Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

TL;DR

This paper extends Ratner's orbit closure theorem to hyperbolic manifolds with Fuchsian ends, showing that certain orbit closures are properly immersed submanifolds and that sequences of geodesic planes can become dense.

Contribution

It generalizes orbit closure results to higher dimensions with Fuchsian ends, overcoming obstacles via the avoidance theorem.

Findings

Closure of any k-horosphere is a properly immersed submanifold

Closure of any geodesic (k+1)-plane is a properly immersed submanifold

Sequences of maximal geodesic (k+1)-planes intersecting the core become dense

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 $M$.