Nondivergence on homogeneous spaces and rigid totally geodesic submanifolds

Let G/Γ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups H of G that are large enough, the orbits of H on G/Γ intersect nontrivially with a fixed compact set. As a consequence, we deduce finiteness results for totally geodesic submanifolds of arithmetic quotients of symmetric spaces that do not admit nontrivial deformation and with bounded volume. Our work generalizes previous work of Tomanov–Weiss and Oh on this topic.