  Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic
lattice. We show that for reductive subgroups $H$ of $G$ that is large enough,
the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact
set. As a consequence, we deduce finiteness result 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.
