Horocycles in hyperbolic 3-manifolds

Curtis T. McMullen, Amir Mohammadi and Hee Oh

22 November 2015

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Configuration spaces and double cosets . . . . . . . . . . . . . 64 Moving to the renormalized frame bundle . . . . . . . . . . . 85 Exceptional frames . . . . . . . . . . . . . . . . . . . . . . . . 106 Classification of U–orbit closures . . . . . . . . . . . . . . . . 117 Classification of AU–orbit closures . . . . . . . . . . . . . . . 13

Research supported in part by the Alfred P. Sloan Foundation (A.M.) and the NSF.

1 Introduction

Let M = Γ\H3 be a complete hyperbolic 3-manifold. A horocycle χ ⊂M isan isometrically immersed copy of R with zero torsion and geodesic curvature1. The torsion condition means that χ lies in an immersed totally geodesicplane.

One can regard χ as a limit of planar circles whose centers have movedoff to infinity. It is natural to ask what the possibilities are for its closure,

χ ⊂M.

When M has finite volume, it is well–known that strong rigidity propertieshold; e.g. χ is always a properly immersed, homogeneous submanifold of M[Sh], [Rn]. Continuing the investigation begun in [MMO], this paper showsthat rigidity persists for horocycles in certain infinite volume 3-manifolds.These are the first examples of Zariski dense discrete groups Γ ⊂ Isom(H3),other than lattices, where the topological behavior of horocycles in Γ\H3

has been fully described.

Horocycles in acylindrical manifolds. To state the main results, recallthat the convex core of M is given by:

core(M) = Γ\ hull(Λ) ⊂M,

where Λ ⊂ C is the limit set of Γ, and hull(Λ) ⊂ H3 is its convex hull. We sayM is a rigid acylindrical manifold if its convex core is a compact submanifoldof M with nonempty, totally geodesic boundary. Our first result describesthe behavior of horocycles in M .

Theorem 1.1 Let χ ⊂ M = Γ\H3 be a horocycle in a rigid acylindrical3-manifold. Then either:

1. χ ⊂M is a properly immersed 1-manifold; or

2. χ ⊂M is a properly immersed 2-manifold, parallel to a totally geodesicsurface S ⊂M ; or

3. χ is the entire 3-manifold M .

Corollary 1.2 The closure of any horocycle is a properly immersed sub-manifold of M .

1

Similar results for geodesic planes in M are obtained in [MMO].

Homogeneous dynamics. To make Theorem 1.1 more precise, we refor-mulate it in terms of the frame bundle FM →M .

Let G denote the simple, connected Lie group PGL2(C). Within G, wehave the following subgroups:

H = PSL2(R),

A =

(a 0

0 a−1

): a ∈ R+

,

K = SU(2)/(±I),

N =

ns =

(1 s

0 1

): s ∈ C

,

U = ns : s ∈ R, and

V = ns : s ∈ iR.

Upon identifying H3 with G/K, we obtain the natural identifications

FM ∼= Γ\G and M ∼= Γ\G/K.

Every (oriented) horocycle χ in M lifts to a unique unipotent orbit xU in theframe bundle FM . Let A+ =

(a 00 a−1

): a ≥ 1 be the positive semigroup

in A, and let

RF+M = x ∈ FM : xA+ ⊂ FM is compact.

This locus is closed and invariant under AN .Our main result may now be stated as follows (see §6).

Theorem 1.3 Let M = Γ\H3 be a rigid acylindrical 3-manifold. Then forany x ∈ FM , either

1. xU is closed;

2. xU = xvHv−1 ∩ RF+M for some v ∈ V ; or

3. xU = RF+M .

It is readily verified that these three alternatives give the three cases inTheorem 1.1, using the fact that the map FM → M is proper and itsrestriction to RF+M is surjective.

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Corollary 1.4 The closure of any U–orbit in RF+M is homogeneous, inthe sense that

xU = xS ∩ RF+M

for some closed subgroup S ⊂ G with U ⊂ S.

Indeed, we can take S = U , vHv−1 or G. As we will see in §7, the classifi-cation of AU–orbits follows from Theorem 1.3 as well:

Corollary 1.5 For any x ∈ RF+M , we have xAU = xH ∩ RF+M .

