Dynamics of totally geodesic submanifolds.

Abstract: I will discuss two results on totally geodesic submanifolds.  In joint work with Minju Lee, we show that in a geometrically finite rank one manifold 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 of them.  In joint work with Subhadip Dey, we construct the first higher rank examples where rigidity fails: we exhibit a Zariski dense surface subgroup $\Gamma<SL(3,\mathbb Z)$ such that the locally symmetric space $\Gamma\backslash SL(3,\mathbb R)/SO(3)$ contains a sequence of geodesic planes whose closures are fractal, with Hausdorff dimensions accumulating to 2.