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Crooked surfaces and anti-de Sitter geometry

Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Gu\'eritaud-Kassel lift to restrictions of crooked surfaces in Einstein space which are adapted under the involution of Einstein space defining anti-de Sitter space.


Introduction
In 1990, Todd Drumm [8] introduced crooked planes to build fundamental polyhedra for free discrete groups acting properly and isometrically on 3-dimensional Minkowski space E 3 . Minkowski space is the Lorentzian analog of Euclidean space, and anti-de Sitter space is the Lorenzian analog of hyperbolic space. Recently Jeff Danciger, François Guéritaud and Fanny Kassel [6] introduced analogous surfaces in 3dimensional anti-de Sitter space AdS 3 .
The conformal compactification of Minkowski space is the Einstein universe Ein 3 , the geometry of which has been extensively studied by Charles Frances [11]. In particular Frances [12] considered the extensions of crooked planes to Ein 3 , which he called generalized crooked planes. (See [2] for an expanded treatment of Einstein geometry and generalized crooked planes, which we renamed crooked surfaces.) The purpose of this note is to interpret the AdS-crooked planes of Danciger-Guéritaud-Kassel in terms of crooked surfaces in Ein 3 .
Here is a precise statement of our main result. 3-dimensional Minkowski space E 3 embeds in the 3-dimensional Einstein universe Ein 3 as the complement of a lightcone. The unique nontrivial double covering AdS of 3-dimensional anti-de Sitter space AdS 3 embeds in Ein 3 as the complement of a 2-dimensional Einstein hypersphere Ein 2 ⊂ Ein 3 . The conformal involution I S of Ein 3 whose fixed set equals Ein 2 realizes the deck transformation of the complement Ein 3 \ Ein 2 = AdS.
Let p ∈ AdS 3 , and letp, −p ∈ AdS be its two lifts under the double covering AdS −→ AdS 3 . Let s ∈ T p (AdS 3 ) be a unit-spacelike vector, and abusing notation, denote the corresponding tangent vector in Tp( AdS) by s as well. Denote the origin in T p (AdS 3 ) by 0 p . The tangent cone of C(p, s) at p is the subset of T p (AdS 3 ) consisting of tangent vectors to smooth paths in C at p. Identify the tangent space T p (AdS 3 ) -as a Lorentzian vector space -with Minkowski space E 3 .
A crooked surface in Ein 3 is determined by an ordered quadruple (p 0 , p ∞ , p 1 , p 2 ) ∈ Ein 3 where p 0 , p ∞ are non-incident and p i is incident to both p 0 , p ∞ for i = 1, 2. (Compare [2,16,4].) Say that the crooked surface C(p 0 , p ∞ , p 1 , p 2 ) is I S -adapted if and only if I S fixes p 1 and p 2 and interchanges p 0 and p ∞ . This is somewhat stronger than the condition that the set C(p 0 , p ∞ , p 1 , p 2 ) is I S -invariant, and rules out potentially interesting hypersufaces which resemble crooked planes, but are not crooked planes. A characterization of this condition in terms of the spine of a crooked surface is given in §4.

