Geometry and Topology Seminar

Let M be a closed hyperbolic manifold. A sequence of closed, smoothly immersed hypersurfaces in M (up to homotopy and commensurability) is called asymptotically geodesic if the hypersurfaces are eventually totally geodesic. We show that if M contains such a sequence, its fundamental group is virtually special and thus a linear group with integer coefficients. If, in addition, M is arithmetic of type I, we build a sequence of asymptotically geodesic, strongly filling hypersurfaces that are equidistributing in the Grassmann bundles, where strongly filling implies a hypersurface having essential intersection with every geodesic.

This partially answers a question of Ben Lowe regarding the gap of hypersurfaces in higher dimensional hyperbolic manifolds and is a joint work with Ruojing Jiang.