The possibilities for xH are recalled in Theorem 2.3 below. (For x 6∈ RF+M ,it is easy to see that the orbit xAU is closed.)

Strategy. The mechanism behind the proof of Theorem 1.3 is the followingdichotomy. Suppose a horocycle χ ⊂ M limits on a properly embedded,totally geodesic surface S (such as one of the boundary components of theconvex core of M). If χ is contained in S then χ is trapped and χ = S;otherwise, χ is scattered by S, and χ = M . In both cases the behaviorof χ is strongly influenced by the behavior of the horocycle flow on S. Tocomplete the proof we show that, up to the action of V , every recurrenthorocycle accumulates on such a surface S. This step uses the classificationof H–orbits from [MMO].

We remark that any connected subgroup of G generated by unipotentelements is conjugate to N , H or U . Theorem 1.3 completes the descriptionof the topological dynamics of these groups acting on FM , since the behaviorof H and N was previously known (see §2).

Outline of the paper. The remainder of the paper is devoted to the proofof Theorem 1.3. In §2 we review existing results about dynamics on FM .In §3 we establish a general lemma about the double coset space U\G/H,and in §4 we prove an approximation theorem for U–orbits. The space ofexceptional frames is introduced in §5, and the proof of Theorem 1.3 iscompleted in §6. Corollary 1.5 is deduced in §7.

Remark: General acylindrical manifolds. When M is a convex cocom-pact, acylindrical manifold that is not rigid, the behavior of horocycles canbe radically different from the rigid case. For example, a horocycle orthogo-nal to a closed leaf of the bending lamination of ∂ core(M) can be properlyembedded, giving rise to a frame x ∈ FM with a compact A–orbit and anonrecurrent U–orbit. The scattering argument also breaks down, due tothe lack of totally geodesic surfaces in M . It is an open problem to developa rigidity theory for these and other infinite–volume hyperbolic 3-manifolds.

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2 Background

In this section we introduce notation and recall known results regardingtopological dynamics on FM .

Figure 1. Limit set of a rigid acylindrical manifold.

Geometry on H3. Notation for G and its subgroups was introduced in§1. We also let AC =

(z 00 z−1

): z ∈ C∗. The action of G on H3 = G/K

extends continuously to a conformal action of G by Mobius transformationson the Riemann sphere,

C = C ∪ ∞ ∼= G/ACN,

and the union H3 ∪ C ∼= B3 is compact. We let R = R ∪ ∞ denote thestandard circle on C. Its orientation–preserving stabilizer in G is H.

Let M = Γ\H3 be a hyperbolic 3-manifold. The natural covering map

FH3 ∼= G→ FM ∼= Γ\G

will be denoted by g 7→ [g]. The limit set of Γ is characterized by Λ = C∩Γp,for any p ∈ H3; the domain of discontinuity is its complement, Ω = C− Λ.The convex hull of Λ is the smallest convex subset of H3 containing allgeodesics with both endpoints in the limit set; and its quotient gives theconvex core of M :

core(M) = Γ\ hull(Λ) ⊂M.

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A discrete group is elementary if it contains an abelian subgroup with finiteindex. We will always assume that Γ ∼= π1(M) is a nonelementary group.

Surfaces in M . There is a natural correspondence between

(i) Closed H–orbits [xH] ⊂ FM ,

(ii) Properly immersed, totally geodesic surfaces S ⊂M , and

(iii) Circles C ⊂ C such that [ΓC] is discrete in the space of all circles,C ∼= G/H.

This correspondence is given, with suitable orientation conventions, by S =the projection of hull(C) ⊂ H3 to M , xH = TS, the bundle of framestangent to S, and [ΓC] = [xH] in Γ\G/H.

Convex cocompact manifolds. Now assume that the convex core of Mis compact. The renormalized frame bundle of M is defined by

RFM = x ∈ FM : xA ⊂ FM is compact.

Replacing A with A+ in the definition above, we obtain the locus RF+M .Note that RFM is invariant under A and RF+M is invariant under AN .

In terms of the universal cover, we have [g] ∈ RF+M if and only ifg(∞) ∈ Λ, while [g] ∈ RFM if and only if g(0), g(∞) ⊂ Λ.