Main Theorem.
(1) Let p ∈ AdS 3 and s ∈ T p (AdS 3 ) be unitspacelike as above. Then the tangent cone to the AdS-crooked plane C(p, s) is a crooked plane C(0 p , s) ⊂ E 3 ≈ T p (AdS 3 ) and C(p, s) = Exp p C(0 p , s) . Then C(p, s) is an I S -adapted crooked surface in Ein 3 . (3) Conversely suppose that C is an I S -adapted crooked surface in Ein 3 . Then C arises from an AdS-crooked plane by the above construction.
The first section of the paper develops anti-de Sitter geometry, defines AdS-crooked planes, describes the double cover, and proves (1). The second section discusses the embedding Ψ of the double cover AdS in Ein 3 and the conformal model of anti-de Sitter space. Using this conformal realization and its symmetries, the final section proves (2) by computing one example explcitly. Symmetry is exploited one last time to prove (3), the converse statement in the Main Theorem.
Our viewpoint follows the spirit of classical projective geometry, but in the conformal context. Conformal models for hyperbolic and elliptic geometry arise from imposing involutions on the sphere S n , the model space for conformal (Riemannian) geometry. The conformal model of H n consists of the two hemispheres of S n complementary to the equator S n−1 ֒→ S n , each given the Poincaré metric. Reflection in the equator is the involution ι defining hyperbolic geometry, in the following sense. Hyperbolic space H n is the quotient of the complement of Fix(ι) = S n−1 by the cyclic group ι . The quotient map S n \ Fix(ι) −→ S n \ Fix(ι) / ι =: H n is a trivial covering space. Either hemisphere is a fundamental domain for the action of ι .
Here anti-de Sitter geometry arises from conformal Lorentzian geometry (on the Einstein universe Ein 3 ) by imposing an involution. Just as the complement of the fixed-point set of conformal inversion on S n is a double covering space of hyperbolic space, the complement of the fixed set of the Lorentzian conformal inversion is a double covering space of anti-de Sitter space. Unlike the Riemannian case above, this covering space is nontrivial.
Metric bisectors in real hyperbolic space H n are totally geodesic hypersurfaces. In this model, totally geodesic hypersurfaces are hyperspheres orthogonal to the equator. Orthogonality to the equator is equivalent to invariance under the reflection in the equator, the involution defining H n . In this way our observation is a direct analog in the crooked context.
A similar viewpoint was adopted in [14] to develop the theory of metric bisectors in complex hyperbolic space in terms of real analytic hypersurfaces in complex projective space P n C which we called extors. Metric bisectors (or equidistant hypersurfaces) in H n C analytically contine to extors in P n C . Furthermore they can be characterized as those extors adapted to the anti-polarity defining H n C ⊂ P n C . Metric bisectors in complex elliptic space (P n C with the Fubini-Study metric) can be similarly characterized as extors in P n C which are adapted to the anti-polarity defining complex elliptic geometry.
A consequence of our observation is that the disjointness criterion for crooked surfaces in Ein 3 proved by Charette, Francoeur, and Lareau-Dussault [4] specializes to a disjointness criterion for the AdS-crooked planes defined by Danciger, Guéritaud, and Kassel. (This extends the disjointness criteria for crooked planes developed in Burelle, Charette, Drumm and Goldman [3] and [9]).
Acknowledgements. I thank Jeff Danciger, François Guéritaud, and Fanny Kassel for many helpful conversations, and especially for sharing their results [6] with me.
These conversations occurred at the special trimester Geometry and Analysis of Surface Group Representations at Institut Henri Poincaré in Paris (January-March 2012), the GEAR Network Retreat at the University of Illinois in August 2012, and the workshop Higher Teichmüller-Thurston Theory at Centre de Recherches Mathématiques in Montréal in October 2012. I am indebted to these venues for their hospitality.
I also express my gratitude to Todd Drumm and Charles Frances for introducing us to crooked planes and crooked surfaces. I am especically grateful to Virginie Charette and Rosemonde Lareau-Dussault, for many illuminating conversations on this subject, and in particular for sharing their results [4,16] with me.
I am grateful to Jeff Danciger, Jean-Philippe Burelle and Greg Laun for reading preliminary versions of this manuscript and suggesting numerous improvements and corrections.
Notation and Terminology. Homogeneous coordinates on projective space will be denoted [X : Y : Z : U : V ], while inhomogeneous coordinates on affine Minkowski space will be denoted (x, y, z). As we predominantly consider Lorentzian 3-manifolds, with two spatial dimensions and one temporal dimension, we shall refer to the ambient dimension: Thus we denote Minkowski space E 2,1 by E 3 , anti-de Sitter space AdS 2,1 by AdS 3 and the Einstein universe Ein 2,1 by Ein 3 , or simply Ein. Denote the double cover of AdS 3 by AdS and the 2-dimensional Einstein universe (one spatial and one temporal dimension) by Ein 2 .
Denote a symmetric space by X = G/H, where G is the isometry group and H is the isotropy group. Denote the exponential map of a complete pseudo-Riemannian manifold M at a point p ∈ M by and the exponential map for a Lie group G with Lie algebra g by Denote the adjoint representation of a Lie group on its Lie algebra by Denote the vector space of n × n real matrices by Mat n (R) and the zero and identity matrices by 0 and 1, respectively. Denote the transpose of a matrix M by M † . Denote equivalence classes in PSL(2, R) of matrices A ∈ SL(2, R) by ±A. Denote the Lie algebra of SL(2, R), consisting of traceless 2 × 2 real matrices, by sl(2, R), and the hyperbolic plane (in the Poincaré upper halfplane model) by H 2 . Denote the geodesic in H 2 corresponding to the positive imaginary axis by H 1 . Denote the closure of a subset S byS and lifts of points, subsets S, etc. from a space M to a double covering space M −→ M by S. Denote the set of points fixed by a transformation f by Fix(f ). Denote the cyclic group generated by f by f .

Anti-de Sitter geometry
We begin with a review of the geometry of AdS, using the PSL(2, R)model. See Danciger [5], §4.8, for an expanded treatment of this model and its projective realization.
2.1. Isometries. A Lorentzian symmetric space is a Lorentzian manifold X such that for every point p, the isometry −1 of the tangent space T p X is the differential of a (necessarily unique) isometry (denoted ι p ) of X. We call ι p the symmetry of X at p. The isometry group acts transitively. For the general theory of (non-Riemannian) symmetric spaces, see Eschenburg [10], Helgason [15], and Wolf [20].
2.1.1. The PSL(2, R) model. A convenient model for 3-dimensional antide Sitter geometry is the group PSL(2, R). Its Lorentzian structure is bi-invariant, and arises from an Ad-invariant inner product on its Lie algebra sl(2, R). In this case AdS 3 corresponds to PSL(2, R) and the identity component of its isometry group is acting faithfully by left-and right-multiplication on PSL(2, R): The most convenient basepoint in a group is of course its identity element e. The isotropy group at e is the subgroup H ⊂ G 0 corresponding to the diagonal g 1 = g 2 . Furthermore H ∼ = PSL(2, R) acting by inner automorphisms of PSL(2, R).

