Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Lee, Minju¹ ; Oh, Hee²

¹ Department of Mathematics, Yale University, New Haven, CT, United States, Department of Mathematics, University of Chicago, Chicago, IL, United States
² Department of Mathematics, Yale University, New Haven, CT, United States

Geometry & topology, Tome 28 (2024) no. 7, pp. 3373-3473.

Résumé

We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma \setminus \mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $\mathcal{ℳ}=\Gamma \setminus {ℍ}^d$ 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 subgroups causes serious issues, but, in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\ge 1$,

the closure of any $k$–horosphere in $\mathcal{ℳ}$ is a properly immersed submanifold;

the closure of any geodesic $(k+1)$–plane in $\mathcal{ℳ}$ is a properly immersed submanifold;

an infinite sequence of maximal properly immersed geodesic $(k+1)$–planes intersecting $\mathrm{core}\mathcal{ℳ}$ becomes dense in $\mathcal{ℳ}$.

Détail

DOI : 10.2140/gt.2024.28.3373

Keywords: orbit closures, geodesic planes, unipotent flows, hyperbolic manifolds, rigidity

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