Parametrizing Shimura subvarieties of ${\mathrm{A}}_1$ Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = ($\mathbf{H}$^2)^a \times ($\mathbf{H}$^3)^b$. A special case describes all Shimura subvarieties of type ${\mathrm{A}}_1$ Shimura varieties. We produce, for any $n\ge 1$, examples of manifolds/Shimura varieties with precisely $n$ commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic $3$-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.