2.1.2.
Components of the isometry group. Isometries not in the identity component are described, for example, in Danciger [5], §4.8 (p.74). Briefly, replace PSL(2, R) by the group PGL(2, R) of all isometries of H 2 , including those which reverse orientation on H 2 . The identity component of PGL(2, R) equals PSL(2, R). The component consisting of orientation-reversing isometries is the coset where, for example, corresponds to a reflection in a geodesic H 1 := Fix(R) ⊂ H 2 (the imaginary axis in the upper halfplane model of H 2 ). Now consider the action of PGL(2, R) × PGL(2, R) by left-and rightmultiplication on PGL(2, R) defined by (1). Then the group G of all orientation-preserving isometries of AdS 3 equals the index-two subgroup of PGL(2, R)×PGL(2, R) which stabilizes the component PSL(2, R) of PGL(2, R). The orientation-preserving isometry group G contains two components; the identity component is G 0 = PSL(2, R)×PSL(2, R). The new component is the coset This coset consists of isometries which preserve the ambient orientation of AdS 3 but reverse time-orientation.
2.1.3. Symmetries in points. The symmetry of the symmetric space AdS 3 at e is given by group-inversion: It is equivariant respecting the permutation automorphism of PSL(2, R)× PSL(2, R): because applying (2) to (1) : The differential of ι e is the antipodal map −1 on the tangent space T e (AdS 3 ) ∼ = sl(2, R), and it reverses orientation. It generates the other cosets of the full group of isometries of AdS 3 , including isometries which reverse orientation as well as possibly reversing time-orientation. In this way AdS 3 is a Lorentzian symmetric space with ι e as the symmetry in e. More generally, the symmetry in the point g ∈ AdS 3 = PSL(2, R) is: 2.2. Geodesics and totally geodesic surfaces. Geodesics in AdS 3 are left-cosets of one-parameter subgroups in PSL(2, R). Since g exp(tξ) = exp t Ad(g)ξ g, geodesics correspond to right-cosets as well. AdS 3 is geodesically complete: since exp(tξ) is defined for all t ∈ R, every geodesic extends indefinitely in its affine parametrization. Furthermore any two points in AdS 3 are connected by a geodesic.
2.2.1. Geometry of the tangent space. Before describing the geodesics and totally geodesic surfaces, we review basic facts about the Lorentzian vector space sl(2, R) ∼ = R 2,1 . Let R 2,1 denote the 3-dimensional Lorentzian vector space with inner product associated to the quadratic form: Under the identification the inner product in R 2,1 identifies with the inner product in sl(2, R) associated to the quadratic form: We denote the corresponding (Lorentzian) inner product by A·B. on sl(2, R) equals −8 times the quadratic form defined by (15).) One reason for this choice of inner product is that if A denotes one of the the matrices respectively, then A · A equals 1, 1, and −1 respectively. Any inner product on a tangent space to a Lie group extends uniquely to a left-invariant pseudo-Riemannian metric. An inner product on the tangent space to the identity element e which is invariant under the adjoint representation Ad extends to a unique pseudo-Riemannian metric invariant under both left-and right-multiplications. In this way the inner product corresponding to (15) defines a bi-invariant Lorentzian metric on PSL(2, R), which has constant curvature −1. This Lorentzian manifold is anti-de Sitter space and denoted AdS 3 .
Under the identification (5), the Lorentzian cross-product corresponds to 1/2 the Lie bracket in sl(2, R). The scalar triple product in R 2,1 identifies with the 3 × 3 determinant: and satisfies, for example, the vector identity is orthogonal to both A and B: (3) implies the Cartan involution is the permutation switching the two summands. The isotropy algebra h is the 1-eigenspace of the Cartan involution and is the diagonally embedded subalgebra tangent to the isotropy group H = Inn PSL(2, R) . Its orthogonal complement m is the (−1)-eigenspace of the Cartan involution and consists of the image of the embedding and models the tangent space

2.2.3.
Geodesics. AdS 3 is a Lorentzian symmetric space and we recall several general facts about symmetric spaces. Choose the identity element e ∈ PSL(2, R) as basepoint in AdS 3 and identify its tangent space T e PSL(2, R) with the Lie algebra sl(2, R) as above.
Geodesics through e in AdS 3 correspond to one-parameter subgroups in PSL(2, R). In particular the Lie-theoretic exponential map agrees with the Levi-Civita exponential map: under the identification of the Lie algebra sl(2, R) with the tangent space: [15], pp. 224-226, and [10]. In particular [15], §V.6 and Exercise 3 (p. 227) describe symmetric spaces arising from Lie groups with the action of left-and right-multiplications.) • Parallel transport along γ is the differential As in §2.2.2, the isotropy algebra h acting on AdS 3 is the diagonal and its orthogonal complement m equals Therefore the transvection along the geodesic γ(t) corresponding to a one-parameter subgroup exp(tξ) ∈ G equals: Totally geodesic subspaces and Lie triples. Identify m ∼ = T e (AdS 3 ) with sl(2, R) to realize subspaces of the tangent space as subspaces of sl(2, R).
Let g = h ⊕ m be the Cartan decomposition. Then totally geodesic subspaces correspond to Lie triples, that is, subspaces s ⊂ m such that  are Borel subalgebras, each of which is conjugate to the algebra b of upper-triangular matrices. Then [s, s] ∼ = [b, b] is a line spanned by a nonzero element (corresponding to a nilpotent upper-triangular matrix in sl(2, R)). Furthermore The remaining two cases follow from the vector identity (7). When s is spacelike, a normal vector is elliptic, and a normal vector v of unit length will satisfy the matrix equation v 2 = −1. Furthermore ad(v) acts on S = v ⊥ by rotation of order 4. When S is timelike, a unit length normal vector satisfies v 2 = 1 and ad(v) acts on S by a linear map with eigenvalues ±1.
Totally geodesic surfaces through the basepoint e ∈ AdS 3 are precisely the images exp(s) = Exp e (s), where we identify T e (AdS 3 ) with m. Thus, for any point g ∈ AdS 3 and every 2-plane s ⊂ T g (AdS 3 ), is the unique totally geodesic surface S ∋ g with T g (S) = s. Corresponding to the three types of planes in sl(2, R), there are three types of totally geodesic surfaces of AdS 3 . (Compare §2.3.1, §2.3.3, §3.5, §2.2, and §3.5.2.) 2.3.1. Hyperbolic planes in AdS 3 . A point g ∈ AdS 3 determines a dual spacelike totally geodesic surface g * ⊂ AdS 3 which is spacelike, as follows.
At the origin e ∈ AdS 3 , timelike geodesics correspond to elliptic oneparameter subgroups, all of the form An elliptic one-parameter subgroup is periodic in SL(2, R) with period 2π and periodic in PSL(2, R) ≈ AdS 3 with period π.
For an arbitrary point g ∈ AdS 3 , the set of timelike geodesics from g correspond to cosets ghE θ h −1 of elliptic one-parameter subgroups, where the parametrization by θ is chosen so that E θ = e if and only θ ∈ πZ. In particular each timelike geodesic in AdS 3 is closed and has length π.
Denote the set of involutions (elements of order two) in PSL(2, R) by Inv. The inverse image Inv of Inv in SL(2, R) consists of elements of order four, which have the form hE θ h −1 where θ ≡ ±π/2 (mod 2π).
The hyperbolic plane dual to g is the isometrically embedded H 2 defined as: Here is an elegant description of this duality. Let g ∈ AdS 3 . Then the fixed set Fix(ι g ) of the symmetry ι g in g equals the disjoint union {g} ∪ g * .
Explicitly, e * = Inv is the image of the isometric embedding of the upper halfplane y > 0. The positive imaginary axis x = 0 < y defines a geodesic