Minimality. We now turn to some dynamical results. Let L be a closedsubgroup of G. We say X ⊂ FM is an L–minimal set if xL = X for allx ∈ X.

Theorem 2.1 (Ferte) If M is convex cocompact, then the locus RF+M isan N–minimal set.

See [Fer, Cor. C(iii)]; a generalization appears in [Win]. We also record thefollowing result from [Da]:

Theorem 2.2 (Dal’bo) If Γ ⊂ H is a nonelementary convex cocompactFuchsian group, then (Γ\H) ∩ RF+M is a U–minimal set.

Rigid acylindrical manifolds. Recall that M is a rigid acylindrical man-ifold if M is convex cocompact, of infinite volume, and ∂ core(M) is totallygeodesic. In this case Ω ⊂ C is the union of a dense set of round disks withdisjoint closures, and Λ is a Sierpinski curve; see Figure 1.

Theorem 2.3 Let M be a rigid acylindrical manifold. Then for any x ∈RFM , either xH is closed or xH = (RF+M)H.

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Proof. Since Ω is a union of round disks, any circle that meets Λ in justone point can be approximated by a circle meeting Λ in two or more points;thus

(RFM)H = (RF+M)H. (2.1)

Let H ′ = PGL2(R) = H ∪ jH, where j =(1 00 −1

). Note that Aj = jA and

hence (RFM)j = RFM .With H ′ in place of H, Theorem 2.3 is proved in [MMO, Cor. 1.7].

Using the H ′ version, we can conclude that either xH is closed or xH ′ =(RF+M)H ′. In the latter case, RFM is contained in xH ∪ xHj. But RFMhas a dense A–orbit [MMO, Thm. 4.3], so RFM is contained in xH or xHj.In either case, we have

RFM = (RFM)j ⊂ xHj2 = xH.

Hence xH = (RF+M)H by equation (2.1) above.

3 Configuration spaces and double cosets

This section and the next present two self–contained results that will beused in §6 below. In this section we will prove:

Theorem 3.1 Suppose gn → id in G − V H, and Tn ⊂ U is a sequence ofK–thick sets. Then there is a K ′–thick set V0 ⊂ V such that

lim supTngnH ⊃ V0.

Double cosets. As motivation for the theorem, we remark that the dou-ble coset space U\G/H is the moduli space of pairs (χ, P ) ⊂ H3, where χis a horocycle and P ∼= H2 is a hyperplane. This moduli space is highlynonseparated near the identity coset, where χ ⊂ P . This means that asχ approaches P , the pair (χ, P ) can have many different limiting config-urations, depending on how we choose coordinates. The Theorem abovedescribes, more precisely, the different limiting configurations that arise.The appearance of multiple configurations is a basic mechanism at work inhomogeneous dynamics.

Limits of sets. We recall that the limsup of a sequence of sets Xn ⊂ Gconsists of all limits of the form g = limxnk

, where nk →∞ and xnk∈ Xnk

.

Thick sets and polynomials. We say T ⊂ R is K-thick if

[1,K] · |T | = [0,∞).

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This notion also makes sense for T inside any Lie group isomorphic to R,such as U or V . A basic fact about thick sets, which will be used below, isthe following. Let p ∈ R[x] be a polynomial of degree d, and let T ⊂ R beK–thick. Then for any symmetric interval I = [−a, a] ⊂ R, we have

maxx∈T∩I

|p(x)| ≥ kmaxx∈I|p(x)|, (3.1)

where k > 0 depends only on K and d. For more details, see [MMO, §8].

-y

yCn

xn+iy

0

Figure 2. The circles Cn → R eventually meet the locus | Im(z)| = y.

Proof of Theorem 3.1. Fix y > 0. We will first show that lim supUgnHcontains v or v−1, where v(z) = z + iy.

