundamental group of the Barth-Nieto space. This article was processed by the author using the Springer-Verlag TEX P Jourlg macro package 1991.