AAn exotic plane in an acylindrical 3-manifold
Yongquan ZhangJanuary 19, 2021
Abstract
Let P be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold M .Assume that P ∗ = M ∗ ∩ P is nonempty, where M ∗ is the interior of the convex core of M . Doesthis condition imply that P is either closed or dense in M ? A positive answer would furnish ananalogue of Ratner’s theorem in the infinite volume setting.In [MMO2] it is shown that P ∗ is either closed or dense in M ∗ . Moreover, there are at mostcountably many planes with P ∗ closed, and in all previously known examples, P was also closedin M .In this note we show more exotic behavior can occur: namely, we give an explicit example ofa pair ( M, P ) such that P ∗ is closed in M ∗ but P is not closed in M . In particular, the answerto the question above is no. Thus Ratner’s theorem fails to generalize to planes in acylindrical3-manifolds, without additional restrictions. This paper is a contribution to the study of topological behavior of geodesic planes in hyperbolic3-manifolds of infinite volume.
Geodesic planes in hyperbolic 3-manifolds.
Let M ∼ = Γ \ H be an oriented, complete hyper-bolic 3-manifold, presented as the quotient of hyperbolic space by a Kleinian groupΓ ⊂ Isom + ( H ) ∼ = PSL(2 , C ) . Let Λ ⊂ S be the limit set of Γ, and Ω = S − Λ the domain of discontinuity. The convex core of M is defined as core( M ) := Γ \ hull(Λ);Equivalently, it is the smallest closed convex subset of M containing all closed geodesics. Let M ∗ be the interior of core( M ). We say M is convex cocompact if M := Γ \ ( H ∪ Ω) is compact, orequivalently core( M ) is compact.A geodesic plane in M is an isometric immersion f : H → M . We often identify f with itsimage P := f ( H ) and call the latter a geodesic plane as well. Given a geodesic plane P , write P ∗ = M ∗ ∩ P . Planes in acylindrical manifolds.
In this paper, we study the topological behavior of geodesicplanes in a convex cocompact, acylindrical hyperbolic 3-manifold M . The topological conditionof being acylindrical means that the compact Kleinian manifold M has incompressible boundaryand every essential cylinder in M is boundary parallel [Thu1]. When M has infinite volume, the1 a r X i v : . [ m a t h . G T ] J a n roperty of being acylindrical is visible on the sphere at infinity: M is acylindrical if and only if Λis a Sierpie´nski curve .When M has finite volume, a geodesic plane P in M is either closed or dense [Sha, Rat]. In theinfinite volume case, if we assume furthermore core( M ) has totally geodesic boundary, it is shownin [MMO1] that any geodesic plane P in M is either closed, dense in M , or dense in an end of M .In other words, geodesic planes in such an M do satisfy strong rigidity properties. In [MMO2], thisis generalized to all convex cocompact acylindrical 3-manifolds if we restrict to the interior of theconvex core M ∗ : Theorem 1.1 ([MMO2]) . Let M be a convex cocompact, acylindrical, hyperbolic 3-manifold. Thenany geodesic plane P intersecting M ∗ is either closed or dense in M ∗ . As a matter of fact, when P ∗ is dense in M ∗ , a stronger statement holds: P is actually densein M . As a complement, it is also shown in [MMO2] that there are only countably many geodesicplanes P so that P ∗ is nonempty and closed in M ∗ . It is natural to ask, `a la Ratner, if thesecountably many planes are well-behaved topologically in the whole manifold. For example, we havethe following question in [MMO2]: if P ∗ is closed in M ∗ , is P always closed in M ? Our maintheorem answers this question: Theorem 1.2.
There exists a convex cocompact, acylindrical, hyperbolic 3-manifold M = Γ \ H and a geodesic plane P in M so that P ∗ is nonempty and closed in M ∗ but P is not closed in M . Therefore, for this concrete acylindrical manifold M , at least one of the closed geodesic planesin M ∗ is not well-behaved topologically in the whole manifold. This supports the idea that theproper setting for generalizations of Ratner’s theorem may be M ∗ rather than M , as suggested in[MMO2]. Exotic planes and circles.
