Fundamental groups of toroidal compactifications

Locally symmetric varieties are quotients of a hermitian symmetric space by a suitable discrete group. They are in general not compact, but there are well-known techniques for compactifying them. The resulting spaces often have major geometric siginificance, so it is important to understand their geometry and topology. I will focus on the most basic topological invariant, the fundamental group. This should be a quotient of the discrete group, but it is not immediately clear which quotient. I will describe joint with with Azniv Kasparian (Sofia) in which we answer this question completely, by largely geometric methods but giving an answer in Lie-theoretic terms.