Horocycle orbit-closure rigidity beyond the rigid acylindrical setting (McMullen–Mohammadi–Oh, 2016)

In 'Horocycles in hyperbolic 3-manifolds' (Geom. Funct. Anal. 2016), McMullen, Mohammadi and Oh classified the closures of horocycles in convex cocompact 'rigid' acylindrical hyperbolic 3-manifolds — those whose convex core is compact with totally geodesic boundary: every horocycle closure is a properly immersed submanifold (a closed or properly embedded horocycle, a surface parallel to a totally geodesic plane, or the entire manifold). They closed by posing the problem of developing a comparable rigidity theory for horocycle dynamics when the convex core boundary is bent ('non-rigid' acylindrical manifolds), where the scattering mechanism off totally geodesic surfaces fails and the behavior of horocycles can be radically different from the rigid case, and more generally for other infinite-volume hyperbolic 3-manifolds. Subsequent work has extended the classification only within the totally geodesic (rigid) world: to geometrically finite manifolds with round Sierpiński limit sets (limit sets whose complement is a union of round disks; Kim–Lee 2025) and to convex cocompact manifolds with Fuchsian ends (totally geodesic boundary) in all dimensions (Lee–Oh 2024); for the neighboring geodesic-plane problem, the general acylindrical case is settled inside the convex core but exhibits chaotic and exotic closures outside it. The classification of one-parameter unipotent orbit closures for bent acylindrical 3-manifolds — including whether all horocycle closures are properly immersed submanifolds — remains open.

Background

A hyperbolic 3-manifold is a quotient $M=\Gamma\backslash\mathbb{H}^3$ of hyperbolic 3-space by a torsion-free discrete group $\Gamma$ of isometries (a Kleinian group). The limit set $\Lambda\subset\partial\mathbb{H}^3=\mathbb{C}\cup\{\infty\}$ is the set of accumulation points of any orbit of $\Gamma$, and the convex core $\mathrm{core}(M)=\Gamma\backslash\mathrm{hull}(\Lambda)$ is the quotient of the convex hull of $\Lambda$. $M$ is convex cocompact when $\mathrm{core}(M)$ is compact, and geometrically finite when a unit neighborhood of $\mathrm{core}(M)$ has finite volume; in both cases $M$ may have infinite volume. A 3-manifold is called acylindrical when its compact core has incompressible boundary (each boundary component injects into the fundamental group of the core) and every essential cylinder in it is boundary-parallel (isotopic into the boundary) — a purely topological condition that singles out a large and structurally tame class of Kleinian groups. Following McMullen–Mohammadi–Oh, a convex cocompact acylindrical $M$ of infinite volume is called rigid acylindrical if the boundary of its convex core is compact and totally geodesic; equivalently (in the geometrically finite case) the limit set is a round Sierpiński gasket, i.e. the complement of $\Lambda$ in the Riemann sphere is a countable union of round open disks with disjoint closures.

A horocycle is an isometrically immersed copy of $\mathbb{R}$ in $M$ with constant geodesic curvature $1$ and zero torsion, i.e. one that lies in an immersed totally geodesic plane; it arises as a limit of planar circles whose centers escape to infinity. The natural home for studying their closures is the frame bundle: with $G=\mathrm{PGL}_2(\mathbb{C})\simeq\mathrm{Isom}^+(\mathbb{H}^3)$ one has $FM\simeq\Gamma\backslash G$, and every oriented horocycle lifts uniquely to an orbit of the one-parameter unipotent subgroup $U=\{n_s:s\in\mathbb{R}\}$, where $n_z$ denotes unipotent translation by $z$ inside a horosphere. The full horospherical subgroup is $N=\{n_z:z\in\mathbb{C}\}$, its complementary one-parameter subgroup is $V=\{n_{is}:s\in\mathbb{R}\}$, the diagonal subgroup $A$ generates the frame flow, and $H=\mathrm{PSL}_2(\mathbb{R})$ corresponds to totally geodesic planes. The renormalized frame bundle $\mathrm{RF}_+M\subset FM$ consists of frames whose forward geodesic remains within a bounded distance of the convex core; outside it the dynamics is elementary, so the problem lives on $\mathrm{RF}_+M$.

When $M$ has finite volume, Ratner's theorem (classifying orbit closures of subgroups generated by unipotent elements on finite-volume quotients $\Gamma\backslash G$) implies that every $U$-orbit closure is homogeneous, so horocycle closures are properly immersed submanifolds. In infinite volume, unipotent orbits need not be recurrent: the orbit closures of the maximal horospherical subgroup $N$ were classified for all geometrically finite hyperbolic manifolds by Dal'bo (surfaces), Ferte (dimension 3) and Winter (general rank one), but the one-parameter subgroup $U$ is non-maximal unipotent and strictly harder. McMullen–Mohammadi–Oh (Geom. Funct. Anal. 2016) proved the first such classification beyond lattices: for $M$ rigid acylindrical, the closure of any horocycle is either a properly immersed 1-manifold ($\chi$ closed or properly embedded), a properly immersed 2-manifold parallel to a totally geodesic surface, or all of $M$; equivalently, for any frame $x\in FM$ the closure $xU$ is closed, equals $xvHv^{-1}\cap\mathrm{RF}_+M$ for some $v\in V$, or equals $\mathrm{RF}_+M$. The proof rests on a scattering dichotomy: a horocycle limiting on a properly embedded totally geodesic surface is either trapped in it or scattered by it, and every recurrent horocycle accumulates on such a surface.