For simplicity, we call P an exotic plane of M if P ∗ is nonemptyand closed in M ∗ but P is not closed in M . We now proceed to describe and visualize the examplein Theorem 1.2 from the perspective of the sphere at infinity.Fix a presentation M ∼ = Γ \ H . Any circle C on the sphere at infinity S determines a uniquegeodesic plane in H , and in turn gives a geodesic plane P in M . Conversely, given a geodesic plane P in M , let ˜ P be any lift to H ; its boundary at infinity is a circle C ⊂ S . We call C a boundarycircle of P . Note that Γ · C gives all the boundary circles of P . In particular, an exotic circle of Γis a boundary circle of an exotic plane in M .The exotic plane in Theorem 1.2 comes from a one-parameter family of acylindrical orbifoldsconstructed in [Zha]. Figure 1 gives some visualizations of the example: Figure 1a depicts the limitset of Γ with an exotic circle C marked, and Figure 1b shows the orbit of C under Γ. Note that C (cid:48) (also marked in Figure 1a) is not a circle in Γ · C , but there exists a sequence γ i ∈ Γ so that γ i C → C (cid:48) . This is reflected in our discussion of the geometry of P (see below), and is quite visiblefrom Figure 1b. Geometry of P , P ∗ and P . We briefly describe here the geometry of the exotic plane P in theorbifold, its restriction P ∗ to the convex core, and its closure P . The plane P is a nonelementary,convex cocompact surface with one infinite end; its restriction to core( M ) cuts the infinite end intoa crown with two tips. In particular P ∗ has finite area. The two tips of the crown wraps aroundand tends to the bending geodesic on the boundary of core( M ). Finally, P = P ∪ P (cid:48) , where P (cid:48) isa closed geodesic plane contained in the infinite end of M . As a matter of fact, P (cid:48) is a cylinder A Sierpi´nski curve is a compact subset Λ of the 2-sphere S such that S − Λ = ∪ i D i is a dense union of Jordandisks with diam( D i ) → D i ∩ D j = ∅ for all i (cid:54) = j . See Figure 1a for an example. a) The limit set Λ and an exotic circle C (b) The orbit Γ · C Figure 1: The example in Theorem 1.2 from the perspective of the sphere at infinity.Note that the orbit Γ · C of the exotic circle C limits on C (cid:48) / ∈ Γ · C . whose core curve is precisely the bending geodesic. Note that P is not a closed submanifold of M ,as it is not locally connected near P (cid:48) .Below is a picture of P near the convex core boundary in the quotient of M by a reflectionsymmetry. We remark that the behavior of P near the convex core boundary only depends on thegeometry of the corresponding quasifuchsian orbifold; see Theorems 2.2 and 3.3, and Figure 5 fordetails. η P P (cid:48) Figure 2: A cross section near the convex core boundary, in aplane orthogonal to P . Here η denotes the bending geodesic. Planes in cylindrical manifolds.
We remark that for cylindrical manifolds, there is no ana-logue of Theorem 1.1 in general. For example, geodesic planes intersecting the convex core ofa quasifuchsian manifold may have non-manifold, even fractal closures, as explained in [MMO1,Appx. A]. Nevertheless, we can still consider exotic planes in a general convex cocompact hyper-3olic 3-manifold M . In fact, we give examples of exotic planes in a concrete family of quasifuchsianmanifolds in § M with incompressible boundary, there may be un-countably many geodesic planes closed in M ∗ . For example, in [MMO2, § M ∗ for a quasifuchsian manifold M . Never-theless, we show in the appendix: Theorem 1.3.
Let M be a convex cocompact hyperbolic 3-manifold with incompressible boundary.Then there are at most countably many exotic planes in M . In particular, the union of all exotic planes in M ∗ has measure zero. Questions.
We conclude the discussion by mentioning the following open questions.1. The example in Theorem 1.2 is fairly tame, but wilder examples may exist when the bendinglamination is nonatomic. For example, is there an exotic plane P so that P ∗ has infinite area?2. In both quasifuchsian and acylindrical cases, we give examples containing one exotic plane.In view of Theorem 1.3, a natural question is if the result is “sharp”. That is, does thereexist a convex cocompact hyperbolic 3-manifold with infinitely many exotic planes? Notes and references.
