Arithmeticity totally geodesic Lie foliations with locally symmetric leaves (Q1002436)

Let \(\mathcal F\) be a smooth foliation on a compact Riemannian manifold \(M\) whose leaves are totally geodesic submanifolds isometrically covered by an irreducible symmetric space \(X\) of noncompact type with rank at least \(2\), and at least one of them is dense. Assume further that \(\mathcal F\) admits a transverse Lie structure, namely its transverse structure is locally modeled on a fixed connected Lie group \(H\). Then the author proves that \(M\) is finitely covered by an arithmetic manifold \(\widehat M\), and that the corresponding foliation \(\widehat{\mathcal{F}}\) of \(\widehat{M}\) is an arithmetic homogeneous Lie foliation. This means that the Lie group \(H\) is semisimple with finite center, and there exists an arithmetic, irreducible lattice \(\Gamma\) in \(H\times \text{Isom}(X)\), such that the foliation \(\widehat{\mathcal{F}}\) on \(\widehat{M}\) is diffeomorphic to the natural foliation of \((H \times X)/\Gamma\) by copies of \(X\), under a diffeomorphism that respects both the leafwise Riemannian metric and the transverse Lie structure.