2.3.2.
Indefinite planes in AdS 3 . Totally geodesic indefinite 2-planes are isometrically embedded copies of AdS 2 , which are homeomorphic to Möbius bands, and therefore nonorientable.
One can see this as follows. The isometry group G 0 acts transitively on pairs (g, s), where g ∈ AdS 3 is a point and S ⊂ T p (AdS 3 ) is an indefinite 2-plane. Thus we may assume that g = e and s is the subspace of T e (AdS 3 ) ∼ = sl(2, R) consisting of matrices of the form 0 * * 0 .
In SL(2, R) this subspace exponentiates to the subset S of matrices having equal diagonal entries with determinant 1: which is a one-sheeted hyperboloid. Its image in AdS 3 is the quotient of this hyperboloid by the antipodal map  with quotient homeomorphic to a Möbius band. Since it is nonorientable, and AdS 3 is orientable, it is one-sided in AdS 3 .

2.3.3.
Null planes in AdS 3 . Tangent to degenerate 2-planes in T g (AdS 3 ) are degenerate totally geodesic subspaces, which we call null planes. A convenient model is the Borel subgroup B = exp(b) consisting of uppertriangular matrices. The geodesics through e in this null plane are the one-parameter subgroups: The commutator subgroup consists of unipotent upper-triangular matrices, the cosets of which are null geodesics. They define a foliation tangent to the line field detecting the degeneracy of the induced Lorentzian structure. The complement of one of these null geodesics is the disjoint union of two null halfplanes. Just as in Minkowski space, the ambient orientation of AdS 3 determines a way to consistently choose a null halfplane bounded by a null geodesic ℓ. (See §2.3.4.) A null geodesic ℓ in AdS 3 lies in a unique null plane P (ℓ). Given any point g ∈ ℓ, the null 2-plane is the unique null plane in the tangent space containing the line T p (ℓ). The totally geodesic subspace is the desired null plane. To see that it is independent of g, choose coordinates so that P consists of the image of upper-triangular matrices in PSL(2, R), and let Φ ℓ (t) denote the transvection along ℓ, taking g ∈ ℓ to h ∈ ℓ. Using the formula (8) for transvections, so that P (h, ℓ) = Φ ℓ (t)P (g, ℓ) = P (g, ℓ) as desired. Henceforth denote P (g, ℓ) by just P (ℓ).

2.3.4.
Consistent choice of orientations. The complement P (ℓ) \ ℓ decomposes as the disjoint union of two null halfplanes; here is how to choose one of them, which we call the wing W(ℓ).
First assume an orientation on AdS 3 , which is given by a volume form Det on the tangent space T g (AdS 3 ). Choose a future-pointing vector v tangent to ℓ, and another future-pointing vector u in the complement of P (ℓ). Then, for p ∈ ℓ, the wing is a null halfplane bounded by ℓ. Compare [3], §3.2.

2.4.
AdS-crooked planes. Now we define crooked planes in anti-de Sitter space, following Danciger-Guéritaud-Kassel [6]. We begin by giving several definitions, the first one of which is analogous to the orginal definition in Minkowski space. From that we given alternate definitions, in terms of the vertex/spine pair, and a dual definition which doesn't have a direct analogue in Minkowski space. Then we describe how an AdS-crooked plane C determines a crooked plane C in Minkowski space, by a tangent cone construction: C is the tangent The main difference between crooked planes in AdS 3 and crooked planes in Minkowski space is that the particles, the timelike geodesics lying on the stem are compact (homeomorphic to circles) in AdS 3 but, in Minkowski space, are lines.
2.4.1. Stems, hinges, wings and the spine. Following [6], given a point g and a unit spacelike vector s ∈ T g (AdS 3 ), we define an AdS-crooked plane, as follows. The orthogonal complement s ⊥ ⊂ T g (AdS 3 ) is a timelike 2-plane, and it exponentiates to a totally geodesic indefinite plane S := Exp g (s ⊥ ) ∋ g. The stem Stem(p, s) is defined as the union of the non-spacelike geodesics in S. We call the timelike geodesics lying in S the particles.
Stem(p, s) is bounded by two null geodesics h 1 , h 2 , the hinges. Each hinge lies in a unique null plane, as in §2.3.3. The stem is the union of the two hinges and the particles.
The wings W i := W(h i ) are defined as the null halfplanes associated to the hinges, as in (14). The crooked plane C(g, s) is then defined as the union: The spine is the unique spacelike geodesic contained in the crooked plane, and equals the image σ of: The spine lies on the union of two wings W 1 ∪ W 2 , bisecting each one. The vertex g and the spine σ completely determine the crooked plane.
Since the identity component of the isometry group G 0 acts transitively on pairs (g, s), all AdS-crooked planes are equivalent under the orientation-preserving isometry group of AdS 3 .

2.4.2.
The dual definition. Using the duality between points in AdS 3 and hyperbolic planes, here is an alternate definition of crooked planes in AdS 3 . Dual to the vertex g is a hyperbolic plane g * ⊂ AdS 3 . The stem Stem(g, s) intersects g * in a geodesic ℓ ⊂ g * . The pair (ℓ, g * ), consisting of a hyperbolic plane g * and a geodesic ℓ ⊂ g * also suffices to determine the crooked plane.
To recover the vertex-spine pair from a pair (H, ℓ), where H is a hyperbolic plane containing a geodesic ℓ, take the vertex to be dual to H. A unique spacelike plane S contains ℓ and is orthogonal to H. Since S is orthogonal to H, it contains g. Indeed, S is characterized as the unique spacelike plane containing ℓ ∪ {g}. The interesection H ∩ S is a spacelike geodesic in S, which is the spine σ of the crooked plane. Thus the data (g, ±s) is equivalent to the data (H, ℓ). This dual description is absent from the classic construction of crooked planes in Minkowski space.