The paper's closing remark observes that both ingredients fail for manifolds that are convex cocompact and acylindrical but not rigid: the convex core boundary is then bent along a measured bending lamination (the measured lamination on $\partial\,\mathrm{core}(M)$ recording the transverse bending angles between adjacent faces of the convex hull boundary), there are no totally geodesic surfaces to scatter off, and new behavior already appears — a horocycle orthogonal to a closed leaf of the bending lamination can be properly embedded, producing a frame with a compact $A$-orbit and a non-recurrent $U$-orbit.

Subsequent progress has stayed inside the totally geodesic (rigid) world. Lee–Oh (Geom. Topol. 2024) classified orbit closures of every unipotent-generated subgroup for convex cocompact hyperbolic $d$-manifolds with Fuchsian ends (totally geodesic core boundary), all $d\ge 2$; for $d=3$ this recovers the McMullen–Mohammadi–Oh theorem. Kim–Lee (Groups, Geometry and Dynamics, 2025) removed convex cocompactness, classifying horocycle orbit closures for geometrically finite manifolds with round Sierpiński limit sets. For the neighboring problem of geodesic planes ($H$-orbits): inside the convex core interior of any acylindrical manifold, closed-or-dense is known (McMullen–Mohammadi–Oh, Duke Math. J. 2022, convex cocompact; Benoist–Oh, Ergodic Theory Dynam. Systems 2022, geometrically finite), while outside the core Zhang (Math. Res. Lett. 2023) exhibited an exotic plane ($P^*$ closed in the core interior while $P$ is not closed in $M$) and Torkaman–Zhang (Adv. Math. 2024) completely classified plane closures there, governed by exotic rays of the bending lamination. In adjacent infinite-volume settings rigidity is known to fail: per Oh's ICM 2026 survey, bent quasi-Fuchsian manifolds (quasiconformal deformations of Fuchsian surface lattices, with Jordan-curve limit sets) contain geodesic planes with chaotic fractal closures (a phenomenon from McMullen–Mohammadi–Oh, Invent. Math. 2017); Farre–Landesberg–Minsky classified horocycle orbit closures in $\mathbb{Z}$-covers (infinite cyclic covers) of compact hyperbolic surfaces, where non-maximal closures are fractal of integer Hausdorff dimension; and Dey–Oh constructed fractal geodesic-plane closures in higher-rank Hitchin manifolds (quotients of higher-rank symmetric spaces by Hitchin representations of surface groups). These adjacent failures shape expectations but do not address the acylindrical horocycle question.

Problem Statement

Let $M=\Gamma\backslash\mathbb{H}^3$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume that is not rigid, i.e. the boundary of the compact convex core $\mathrm{core}(M)$ is a bent surface carrying a nontrivial measured bending lamination (the measured lamination recording the bending angles of $\partial\,\mathrm{core}(M)$). Determine the possible horocycle closures in this non-rigid setting, the question left open by McMullen–Mohammadi–Oh (2016). Concretely:

(1) Classify the orbit closures of the one-parameter unipotent subgroup $U=\{n_s:s\in\mathbb{R}\}\subset\mathrm{PGL}_2(\mathbb{C})$ — the subgroup whose orbits in the frame bundle $FM\simeq\Gamma\backslash\mathrm{PGL}_2(\mathbb{C})$ are the lifts of horocycles — acting on $FM$, at least for frames in the renormalized frame bundle $\mathrm{RF}_+M$ (the set of frames whose forward geodesic remains within a bounded distance of the convex core), and equivalently the closures $\overline{\chi}\subset M$ of horocycles $\chi\subset M$.

(2) In particular, determine whether an analogue of the rigid-case trichotomy holds — for rigid $M$, the closure of a horocycle is either a closed or properly embedded horocycle, a properly immersed surface parallel to a totally geodesic plane, or all of $M$: is every horocycle closure in such a manifold a properly immersed submanifold, or do new phenomena arise (e.g. closures that are fractal, chaotic, or otherwise not of rigid type)? If new phenomena occur, describe them completely. Any answer must be consistent with the source example: a horocycle orthogonal to a closed leaf of the bending lamination can be properly embedded, giving a frame with compact $A$-orbit and non-recurrent $U$-orbit.

(3) Beyond the convex cocompact bent setting required in (1)–(2), the source frames this within the broader problem of understanding horocycle dynamics in other infinite-volume hyperbolic 3-manifolds; the natural next case, explicitly stated as open by Kim–Lee (2025), is the same classification for a general geometrically finite acylindrical hyperbolic 3-manifold (bent core boundary, possibly with cusps). This broader branch is recorded as programmatic context, not as a required deliverable of (1)–(2).

References

1. Curtis T. McMullen, Amir Mohammadi, Hee Oh: Horocycles in hyperbolic 3-manifolds. Geometric and Functional Analysis 26(3):961–973, 2016. DOI: 10.1007/s00039-016-0373-8. URL: https://doi.org/10.1007/s00039-016-0373-8 (author PDF: https://people.math.harvard.edu/~ctm/home/text/papers/horo/horo.pdf)
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