The rigidity results of [MMO1, MMO2] have recently been extended tocertain geometrically finite acylindrical manifolds in [BO].This paper is organized as follows. Section 2 is devoted to examples of exotic planes in a concretefamily of quasifuchsian orbifolds. We then leverage the quasifuchsian examples to construct anacylindrical example, giving a proof of Theorem 1.2 in Section 3. Finally, we calculate explicitlythe parameters for our example in Section 4; they are needed to produce Figure 1.
Acknowledgements.
I would like to thank C. McMullen for enlightening discussions, and forsharing his example of a quasifuchsian exotic plane, which appears in Section 2. Figure 6c wasproduced by
Geomview [Geo], and computer scripts written by R. Roeder [Roe]. Figures 1, 3aand 4 were produced by McMullen’s program lim [McM].
In this section, we give examples of exotic planes in a concrete family of quasifuchsian orbifolds.Fix an integer n ≥
3. Let R be a quadrilateral in the extended complex plane whose sidesare either line segments or circular arcs, and whose interior angles are all π/n . Reflections inthe sides of R generate a discrete subgroup of PSL(2 , C ), and let Γ R be its index 2 subgroup oforientation preserving elements. We note that Γ R is a quasifuchsian group, and the correspondingquasifuchsian orbifold N R := Γ R \ H is homotopic to a sphere with 4 cone points of order n .Denote by C i , ≤ i ≤ R lie, so that C and C containopposite sides. The corresponding hyperbolic planes in H are also denoted by C i . To construct N R , one can take two copies of the infinite “tube” bounded by these four planes and identifycorresponding faces. A fundamental domain for Π R is thus two copies of the tube, although formost discussions we consider the full reflection group with a fundamental domain simply being thetube.Two copies of the geodesic segment perpendicular to both C and C glue up to a closed geodesic ξ in N R , and similarly the other two planes give a closed geodesic η . Let 2 cosh − ( s ) and 2 cosh − ( t )be the lengths of ξ and η respectively. Then 4 roposition 2.1. For any quadrilateral R , we have ( s − t − ≤ ( π/n ) , with equality ifand only if Γ R is Fuchsian. When the inequality is strict, the convex core of N R is bent along ξ onone side and η on the other. This proposition is a combination of Proposition 4.1 and Lemma 4.3 in [Zha]. See Figure 3 forsome visualizations with n = 3. Note that Figure 3a is drawn so that R is centered at the origin, (a) Quadrilateral R in gray and limit set of Γ R . ξη (b) A schematic picture of core( N R ) Figure 3: An example of N R . Here η is the bending geodesic on the side of thedomain of discontinuity containing R , and ξ the bending geodesic on the otherside. A lift of η has end points p, p (cid:48) , and a lift of ξ has end points q, q (cid:48) . and symmetric across real and imaginary axes. The end points p, p (cid:48) of a lift ˜ η of η then lie onthe imaginary axis; the corresponding hyperbolic element, also denoted by ˜ η , is given by reflectionacross C followed by refection across C . Similarly, the end points q, q (cid:48) of a lift ˜ ξ of ξ lie on thereal axis, see Figure 3a. (a) Axes of bending geodesics and an exotic circle (b) Orbit of the exotic circleFigure 4: An exotic circle for quasifuchsian group Γ R Let Λ be the limit set of Γ R . Take the circle C passing through the end points q, q (cid:48) of ˜ ξ andthe repelling fixed point p of ˜ η ; see Figure 4a. It is easy to see that p is an isolated point in C ∩ Λ.As C is stabilized by reflections in C and C , it also passes through the orbit of p under the groupgenerated by these reflections. It is clear that C ∩ Λ consists of points in this orbit together withthe end points q, q (cid:48) of ˜ ξ . We have Theorem 2.2.