2.4.3.
The tangent cone crooked plane. An AdS-crooked plane with vertex g is star-shaped about g: that is, it is the union of geodesic rays emanating from g. By definition, its tangent cone C ⊂ T g (AdS 3 ) at g consists of all vectors v ∈ T g (AdS 3 ) tangent to smooth rays emanating from g. (It is a cone in that it is invariant under positive scalar multiplications in the tangent space.) In particular, Moreover, under the identication of the tangent space T g (AdS 3 ) with Minkowski space, C ⊂ T g (AdS 3 ) identifies with a crooked plane with vertex 0 ∈ T g (AdS 3 ). with ad − bc = 1. Since AdS = SL(2, R) is a group, it has a natural basepoint, the identity matrix 1 ∈ SL(2, R). The deck transformation is the involution given by multiplication by −1, which we denote bŷ The action of G 0 lifts to the action of the group SL(2, R) × SL(2, R) by left-and right-multiplication (1), whose kernel is the diagonally embedded central subgroup of order two generated by (−1, −1).
In AdS, the timelike geodesics are all closed and have length 2π. Unlike AdS 3 , not every pair of points in AdS is connected by a geodesic. Let 1 ∈ AdS be the identity matrix. Then the point corresponding to g ∈ SL(2, R) is connected by a geodesic to 1 if and only if tr(g) > −2 or g corresponds to −1.
The duality between points and hyperbolic planes is somewhat different as well. Choose a pointĝ ∈ AdS covering g ∈ AdS 3 and a lift g * ⊂ AdS of the dual hyperbolic plane g * ⊂ AdS. The preimage in AdS of g * is the disjoint unionĝ * ∪(−ĝ * ), whereĝ * is an (arbitrarily chosen) component.
Then the symmetry ιĝ inĝ fixes bothĝ and −ĝ but interchanges the planesĝ * and −ĝ * . On the other hand, is another reflection which fixes each of the two planes ±ĝ * obtained by lifting g * , but interchanges the pointsĝ and −ĝ. Each of the two wings of C(ĝ, s) has two preimages as well, determined by which hinge or cohinge bounds them. We call the lifts of the wings which pass through the vertex wings of the AdS-crooked plane, and the lifts of the wings through the covertex the cowings. Similarly the spine lifts to two spacelike geodesics; the one containing the vertex we call the spine of the AdS crooked plane and the one containing the covertex its cospine.
Although AdS-crooked planes are star-shaped about their vertices, their lifts are not. If C is the tangent cone of an AdS crooked plane C at a vertexĝ, then Expĝ(C) contains neither cohinges nor cowings. However, where −C denotes the tangent cone of AdS at the covertex −ĝ.

Embedding in the Einstein universe
The double covering space AdS of AdS 3 embeds in Ein 3 . This is analogous to the embedding of a double covering space of H 2 in S 2 as the upper and lower hemispheres. Reflection in the equator permutes these hemispheres, acting as the deck involution in this covering space. The equator is the fixed set of the reflection, and models conformal Riemannian geometry in one dimension lower. Similarly conformal inversion I S in the unit (Lorentzian) sphere is the deck involution in AdS for the double covering AdS → AdS 3 . The complement of AdS in Ein 3 is exactly the set Fix(I S ) of fixed points, which is just Ein 2 ⊂ Ein 3 , analogous to the Riemannian case.
3.1. Embeddings in Ein 3 . Following [2], Einstein n-space Ein n is the projectivized nullcone P(N) ⊂ P(R n,2 ) in the real inner product space R n,2 . We follow the notation from [2] for n = 3: points in R 3,2 have coordinates (X, Y, Z, U, V ) and the quadratic form is: Thus the nullcone in R 3,2 equals: The Minkowski patch Min(p ∞ ) is the image of the embedding of threedimensional Minkowski space in Ein 3 (see [2]): In homogeneous coordinates, this Minkowski patch Min(p ∞ ) is defined by V = 0, and the point p 0 corresponds to the origin (0, 0, 0) ∈ E 3 . The ordered pair (p ∞ , p 0 ) makes E 3 into a Lorentzian vector space. The affine lightcone (the "usual" light cone in Minkowski space) Analogous to inversion in Euclidean geometry, inversion in S : is conformal, and we use this inversion to define a coordinate atlas on Ein 3 . It is the restriction of the involution of Ein 3 defined by: interchanging p 0 and p ∞ . The image consisting of ideal points of photons through the origin p 0 . It (together with the improper point p ∞ ) compactifies both the unit-sphere S and the lightcone L aff (p 0 ). Alternatively, C ∞ consists of ideal points of photons through the improper point p ∞ ).
3.2.1. Topology of Ein 3 . Topologically, Ein 3 is a mapping torus of the antipodal map of S 2 , decomposing into four strata: • (Dimension 3) The Minkowski patch E 3 is a 3-cell; • (Dimension 2) The inverse cone I consists of two disjoint annuli; • (Dimension 1) The ideal circle C ∞ is a 1-sphere; • (Dimension 0) The improper point p ∞ is a 0-cell.
For more details see [3,16]. Because of (18), the union Min(p 0 ) ∪Min(p ∞ ) equals the complement Ein 3 \C ∞ . Since Ein 3 is not a sphere, one needs at least three Minkowski patches to cover it. If p ∈ Ein 3 , then if and only if p lies on a timelike line through p 0 . This is the condition that the Maslov index of the triple (p 0 , p ∞ , p) equals ±1.
The involution I S extends to Ein 3 by mapping p 0 to p ∞ , mapping L aff (p 0 ) \ {p 0 } to I, and fixing C ∞ pointwise. Denote this involution of Ein 3 by I S as well, when there is no danger of confusion.