Let Cn = gn(R). Since gn → id, we have Cn → R in the Hausdorfftopology on closed subsets of C. Note that for n 0, Cn ∩ C is either acircle of large radius or a straight line of nonzero slope (since gn 6∈ V H).Thus Cn meets the locus L = z : | Im(z)| = y for all n 0. Passing toa subsequence, we can assume that Cn ∩L 6= ∅ for all n, and that the pointof Cn ∩ L closest to the origin has the form xn + εy for a fixed ε = ±1 (seeFigure 2). Let un(z) = z − xn; then

ungn(R)→ R + iεy

as n → ∞. It follows that ungnhn(z) → z + iεy for suitable hn ∈ H, sincethe latter group can be used to reparameterize R. Equivalently, v or v−1

belongs to lim supUgnH.We now take into account the thick sets Tn. Note that at the scale |xn|,

the arc of gn(R) close to R is well–modeled by a parabola, i.e. the graph of aquadratic polynomial. Applying equation (3.1) to this polynomial, we findthere is a K ′ depending only on K, and a sequence x′n + iy′n ∈ Cn, such that

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u′n(z) = z − x′n ∈ Tn, and 1 ≤ |y/y′n| ≤ K ′. Passing to a subsequence andarguing as above, we conclude that v(z) = z + iy′ belongs to lim supTngnHfor some y′ with 1 ≤ |y/y′| ≤ K ′. Since y > 0 was arbitrary, this shows thatV ∩ (lim supTngnH) is a K ′–thick subset of V .

Remark. Theorem 3.1 is a strengthening of [MMO, Lem. 8.2]; the proofhere is more geometric.

4 Moving to the renormalized frame bundle

In this section we describe how to use U to move points close to RFM intoRFM . The boundary of the convex core of M gives rise to an exceptionalcase.

Theorem 4.1 Suppose xn ∈ (RFM)U and xn → y ∈ RFM . Then thereexists a sequence un ∈ U such that xnun ∈ RFM and

1. We have un → id, and hence xnun → y; or

2. There is a component S of ∂ core(M) such that yH = TS, and xnunaccumulates on TS as n→∞.

Rp

Figure 3. If d(γ, χ) ≤ R− 1, with R 0, then d(p, γ) < R.

The proof relies on the following fact from planar hyperbolic geometry.

Lemma 4.2 Let γ, χ ⊂ H2 be a geodesic and a horocycle respectively, letδ be a geodesic joining the base of χ to one of the endpoints of γ, and letp = δ ∩ χ. Then for all R 0, if d(χ, γ) < R− 1, then d(p, γ) < R.

8

The proof is indicated in Figure 3, where the endpoint in common to γ andδ is at infinity. Note that an R–neighborhood of γ ⊂ H2, for R 0, isbounded by a pair of rays meeting at an angle of nearly 180.

Proof of Theorem 4.1. Choose gn → g0 in G such that [gn] = xn and[g0] = y, and let Cn = gn(R).

Recall that [g] ∈ RFM if and only if g(0), g(∞) ⊂ Λ. By assumption,gn(∞) ∈ Λ for all n, and g0(0) ∈ Λ. Moreover, since xn ∈ (RFM)U , thereexist sn ∈ R such that gn(sn) ∈ Λ. Let us arrange that |sn| is as small aspossible; then gn(In) ⊂ Ω, where In = (−sn, sn). Setting un(z) = z− sn, wethen have [gnun] = xun ∈ RFM .

It remains to verify that (1) or (2) is true. If |sn| → 0, then clearly weare in case (1), so let us assume that s = lim sup |sn| > 0. In this case,we claim C0 bounds a component Ω0 of Ω. To see this, recall that Ω is aunion of round disks with disjoint closures. The arc J = g0(−s, s) is thelimit, along a subsequence, of arcs gn(In) ⊂ Ω; since Ω has only finitelymany components with diameter greater than diam(J)/2, there is a uniquecomponent Ω0 of Ω such that g0(0) ∈ ∂Ω0. In fact the entire circular arc Jmust lie in Ω0, and hence C0 ⊂ Ω0. Since |C0 ∩ Λ| ≥ 2, we have C0 = ∂Ω0.

Consequently the plane H0 = hull(C0) ⊂ H3 covers a component S of∂ hull(M). In particular, we have y ∈ TS.