For any quadrilateral R , the circle C is exotic. That is, the corresponding geodesicplane P is closed in N ∗ R but not in N R . roof. For simplicity, we will suppress the subscript R . Since C passes through the repelling fixedpoint of ˜ η , the sequence ˜ η n · C tends to a circle C (cid:48) passing through both end points of ˜ η ; inparticular, the geodesic plane P determined by C is not closed in N , and accumulates on the plane P (cid:48) determined by C (cid:48) . This is clearly visible in Figure 4b, where the orbit of C under Γ is drawn.˜ ξC C p p p The slice in (b) → (a) A fundamental domain inside ˜ P ˜ η C C ˜ ξ ˜ P (b) A perpendicular slice of ˜ P Figure 5: Some visualizations for the proof of Thm. 2.2
It remains to show P ∗ is closed in N ∗ . The hyperbolic plane ˜ P in H determined by C is dividedinto two half planes by ˜ ξ . One half descends to a half cylinder contained in an end of N ; indeed,this half plane is stabilized by ˜ ξ and a fundamental domain for the action of this hyperbolic elementis compact in H ∪ Ω, where Ω is the domain of discontinuity of Γ, and thus our assertion followsfrom proper continuity of the action.For the other half, again since it is stabilized by ˜ ξ , it suffices to consider a fundamental do-main under the action of this hyperbolic element. As a matter of fact, we may even consider afundamental domain under the full reflection group. One choice of this is the portion of the halfplane sandwiched between C and C . Let p and p be the images of p under reflections across C and C respectively. The geodesic with end points p and p descends to a complete geodesiccontained in the convex core boundary, and same for the geodesic with end points p and p . Henceto understand P ∗ we only need to consider the portion mentioned above bounded between thegeodesics pp and pp ; see Figure 5a.This portion is divided by orbits of Faces 3 and 5 into countably many pieces, each one compactand descends to a piece contained entirely in core( N ). Moreover, the maximum distance of a pointon the piece to η goes to 0; see Figures 5a and 5b. This implies that P ∗ only accumulates on η , soit is closed in N ∗ . 6he proof above gives a clear picture of the topology of P ∗ ; indeed, we described a fundamentaldomain under the action of the full reflection group (the gray region in Figure 5a). Two pieces ofthis gives a crown with two tips, and P ∗ is precisely the interior of this crown properly immersed in N ∗ R . It is also easy to understand the behavior of P in N R . Note that the limit plane P (cid:48) describedin the proof above is closed, and P only accumulates on this plane. These assertions can be provedusing similar arguments to the proof above. In particular we have P = P ∪ P (cid:48) . Any neighborhoodof a point on P (cid:48) intersects P in infinitely many pieces. In particular we have Corollary 2.3. P is not locally connected, and thus not a submanifold of N R . In this section, we briefly review the acylindrical orbifolds constructed in [Zha], which are coveredby the quasifuchsian orbifolds described in the last section. We then construct an exotic plane inone of these acylindrical orbifolds by projecting down the corresponding quasifuchsian example.
The example manifold and its deformations.
One way to construct explicit examples of hy-perbolic 3-manifolds is gluing faces of hyperbolic polyhedra (with desired properties) via hyperbolicisometries. See [Thu2, § Q be the hyperbolic polyhedron with an infinite endobtained by extending across Face 1 to infinity; see Figure 6c. Reflections in all faces of ˜ Q generatea discrete subgroup of hyperbolic isometries; a subgroup of index 2 gives an acylindrical hyperbolic3-orbifold with Fuchsian end. We can deform ˜ Q by pushing closer or pulling apart Faces 3 and 5,fixing the dihedral angles, and obtain deformations of the orbifold. This gives [Zha, Thm. 1.2]:6 234 5 17 8910 (a) The Coxeter diagram
15 234 6107 8 9 (b) The corresponding polyhe-dron (c) The hyperbolic polyhedron in theunit ball modelFigure 6: Combinatorial data and visualization of the polyhedron heorem 3.1. For each t ∈ (cid:32) , √ (cid:33) , there exists a unique hyperbolic polyhedron ˜ Q ( t ) so thatthe hyperbolic distance between Faces and is cosh − ( t ) . The corresponding hyperbolic orbifold M ( t ) is acylindrical convex cocompact, whose convex core boundary is totally geodesic if and onlyif t = 2 . See [Zha, Figure 3] for some samples of the deformation. One way to explicitly construct M ( t )is to take two copies of ˜ Q ( t ) and identify corresponding faces. A fundamental domain for thecorresponding Kleinian group is thus two copies of ˜ Q ( t ), although for most discussions we oftenconsider the full reflection group with a fundamental domain simply being ˜ Q ( t ).It is clear from construction that the quasifuchsian orbifold N ( t ) corresponding to the boundaryof M ( t ) is an example of those described in Section 2. Consistent with the notations there, let η be the simple closed geodesic in M ( t ) coming from two copies of the geodesic segment orthogonalto both Faces 3 and 5, and ξ be that from Faces 2 and 4. We have [Zha, Thm. 1.3]: Theorem 3.2.
