[1902.06621] Orbital closure of a unipotent flow of a hyperbolic manifold with a Fuchsian edge

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overview: Ratner’s orbit closure for the connected closed subgroup generated by the unipotent elements of $\operatorname{SO}(d,1)$ acting on the space $\Gamma\backslash \operatorname{SO}(d,1) We establish an analogue of the theorem. Suppose $, the associated hyperbolic manifold $M=\Gamma\backslash \mathbb H^d$ , is a convex cocompact manifold with a Fuchsian edge. For $d=3$ this was previously proved by McMullen, Mohammadi, and Oh. For higher dimensions, the possibility of accumulation in closed orbits of the intermediate group poses a very serious obstacle, and overcoming these by means of the avoidance theorem (Theorem 7.13) is the crux of this paper. Our result means that: for any $k\ge 1$ ,

(1) The closure of any $k$-holosphere in $M$ is a properly immersed submanifold.

(2) The closure of any geodesic $(k+1)$ plane in $M$ is a well-immersed submanifold.

(3) The infinite sequence of maximal well-embedded geodesic $(k+1)$ planes intersecting $\operatorname{core}M$ is dense at $M$.