The greatest common quotient of Borel-Serre and the toroidal compactifications of locally symmetric spaces (Q1277078)

Let \(G={\mathcal G}(\mathbb{R})\), where \({\mathcal G}\) is a simple linear algebraic group over \(\mathbb{Q}\). Assume that the symmetric space \(X=G/K\) attached to \(G\) is Hermitian, and let \(\Gamma\) be an arithmetic subgroup of \({\mathcal G} (\mathbb{Q})\). The space \(\Gamma\setminus X\) admits several interesting compactifications. There is the Bailey-Borel compactification \(\overline {\Gamma \setminus X}^{BB}\) which is a projective variety with singularities, the Borel-Serre compactification \(\overline{\Gamma \setminus X}^{BS}\), which is a manifold with corners being homotopy equivalent to \(\Gamma\setminus X\). Further, there are various toroidal compactifications \(\overline{\Gamma \setminus X}^{\text{tor}}\), depending on a local geometric construction, which are varieties with toric singularities. The Borel-Serre compactification and all toroidal compactifications dominate the Bailey-Borel compactification, so we have the following picture: \[ \begin{matrix} \overline {\Gamma\setminus X}^{BB} & & & & \overline {\Gamma \setminus X}^{\text{tor}}\\ & \searrow & & \swarrow\\ & & \overline {\Gamma \setminus X}^{BB} \end{matrix}. \] Fix a toroidal compactification \(\overline{\Gamma \setminus X}^{\text{tor}}\). In [Invent. Math. 116, 243-308 (1994; Zbl 0860.11031)] \textit{M. Harris} and \textit{S. Zucker} conjectured that \(\overline{\Gamma \setminus X}^{BB}\) is indeed the maximal compactification dominated by the Borel-Serre compactification and \(\overline{\Gamma\setminus X}^{\text{tor}}\). In the paper under review, it is shown that the maximal compactification dominated by both, written \(\overline {\Gamma \setminus X}^{BB} \wedge\overline {\Gamma\setminus X}^{\text{tor}}\), exists and is independent of the choice of \(\overline{\Gamma \setminus X}^{\text{tor}}\). Indeed, the author gives a direct construction of this compactification. Using this, he shows that if \({\mathcal G}\) is not absolutely simple then the conjecture of Harris and Zucker holds. Further, if \({\mathcal G}\) is \(\mathbb{Q}\)-split then the conjecture does not hold.