The convex core boundary of M ( t ) is bent along η for t ∈ (1 , , and ξ for t ∈ (2 , (5 + √ / , with hyperbolic lengths l η ( t ) = 2 cosh − ( t ) and l ξ ( t ) = 2 cosh − ( s ) , where s = φ ( t ) is an explicit monotonic function. Bending angles λ η ( t ) and λ ξ ( t ) can also be explicitly calculated.On the other side of the corresponding quasifuchsian orbifold, the boundary is bent along ξ for t ∈ (1 , and η for t ∈ (2 , (5 + √ / , and lengths and angles can also be similarly calculated. ξη Figure 7: A schematic picture of core( M ( t )) for t ∈ (1 , .Note that a copy core( N ( t )) is embedded in the polyhedron We refer to [Zha] for proofs of these statements and explicit calculations.
Existence of an exotic plane.
We wish to make use of the quasifuchsian examples in Section 2to construct an acylindrical example. In particular, if we take the same plane P ( t ) in N ( t ) andproject it down to M ( t ) via the covering map N ( t ) → M ( t ) of infinite degree, we have to makesure that the cylinder contained in the bottom end of N ( t ) projects to a closed surface in M ( t ).For this, we have Theorem 3.3.
There exists t = t ∈ (1 , so that the image of P ( t ) under the projection N ( t ) → M ( t ) is an exotic plane. Moreover, the closure of the plane in M ( t ) is not locally connected, andhence not a submanifold.Proof. For this, we look at how the plane intersects a fixed fundamental domain, the polyhedron˜ Q ( t ). A lift ˜ P ( t ) of the plane P ( t ) passes through the geodesic segment orthogonal to Faces 2 The top end of N ( t ) is the infinite end shared with M ( t ), and the other end is the bottom end . t →
2, ˜ P ( t ) tends to Face 1; when t → P ( t ) tends to a plane orthogonal to Face 6, dividing the polyhedron ˜ P (1) into two equal parts. Bycontinuity, for some subinterval of (1 , P ( t ) intersects the edge shared by Faces 7 and 8. Anothercontinuity argument guarantees the existence of a t = t ∈ (1 ,
2) so that ˜ P ( t ) intersects this edgeorthogonally. This implies that ˜ P ( t ) intersects both Faces 7 and 8 orthogonally, and disjoint fromFace 5. See Figure 8 for a schematic picture of ˜ P ( t ) ∩ ˜ Q ( t ).247 8 Figure 8: A piece of the exotic plane P inside the fundamental domain Clearly, as ˜ P ( t ) is orthogonal to Faces 2, 4, 7 and 8, the half plane that projects to a cylinderin the bottom end in N ( t ) further projects down to a orbifold surface in M ( t ), with one geodesicboundary component, two cone points of order 2 and one cone point of order 3. The other halfbehaves exactly as that in the quasifuchsian example (recall that a copy of core( N ( t )) is embeddedin core( M ( t )), see Figure 7), so it is indeed an exotic plane in M ( t ). Following Corollary 2.3, theclosure of this plane is not locally connected.Theorem 1.2 is then a direct consequence of Theorem 3.3. In this section, we calculate the explicit value of t predicted in the previous section. We refer tothe calculations in [Zha, §
6] freely.We can uniquely determine a hyperbolic plane in H using its unit normal in the hyperbloidmodel. Set u = (cid:112) ( t + 1) /
2. One choice of normals for Faces 2 and 4 may be (cid:32) u − √ u + 4 v − − u v u − , ± v, , − u √ u + 4 v − − u v u − (cid:33) where v = u + √ ( u +2)(16 u − u − . Correspondingly, the normals for Faces 7 and 8 are (cid:32) u √ − √ u + 22 u − , ∓ √ , √ , u √ u + 2 − u √ u − (cid:33) . Suppose the unit normal to the plane ˜ P is ( x , x , x , x ). Then this vector is orthogonal to thenormals listed above. Therefore x = 0 and x = √ u + 2 − u √ √ u + 4 v − − u v √ − u √ u + 4 v − − u v ) x , x = u − √ u + 4 v − − u v − u √ u + 4 v − − u v x .
