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Since the pioneering work of McMullen–Mohammadi–Oh, there have been many exciting developments concerning rigidity phenomena for totally geodesic submanifolds in rank-one locally symmetric spaces of infinite volume. In general, whether one can expect such rigidity depends strongly on the geometry of the ambient manifold. 

In recent joint work with Oh, we show that a strong form of rigidity hold for totally geodesic submanifolds contained in the convex core of a geometrically finite rank-one manifold M of infinite volume: every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed, has finite volume, and there are only finitely many such submanifolds. Moreover, any totally geodesic submanifold of M containing a maximal totally geodesic submanifold of finite volume (and of dimension at least two) is either properly immersed or dense in M.