Modular subvarieties of arithmetic quotients of bounded symmetric domains

A reductive Q-simple algebraic group G is of hermitian type, if the symmetric space D defined by G(R) is a hermitian symmetric space. A discrete subgroup Γ ⊂ G(Q) is arithmetic, if for some faithful rational representation ρ : G −→ GL(V ), and for some lattice VZ ⊂ VQ, Γ is commensurable with ρ−1(GL(VZ)). A non-compact hermitian symmetric space is holomorphically equivalent to a bounded symmetric domain. If this is the case, Γ acts on D preserving the natural Bergmann metric, and XΓ = Γ\D is, if Γ is torsion-free, a complex manifold, in general not compact (it is compact exactly when G is anisotropic). We call spaces XΓ arithmetic quotients of bounded symmetric domains even when Γ has torsion; it is known that in this case XΓ is a V -manifold (in the sense of Satake), locally the quotient of a smooth space by a finite group action. There is a natural compactification of XΓ, the Satake compactification X∗ Γ, which has the property: the complement X ∗ Γ −XΓ is a finite disjoint union of arithmetic quotients of bounded symmetric domains of lower dimension. A natural problem in this respect is to consider, in addition to the data above, a symmetric subdomain D′ ⊂ D which has the property that the restriction of the action of Γ to D′ is discrete, say by a discrete subgroup Γ′, resulting in a commutative square D′ →֒ D ↓ ↓ XΓ′ = Γ ′\D′ →֒ Γ\D = XΓ.