9n the other hand, the sequence of planes ˜ η n · ˜ P → ˜ P (cid:48) , where ˜ P (cid:48) has unit normal ( √ u − , , , − u √ u − ).This can be calculated using the formula in [Zha, § η , and the factthat the circle C (cid:48) corresponding to ˜ P (cid:48) passes through the fixed points of ˜ η and is symmetric acrossthe imaginary axis. Since ˜ P is tangent to ˜ P (cid:48) at infinity, the inner product of their unit norms is 1(or −
1, but we can always change the orientation of ˜ P ), so − x √ u − − x u √ u − . Hence we have x = 1 − u √ u + 4 v − − u v √ u − ,x = −√ u + 2 + u √ √ u + 4 v − − u v √ √ u − ,x = √ u + 4 v − − u v − u √ u − . Since − x + x + x = 1, we have4 u + 4 v − − u v − √ u (cid:112) u + 2 (cid:112) u + 4 v − − u v + u + 23 = 0 , and thus9(4 u + 4 v − − u v ) + ( u + 2) + 6(4 u + 4 v − − u v )( u + 2) − u ( u + 2)(4 u + 4 v − − u v ) = 0 . Plugging in the expression of v in terms of u , the left hand side gives u − u − (cid:16) f ( u ) + g ( u ) (cid:112) u + 29 u − (cid:17) where f ( u ) = −
625 + 11153 u − u + 65632 u − u + 22144 u − u ,g ( u ) = 900 u − u + 9072 u − u + 1152 u . Thus we have0 = f ( u ) − g ( u ) (16 u − u − u − ( −
625 + 944 u − u + 576 u )( −
625 + 3586 u − u + 3112 u − u + 576 u )= 14 (1 + 2 t ) ( −
325 + 200 t − t + 72 t )( − − t − t + 916 t + 20 t + 72 t )Set h ( t ) := − − t − t +916 t +20 t +72 t . Then h (cid:48)(cid:48) ( t ) = − t +240 t +1440 t > t ∈ (1 , h ( t ) attains maximum at t = 1 or t = 2. But h (1) = − < h (2) = − <
0, so h ( t ) < t ∈ (1 , t ∈ (1 ,
2) satisfying the equation above, wemust thus solve −
325 + 200 t − t + 72 t = 0 . t = t = 154 − (cid:32) √ − (cid:33) / + (cid:32) √
773 + 6547612 (cid:33) / ≈ . , in (1 ,
2) as desired.
A Appendix: Countably many exotic planes
In this appendix, we give a proof of Theorem 1.3; that is, there are at most countably many exoticplanes in a convex cocompact hyperbolic 3-manifold M = Γ \ H with incompressible boundary.We start by recalling the following theorem in [MMO2]: Theorem A.1 ([MMO2]) . If M is convex cocompact with incompressible boundary, then the fun-damental group of any plane P with P ∗ nonempty and closed in M ∗ is nontrivial. In particular, either P ∗ is a cylinder, or P ∗ is a nonelementary surface. For the latter case, thefundamental group of P ∗ contains a free group on two generators. Following the same argumentsof the proof of [MMO2, Cor. 2.3], we conclude that there are only countably many nonelementary P ∗ .For the former case, P ∗ contains a closed geodesic γ of M . Let ˜ γ be any lift, with end points p, q on the sphere at infinity. Let C be the boundary circle of P passing through p, q . We have Lemma A.2.
