Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Lee, Minju; Oh, Hee

doi:10.2140/gt.2024.28.3373

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Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Minju Lee and Hee Oh

Geometry & Topology 28 (2024) 3373–3473

DOI: 10.2140/gt.2024.28.3373

Abstract

We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $SO (d, 1)$ acting on the space $Γ ∖ SO (d, 1)$ , assuming that the associated hyperbolic manifold $ℳ = Γ ∖ ℍ^{d}$ 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 $k \geq 1$ ,

the closure of any $k$ –horosphere in $ℳ$ is a properly immersed submanifold;
the closure of any geodesic $(k + 1)$ –plane in $ℳ$ is a properly immersed submanifold;
an infinite sequence of maximal properly immersed geodesic $(k + 1)$ –planes intersecting $core ℳ$ becomes dense in $ℳ$ .

Keywords

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

Mathematical Subject Classification

Primary: 37A17

Secondary: 22E40

References

Publication

Received: 3 September 2022

Revised: 23 February 2023

Accepted: 24 March 2023

Published: 25 November 2024

Proposed: David Fisher

Seconded: Anna Wienhard, Benson Farb

Authors

Minju Lee
Department of Mathematics Yale University New Haven, CT United States	Department of Mathematics University of Chicago Chicago, IL United States

Hee Oh
Department of Mathematics Yale University New Haven, CT United States

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