Now even in this case, we have sn → 0 along the subsequence wheregn(0) 6∈ Ω0. Thus to complete the proof, it suffices to show that (2) holdsunder the assumption that gn(0) ∈ Ω0 for all n. Under this assumption,Cn∩Ω0 is a circular arc with two distinct endpoints, one of which is gn(sn).Equivalently, Hn = hull(Cn) meets H0 along a geodesic γn ⊂ H3, with oneend converging to gn(sn).

Let χn ⊂ Hn denote the horocycle resting on gn(∞) whose natural liftto FH3 gives the orbit gnU . Let δn denote the geodesics in H3 connectinggn(∞) to gn(sn). Note that δn and χn both lie in the plane Hn, and crossat a unique point pn.

We claim that d(pn, H0) → 0. To see this, fix ε > 0. It is easy to seethat the set of points in Hn that are within hyperbolic distance ε of H0 isconvex and invariant under translation along γn; thus

Hn(ε) = p ∈ Hn : d(p,H0) < ε = p ∈ Hn : d(γn, p) < Rn

for some Rn > 0. Since xn → y, we have Hn → H0 and hence Rn → ∞;moreover, χn converges to a horocycle in H0, so eventually d(γ, χn) < Rn−1.By Lemma 4.2, this implies that d(pn, γ) < Rn, and hence d(pn, H0) < ε forall n 0.

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By construction we have gnun ∈ FpnH3. Since the frame gnun is tangentto the geodesic δn, whose endpoints lie in the limit set, we have [gnun] ∈RFM ; and since d(pn, H0) → 0 (and indeed Hn and H0 are nearly parallelnear pn), the frames gnun accumulate on TH0 and hence the frames xnun =[gnun] accumulate on TS.

5 Exceptional frames

Let M be a rigid acylindrical manifold. We define the locus of exceptionalframes in FM by

EM =⋃xHV : x ∈ RFM and xH ⊂ FM is closed.

In this section we develop some basic properties of the exceptional locus.

Immersed surfaces. As we remarked in §1, when x ∈ RFM and xH isclosed, its projection to M is a properly immersed, totally geodesic surfaceS passing through the convex core of M . For v ∈ V , the projection of xHvto M is a surface equidistant from S. The exceptional locus accounts for allthe horocycles that lie on such surfaces.

Like RF+M , the locus EM is invariant under the action of AN . In termsof the universal cover, we have [g] ∈ EM iff g(R) is tangent, at g(∞), to acircle C such that |C ∩ Λ| ≥ 2 and ΓC is discrete. Note that

EM ∩ RFM 6= ∅, (5.1)

since EM contains the compact H–orbits coming from the totally geodesicboundary components of the convex core of M .

Lemma 5.1 If x ∈ RFM , then xAU meets EM .

Proof. If xH is closed, then we have x ∈ EM already. Otherwise, wehave xH = (RF+M)H by Theorem 2.3, and xAUH = xH, since AU\His compact. Thus xAUH = (RF+M)H contains one of the compact orbitsyH ⊂ EM coming from the boundary of the convex core of M , so xAUmust meet this orbit as well.

Lemma 5.2 For any x ∈ EM ∩RF+M , the locus Y = xU is a U–minimalset, and Y = xvHv−1 ∩ RF+M for some v ∈ V .

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Proof. Since U commutes with the action of V , it suffices to treat thecase where xH is closed in FM . In this case, xH = TS for some properlyimmersed, totally geodesic surface S ⊂ M . The subgroup π1(S) ⊂ π1(M)determines a covering space M ′ → M , which we can normalize so thatM ′ = Γ′\H3 with Γ′ ⊂ H. (If S happens to be nonorientable, we pass tothe orientation–preserving subgroup of index two.)

Since S is properly immersed, M ′ is convex cocompact; and since M isacylindrical, M ′ is nonelementary. It is now easy to check that the coveringmap FM ′ → FM sends (Γ′\H) ∩ RF+M

′ isomorphically to Y = (xH) ∩RF+M , respecting the action of U (cf. [MMO, Thm. 6.2, Prop. 7.2]). Theresult then follows from Dal’bo’s minimality Theorem 2.2.

Lemma 5.3 For any x ∈ RF+M − EM , the orbit xU meets RFM .