Equivariance under the deck involution. The double covering
AdS −→ AdS 3 corresponds to the double covering SL(2, R) −→ PSL(2, R). In this model, the deck involution is multiplication by −1 in SL(2, R), and Ψ relates this involution to the inversion I S defined by (17). To see this, write: The hyperplane in R 3,2 defined by U = V has signature (2, 2) and therefore its image is a 2-dimensional Einstein subspace, which we henceforth call Ein 2 . It equals the union Although the unit sphere S (that is, the one-sheeted hyperboloid x 2 + y 2 − z 2 = 1) disconnects E 3 , its closure Ein 2 does not disconnect Ein 3 .

3.3.3.
Ideal points for the embedding of AdS ֒→ Ein 3 . The lightcone is the complement of Minkowski space E 3 in Ein 3 . It is the union of all null geodesics through the improper point p ∞ = I S (p 0 ). Since p ∞ corresponds to the matrix −1 ∈ SL(2, R), the lightcone L(p ∞ ) corresponds to the cosets of unipotent one-parameter subgroups through −1. These are precisely the elements of SL(2, R) having trace −2.
As in [2], this lightcone meets L(p 0 ) in the ideal circle C ∞ . Thus the boundary of AdS is the Lorentzian unit sphere S ⊂ E 3 compactified with the ideal circle C ∞ . The union S ∪ C ∞ is Ein 2 , which is topologically a torus, which is a 1-sided surface in the nonorientable 3-manifold Ein 3 . Since this theory seems not to be so well-known, we begin with a brief exposition of singular projective transformations. This is a simple case of the theory of wonderful compactifications developed by de Concini-Procesi [7]. For more details see [13]  Points on the boundary ∂SL(2, R) ≈ S 1 × S 1 can be geometrically interpreted as follows. Choose an orientation O R 2 on R 2 . The double covering RP 1 of RP 1 consists of oriented lines, that is, pairs (ℓ, O ℓ ),

Einstein hyperspheres and the boundary of
where ℓ ∈ RP 1 is a line in R 2 and O ℓ is an orientation on ℓ. Then the boundary ∂SL(2, R) consists of ordered pairs of oriented lines such that: 3.4.2. Rulings of Ein 2 . The fibers of the two projections correspond to the rulings of ∂ AdS = Ein 2 . (Compare Danciger [5].) In terms of the affine piece of Ein 2 , the Lorentzian unit-sphere S defined by x 2 + y 2 − z 2 = 1, these are just the two families of null lines

3.5.
AdS totally geodesic subspaces in Ein 3 . To relate AdS-crooked planes, crooked surfaces in Ein 3 and classic crooked planes in Minkowski space E 3 , we first describe how AdS totally geodesic subspaces extend to submanifolds of Ein 3 .
3.5.1. Geodesics. As timelike geodesics in AdS correspond to elliptic one-parameter subgroups in SL(2, R), they are all closed and do not approach the boundary of AdS in Ein 3 . Null geodesics in AdS have a single endpoint in ∂ AdS = Ein 2 . In our conformal Minkowski model of Einstein space, there are two types, depending on whether or not the null geodesic ν passes through the improper point. hIf p ∞ ∈ ν, then ν lies on I ∪ {p ∞ }; its single endpoint lies on the ideal circle C ∞ . If ν does not contain p ∞ , then it misses the ideal circle and contains a point of I. Its endpoint lies on S .
The embedding Ψ defined in (19) maps the (cosets of) unipotent one-parameter subgroups to photons: Their images h + 1 , h + 2 , correspond to null lines x = y − z = 0 and x = y − z = 0 respectively. Inverting in S yield h − 1 , h − 2 respectively, which lie on the inverse cone.
Finally, Ψ maps cosets of hyperbolic one-parameter subgroups in SL(2, R) = AdS to spacelike arcs. An illustrative model is the intersection of the x-axis with the one-sheeted hyperboloid S . The geodesic in AdS emanating from 1, which corresponds to the origin (0, 0, 0) ∈ E 3 , is just the line segment The geodesic in AdS emanating from −1, which corresponds to the improper point p ∞ , divides into the union of two rays which under (16) corresponds to the x-axis y = z = 0, with The other geodesic corresponds to its I S -image: which under (16) corresponds to the x-axis y = z = 0, with These two AdS geodesics, together with their two endpoints on S constitute a spacelike circle in Ein 3 , as defined in [2].

3.5.2.
Totally geodesic surfaces. Now we see how totally geodesic surfaces of AdS embed in Ein 3 . Recall three types of totally geodesic surfaces in AdS 3 exist: definite planes, isometric to H 2 ; degenerate planes, corresponding to cosets of Borel subgroups of SL(2, R); indefinite planes, isometric to AdS 2 .
A definite plane is dual to a point g ∈ AdS 3 as discussed in §2.3.1. Here is a simple example. The plane e * dual to the identity element e ∈ PSL(2, R) is the set of involutions Inv ⊂ PSL(2, R); see (10) for an explicit embeddiing of H 2 in AdS 3 . In Ein 3 the two lifts of this hyperbolic plane are the two components of the hyperboloid x 2 + y 2 − z 2 = −1; their common boundary is the ideal circle C ∞ ⊂ ∂ AdS.
Another example of a hyperbolic plane is the plane dual to an involution itself. For example, let g ∈ AdS 3 be the involution J(0) defined in (11) and corresponding to i ∈ H 2 . Then one lift of g * to AdS is the unit disc D 2 in the xy-plane defined by z = 0. The other lift is the exterior of D 2 in the xy-plane, which is the image I S (D 2 ). The closure of each disc intersects S in the unit circle in the xy-plane.
An example of a degenerate totally geodesic surface is the Borel subgroup B = exp(b) ⊂ PSL(2, R) consisting of the images of uppertriangular matrices in SL(2, R); compare the discussion in §2.3.3. Lift B to the connected subgroup B ⊂ SL(2, R), which embeds in Ein 3 by: The image is the subspace defined by Y = Z but U = V . In affine coordinates (x, y, z) on E 3 , this is the plane y = z. Its closure in Ein 3 is the lightcone L(p 1 ) where Furthermore the boundary of Ψ( B) consists of two rulings S + 0 (given by x = 1) and S − π (given by x = −1), as defined in (20). Another degenerate plane is defined by the subgroup B † consisting of the images of lower triangular matrices. It embeds via The image is the subspace defined by Y = −Z but U = V . In affine coordinates (x, y, z) on E 3 , this is the plane y = −z. Its closure in Ein 3 is the lightcone L(p 2 ) where Furthermore the boundary of Ψ( B) consists of two rulings S − 0 (given by x = 1) and S − 0 (given by x = −1), as defined in (20). Finally we consider an indefinite plane discussed in §2.3.2. It maps to Ein 3 by: where a 2 − bc = 1. The image Ψ(S) is defined by X = 0 and U = V .

