Nondivergence on homogeneous spaces and rigid totally geodesics

TL;DR

This paper proves that large enough reductive subgroups in arithmetic quotients of semisimple Lie groups have orbits intersecting fixed compact sets, leading to finiteness results for certain rigid totally geodesic submanifolds.

Contribution

It generalizes previous results by establishing nondivergence of orbits for larger classes of subgroups and deriving finiteness of rigid totally geodesic submanifolds.

Findings

Orbits of large enough reductive subgroups intersect fixed compact sets.

Finiteness of non-deformable, volume-bounded totally geodesic submanifolds.

Extension of previous nondivergence and rigidity results.

Abstract

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.