Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Abstract: We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\operatorname{SO}(d,1)$ acting on the space $\Gamma\backslash \operatorname{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash \mathbb H^d$ is a convex cocompact manifold with Fuchsian ends. For $d=3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate groups causes very serious obstacles, and surmounting these via the avoidance theorem (Theorem 7.13) is the heart of this paper. Our results imply the following: for any $k\ge 1$, (1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold; (2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold; (3) any infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\operatorname{core} M$ becomes dense in $M$.