Proof. Suppose x ∈ RF+M but xU does not meet RFM . Then x = [g]where C = g(R) meets Λ in just one point. Therefore C is tangent toD = ∂Ω0 for some component Ω0 ⊂ Ω, and ΓD is discrete, so x ∈ EM .

6 Classification of U–orbit closures

We can now complete the proof of Theorem 1.3. The interaction betweenxU and the exceptional locus EM plays a leading role in the proof.

Lemma 6.1 For any x ∈ RF+M , the orbit closure X = xU meets EM .

Proof. Note that the result holds for x if and only if it holds for somex′ ∈ xAN . Thus we are free to adjust x by elements of AN in the course ofthe proof.

Suppose X is disjoint from EM . By Lemma 5.3, after replacing x withan element of xU , we may assume x ∈ RFM . Then X contains a closed,U–invariant set Y such that Y L+ ⊂ Y for some 1-parameter semigroupL+ ⊂ AV , by [MMO, Prop. 9.3 and Thm. 9.4]. Let L ⊂ AV be the groupgenerated by L+. Note that either L = V or L = vAv−1 for some v ∈ V .

Choose `n → ∞ in L+∼= R+. Then L =

⋃`−1n L+. The locus Y `n ⊂ X

is U–invariant, so by Lemma 5.3 again we can find yn ∈ RFM ∩ Y `n. Passto a subsequence such that yn → z ∈ RFM . We have yn`

−1n L+ ⊂ X for all

n, so in the limit we obtain zL ⊂ X.

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If L = V , then we have zN ⊂ X, so X = RF+M by Theorem 2.1, andthus X meets EM by equation (5.1). Otherwise, L = vAv−1 for some v ∈ V .Therefore

X ⊃ zvAUv−1.

Again, we can find u ∈ U such that y = zuv ∈ RFM . Then yAU = zvAU .By Lemma 5.1, yAU meets EM , so X meets EM as well.

Typical orbits. Using the results of §3 and §4, we can now finally describethe behavior of U–orbits outside of the exceptional locus.

Theorem 6.2 Suppose x ∈ RF+M − EM . Then xU = RF+M .

Proof. Let X = xU . Choose y ∈ X ∩ EM , using Lemma 6.1. By Lemma5.2, there is a v ∈ V such that Z = yvHv−1 is closed, we have

X ⊃ Y = yU = Z ∩ RF+M,

and Y is a U–minimal set. Replacing x with xv, we can assume that v = id,and hence Z = yH. Then Y ∩ RFM 6= ∅, so we can also assume thaty ∈ RFM . By Lemma 5.3, after replacing x with xu for some u ∈ U , wecan further assume that x ∈ RFM .

Let X∗ = X ∩ RFM , and let

G0 = g ∈ G : Zg ∩X∗ 6= ∅.

We claim there is a sequence gn → id in G0 − HV . To see this, firstnote that since y ∈ X, we can find un ∈ U and gn → id in G such thatxun = ygn. In particular, we have xun → y. We now apply Theorem4.1. This Theorem implies that after changing our choice of un ∈ U , we canassume that xun ∈ X∗ and either (i) xun → y, or (ii) Z = Y is compact, andxun accumulates on Y . In either case, after passing to a subsequence and(in case (ii)) possibly changing our choice of y ∈ Y , we still have xun = ygn.Then clearly gn ∈ G0, we have gn → id, and gn 6∈ HN = HV becauseyH ⊂ EM while x 6∈ EM .

Since Z is H–invariant, we have HG0 = G0. By [MMO, Lem. 9.2], thereis also a K > 1 and a sequence of K–thick sets Tn such that gnTn ⊂ G0

for all n. Applying Theorem 3.1 (with the order of factors reversed) to thesequence HgnTn ⊂ G0, we find that G0 contains a thick subset V0 ⊂ V .In particular, we can choose vn → ∞ in V ∩ G0. Then Zvn meets X∗ bythe definition of G0. But Zvn ∩ RF+M = Y vn, so the U–minimal set Y vn

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also meets X∗, and thus Y vn ⊂ X for all n. Now Y vn is invariant underthe closed subgroup v−1n AUvn of AN , which converges to N as n→∞. Bycompactness of X∗, we conclude that X contains the N–orbit of a point inX∗, and hence X = RF+M by Theorem 2.1.