Crooked surfaces adapted to an involution
This section develops the theory of crooked surfaces, especially as they relate to an involution I S in an Einstein hypersphere. The main goal is to show that classic AdS-crooked planes correspond to crooked surfaces in Ein 3 which are adapted to I S .
A useful invariant of a crooked plane is its spine: the unique spacelike geodesic σ lying on the crooked plane. The spine contains the vertex p and the pair (p, σ) determines the crooked plane.
In a similar way, we define a spine σ for a crooked surface C. When C is the closure of a crooked plane C ∈ E 3 , then σ equals the union σ ∪ {p ∞ }, where p ∞ is the improper point. We relate this invariant to the parametrization of crooked surfaces by stem-configurations discussed in [2]. The spine σ is a spacelike circle containing the vertex p 0 and the improper point p ∞ . In general triples (σ, p 0 , p ∞ ), where σ is a spacelike circle in Ein 3 and p 0 , p ∞ ⊂ σ are distinct points, provide an alternate parametrization of crooked surfaces.
We say that C is adapted to I S if and only if its spine σ is invariant under I S and p 0 , p ∞ are interchanged under I S .

Crooked surfaces.
A crooked surface is a piecewise smooth hypersurface in Ein 3 constructed as follows. Take two points q 0 , q ∞ in Ein 3 which do not lie on a photon. Call one of them, say q 0 , the vertex . Call the other one, q ∞ , the covertex. The intersection of the lightcones L(q 0 )∩L(q ∞ ) is a spacelike circle in Ein 3 disjoint from {q 0 , q ∞ }. Choose two distinct points q 1 , q 2 ∈ Ein 3 which lie on L(q 0 ) ∩ L(q ∞ ). Call the points q 1 , q 2 the hingepoints of the crooked plane.
For i = 0, ∞ and j = 1, 2, let φ j i be the unique photon containing q i and q j . Call the photons φ 1 0 , φ 2 0 hinges and the photons φ 1 ∞ , φ 2 ∞ cohinges. Call such a quadruple (q 0 , q ∞ ; q 1 , q 2 ) a stem configuration. Such a configuration lies on a unique Einstein hypersphere W . The hinges and cohinges are rulings of W and decompose W into four squares, each of which have the same four vertices q 0 , q 1 , q ∞ , q 2 . Two of these squares contain timelike directions emanating from q 0 or q ∞ , and constitute the two components T 1 , T 2 of the stem.
The wings are obtained from the lightcones L(q j ) as follows. The photons φ j 0 , φ j ∞ separate L(q j ) into two bigons, one of which attaches to T 1 ∪ T 2 . 4.1.1. The spine of a crooked surface. Spacelike circles in Ein 3 correspond to projectivized nullcones of three-dimensional subspaces F ⊂ R 3,2 of signature (2,1). That is, F is the image of an isometric embedding R 2,1 ֒→ R 3,2 of inner product spaces. Its orthogonal complement F ⊥ is nondegenerate as well, and has signature (1, 1) and together F, F ⊥ define an orthogonal decomposition The projectivized null cone of R 1,1 defines an unordered pair of nonincindent points in Ein 3 . Orthogonal complementation defines a natural bijection between spacelike circles and unordered pairs of nonincident points. When applied to such a pair {p 1 , p 2 } ⊂ Ein 3 , the corresponding spacelike circle equals the intersection of lightcones L(p 1 ) ∩ L(p 2 ). Suppose that C = C(p 0 , p ∞ ; p 1 , p 2 ) is a crooked surface. Define the spine σ of C as the spacelike circle dual to the pair of hingepoints p 1 , p 2 : σ := L(p 1 ) ∩ L(p 2 ).
Since p 0 and p ∞ are each incident to p i for i = 1, 2, both p 0 and p ∞ lie in σ. The triple (p 0 , p ∞ ; σ) is equivalent to the stem-configuration (p 0 , p ∞ ; p 1 , p 2 ) and equally well determines the crooked surface C.

4.1.2.
Invariant crooked surfaces and adapted crooked surfaces. Suppose that C is a crooked surface invariant under the involution I S . Then I S preserves the stem-configuration and permutes the pairs p 0 , p ∞ and p 1 , p 2 . The points p 0 , p ∞ are characterized by the property that the tangent cones of C at these points are crooked planes. We say that C is adapted to I S if I S interchanges p 0 and p ∞ but fixes p 1 and p 2 .
This condition is stronger than being invariant. The only spacelike circles C invariant under I S meet S in two points. Thus the restriction of I S is an orientation-reversing involution of C fixing this pair. In particular I S -invariant crooked surfaces exist where I S do not fix the hingepoints, and these will not be adapted in our definition.