Each component of C − { p, q } is contained in the closure of a connected componentof the domain of discontinuity Ω .Proof. We follow many of the arguments in [MMO2, § P is the image of hull( C ) ∼ = H under an isometric immersion, which descends to a map f : S → M where S = Stab Γ ( C ) \ hull( C )is a hyperbolic cylinder. Let S ∗ = f − ( M ∗ ). Then S ∗ is a convex subsurface of S , also a topologicalcylinder, and the restriction f : S ∗ → P ∗ ⊂ M ∗ is proper.Since core( M ) is homeomorphic to M , M ∗ deformation retracts to a compact submanifold K ⊂ M ∗ . One can also arrange that K is transverse to f , and so S := f − ( K ) ⊂ S ∗ is a compact,smoothly bounded region in S ∗ , although not necessarily connected.As in the proof of Theorem 2.1 in [MMO2], after changing f by a compact deformation , wemay assume that the inclusion of S into S ∗ is injective on π . So the components of S are eithercylinders, or disks; see Figure 9.Boundaries of the cylinder components of S are essential curves of S ∗ , all homotopic; let S (cid:48) be the region of S ∗ bounded by the two of these boundary curves farthest into either end of S . Inparticular S (cid:48) is also a cylinder, and include all cylinder components of S . For each component of S not contained in S (cid:48) , since it is a disk with boundary in ∂K , we may homotope it rel boundaryinto ∂K , as K ∼ = M has incompressible boundary.Therefore after changing f further by a compact deformation, we may divide S into S (cid:48) andtwo ends S , S so that f ( S (cid:48) ) is a cylinder with both boundary components on ∂K , and f ( S ) and f ( S ) are contained completely in M − K .Let E , E be the two components of M − K containing f ( S ) and f ( S ). Note that the closedgeodesic γ contained in P is homotopic to an essential curve γ i on ∂E i for i = 1 ,
2. Moreover,11 igure 9: The hyperbolic cylinder S , the convex subsurface S ∗ bounded by red, and S E i differs from an actual end of M by a compact set, so any lift of E i is bounded at infinityby a connected component of Ω. If we take the lift corresponding to ˜ γ , let Ω i be the connectedcomponent bounding the lift of E i at infinity. Since p, q are the end points of ˜ γ i , we must have p, q ∈ ∂ Ω i . Finally, since f differs from an isometric immersion by compact deformation, the liftof f ( S i ) is a half plane bounded by ˜ γ i and a component of C − { p, q } at infinity; this half planeis totally geodesic except in a band of bounded width near ˜ γ i . This lift is contained completely inthe lift of E i , so the corresponding component of C − { p, q } is contained in Ω i .Given a closed geodesic γ ⊂ M , there is an S -family of planes passing through γ . Fix anycontinuous parametrization P t of this family by t ∈ R invariant under translation by an integer.Consider the set L := { t ∈ R : P ∗ t is a properly immersed cylinder in M ∗ } . We claim
Proposition A.3.
Suppose a sequence { t i } ⊂ L satisfies t i > t and t i → t for some t ∈ L . Thenthere exists (cid:15) > so that [ t, t + (cid:15) ] ⊂ L and P s is not exotic for any s ∈ ( t, t + (cid:15) ) .Proof. Choose any lift ˜ γ of the closed geodesic γ , and a boundary circle C of P t passing through theend points p, q of Γ. By the previous lemma, each of the two components of C − { p, q } is containedin the closure of a connected component of the domain of discontinuity Ω, say Ω and Ω .For each i , a boundary circle C i of P t i also passes through p, q . Since C i → C as i → ∞ , when i is large enough, the two components of C i − { p, q } intersect Ω and Ω respectively. As P ∗ t i is alsoa cylinder, those two components must be contained in Ω and Ω respectively.Finally, since M has incompressible boundary, Ω and Ω are both simply connected. In par-ticular, for any s ∈ ( t, t i ), a boundary circle C s of P s is sandwiched between C and C i , and C s intersects the limit set at exactly p and q . It is easy to see P s itself is closed in M .Similarly, a sequence { t i } ⊂ L tending to t ∈ L from below gives a corresponding interval with t as the right end point. We have Corollary A.4.
Every point in the set { t ∈ L : P t is exotic } is isolated, and therefore this set iscountable. M contains countably many closed geodesics, we conclude that there are only countablymany exotic cylinders. This, together with the fact that there are only countably many nonele-mentary P ∗ , gives Theorem 1.3. References [BO] Y. Benoist and H. Oh. Geodesic planes in geometrically finite acylindrical 3-manifolds.
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