Proof of Theorem 1.3. Let x be an element of FM .(1) If x 6∈ RF+M , then xU is closed. Indeed, in this case xU corresponds

to a horocycle χ ⊂ H3 resting on a point of Ω, and the projection of χ toM is a proper immersion.

(2) If x ∈ EM∩RF+M , then we xU = xvHv−1∩RF+M for some v ∈ V ,by Lemma 5.2.

(3) Finally, if x ∈ RF+M −EM , then xU = RF+M by Theorem 6.2.

7 Classification of AU–orbit closures

In this final section we use the classification of U–orbits to show that

xAU = xH ∩ RF+M (7.1)

for all x ∈ RF+M , as stated in Corollary 1.5.

Generic circles. Let M = Γ\H3 be a rigid acylindrical manifold. LetC = G/H be the space of oriented circles in C, let

C0 = C ∈ C : |C ∩ Λ| ≥ 2,

and letC1 = C ∈ C0 : ΓC is discrete in C.

Lemma 7.1 The set C1 is countable.

Proof. A circle C ∈ C1 corresponds to a properly immersed, totally geodesicsurface S with fundamental group π1(S) ∼= ΓC . Thus ΓC is a finitely gen-erated, nonelementary group and C is the unique circle containing Λ(ΓC).Since Γ is countable, there are only countably many possibilities for ΓC , andhence only countably many possibilities for C.

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Lemma 7.2 There is a circle C ∈ C0 that is not tangent to any circle inC1.

Proof. It is easy to see that C0 has nonempty interior, while the set ofcircles tangent to a given C ∈ C1 is nowhere dense. Since C1 is countable,the result follows from the Baire category theorem.

Rephrased in terms of Γ\G, this shows:

Corollary 7.3 There is an orbit yH ⊂ FM − EM that meets RFM .

Proof of Corollary 1.5. The argument is similar to the proof of Lemma5.1. Consider x ∈ RF+M . We always have xAU ⊂ RF+M , since the latterset is closed and AU invariant.

If xU meets RFM , then we can reduce to the case where x ∈ RFM .Under this assumption, if xH is closed, then xU = xH ∩ RF+M by The-orem 1.3; since the latter set is A–invariant, it also coincides with xAU .Otherwise, by Theorem 2.3 and compactness of AU\H, we have

xAUH = xH = (RF+M)H.

In particular, by Corollary 7.3, xAU meets RF+M − EM . Let y denote apoint in their intersection. Then we have

RF+M = yU ⊂ xAU

by Theorem 6.2, so equation (7.1) holds in this case as well.Finally, if x ∈ RF+M but xU does not meet RFM , then xH corresponds

to a circle tangent to Λ in just one point, and (7.1) is easily verified usingminimality of the horocycle flow on a compact hyperbolic surface (cf. [MMO,Thm. 1.5]).

References

[Da] F. Dal’bo. Topologie du feuilletage fortement stable. Ann. Inst.Fourier 50(2000), 981–993.

[Fer] D. Ferte. Flot horospherique des reperes sur les varietes hyper-boliques de dimension 3 et spectre des groupes kleiniens. Bull.Braz. Math. Soc. 33(2002), 99–123.

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[MMO] C. McMullen, A. Mohammadi, and H. Oh. Geodesic planes in hy-perbolic 3–manifolds. Preprint, 2015.

[Rn] M. Ratner. Raghunathan’s topological conjecture and distributionsof unipotent flows. Duke Math. J. 63(1991), 235–280.

[Sh] N. A. Shah. Unipotent flows on homogeneous spaces of SL(2,C).M.Sc. Thesis, Tata Institute of Fundamental Research, Mumbai,1992.

[Win] D. Winter. Mixing of frame flow for rank one locally symmetricspaces and measure classification. Israel J. Math. 210(2015), 467–507.

Mathematics Department, Harvard University, 1 Oxford St,Cambridge, MA 02138-2901

Mathematics Department, University of Texas at Austin, 1University Station C1200, Austin, TX 78712

Mathematics Department, Yale University, 10 Hillhouse Av-enue, New Haven, CT 06511, and Korea Institute for AdvancedStudy, Seoul, Korea

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