4.2.
Completing AdS crooked planes to crooked surfaces. Next we show ((2) of the Main Theorem) that an AdS-crooked plane lifts to an object in AdS whose closure in Ein 3 is a crooked surface, as defined in [11,2,16]. Call the special point corresponding to the improper point in a crooked surface arising from a crooked plane in Minkowski space) the covertex. As shown in [2], the vertex and covertex are related by an automorphism of Ein 3 which preserves the crooked surface. The condition that the crooked surface in Ein 3 arises from an AdS-crooked plane is that the vertex and covertex are related by the involution I S .
Begin with an AdS-crooked plane C(g, s) as in §2.4. Working in the PSL(2, R)-model, we may assume that the vertex g equals the identity element e ∈ PSL(2, R). Now lift the crooked plane to an AdS-crooked plane C(p, s) in AdS as in § 2.5.1. Then the vertexĝ is the identity matrix 1 ∈ SL(2, R). The covertex is −1 ∈ SL(2, R), the image of 1 under the deck involution. We identify the tangent spaces T e (AdS 3 ) with T 1 ( AdS) and denote the unit spacelike vector in T 1 ( AdS) corresponding to s byŝ.
Under the embedding Ψ of AdS in Ein 3 , the vertex 1 corresponds to the origin 0 = (0, 0, 0) in Minkowski space E 3 andŝ corresponds to a unit spacelike vector (still denotedŝ) in T 0 E 3 ∼ = V. By applying yet another isometry of AdS 3 , we can assume thatŝ is dual to the geodesic H 1 ⊂ H 2 defined by (11), corresponding to the matrices in SL(2, R) with zero diagonal entries. Now we build up the picture of the AdS-crooked plane and show that its closure in Ein 3 is the crooked surface which is the closure of a standard crooked plane in E 3 . The main difference are the parametrizations of particles and the spine, since these non-null geodesics are complete in both the realization in Minkowski space and anti-de Sitter space. In Minkowski space these geodesics limit to the improper point p ∞ while in AdS 3 these geodesics limit to Ein 2 = S ∪ I.

4.2.1.
Particles and the stem. We begin with a particle, that is a timelike geodesic, and show that it is just a reparametrized particle in Minkowski space compactified by the improper point p ∞ . The stem is a union of particles, whose intersection points with p * form a geodesic in the hyperbolic plane p * . As in §3.5, the two lifts of p * correspond to the components of the 2-sheeted hyperboloid x 2 + y 2 − z 2 = −1. The geodesic σ ⊂ p * corresponding to H 1 ⊂ H 2 is represented by the hyperbola x = y 2 − z 2 + 1 = 0.
Under the embedding AdS ֒→ Ein 3 in (19), this is an open subset in the hypersurface X = 0, that is the subset defined by: which defines the stem of the standard example of a crooked plane, as in [9]. The embedding of this totally geodesic submanifold of AdS is explicitly given in (29).

4.2.2.
Hinges and cohinges. The stem is bounded by two hinges, which in AdS 3 is the pair of unipotent one-parameter subgroups which respectively correspond to the hinges h 1 , h 2 ⊂ AdS 3 . By (30), their AdS-lifts map via Ψ to arcs of photonŝ In Minkowski space these hinges are just the lines consisting of points (0, y, z) ∈ E 3 where y = z and y = −z respectively. and the cohinges are their I S -images in the inverse cone I. The hingepoints p i = ∂ĥ ± 1 (for i = 1, 2) each lie on the ideal circle C ∞ . 4.2.3. Wings and cowings. From the hinges h 1 , h 2 we determine the halfplanes h ⊥ i (where i = 1, 2) in Minkowski space. These will intersect AdS in totally geodesic null surfaces, and the complements of the hinges in these planes will be halfplanes, one of which will be the wings of the crooked plane. As the hinges h ± correspond to unipotent one-parameter subgroups U ± (as in (30) in §4.2.2), the associated null planes h ⊥ ± correspond to the normalizers of these subgroups B ± := Norm(U ± ): (32) ± a b 0 a −1 , ± a 0 c a −1 which define totally geodesic surfaces in AdS which are degenerate. The complement h ⊥ ± \ h ± is a disjoint union of two null halfplanes.
The unipotent one-parameter subgroup U (which is the commutator subgroup of B ± ) is defined by a = 1. While U is the stabilizer of the vector 1 0 ∈ R 2 , its normalizer B is the stabilizer of the corresponding point (denoted ∞) in the projective line P(R 2 ). Furthermore U separates B into two subsets depending on whether a > 1 or a < 1. The complement B \ U consists of hyperbolic elements in B, and the two subsets are distinguished by whether ∞ is an attracting (respectively repelling) fixed point. The corresponding subsets in Ein 3 are, respectively: Let (q 0 , q ∞ ; q 1 , q 2 ) be a stem configuration and C := C(q 0 , q ∞ ; q 1 , q 2 ) ⊂ Ein 3 the corresponding crooked surface. Suppose that C is I S -adapted. Then I S interchanges the vertex q 0 and the covertex q ∞ and fixes the hingepoints q 1 and q 2 . Then, since I S fixes neither q 0 nor q ∞ , both points lie in Ψ( AdS). Since the group of isometries Isom( AdS) acts transitively on AdS, we may apply an automorphism of AdS 3 to assume that the covertex of C is the improper point, that is, q ∞ = p ∞ . Since I S interchanges the origin and the improper point, and interchanges the covertex and the vertex (because C is I S -adapted), the vertex equals the origin: q 0 = p 0 . Because the covertex of C equals p ∞ , the crooked surface is the closure of a crooked plane C in Minkowski space: and the vertex of this crooked plane is the origin p 0 . Now we adjust the hingepoints. The group of orientation-preserving isometries AdS which fix 1 and −1 is just the group H of inner automorphisms of SL(2, R), which is isomorphic to the identity component of SO (2,1). Similarly the group of orientation-preserving isometries of E 3 which fix the origin equals SO(2, 1), and the embedding Ψ is equivariant with respect to the isomorphism Inn(SL(2, R)) ∼ = − → SO(2, 1) 0 ∼ = Isom(H 2 ) 0 .
Since Isom(H 2 ) 0 acts transitively on geodesics in H 2 , the group SO(2, 1) acts transitively on ordered pairs of distinct null lines in R 2,1 . Therefore there exists an isometry of AdS which maps the hingepoints q i to p i for i = 1, 2 respectively. Apply this isometry to assume that q i = p i . Thus C is the closure of the double cover C of an AdS-crooked plane C. The proof is complete.