Lorentz Dynamics on Closed 3-Manifolds

In this paper, we give a complete topological, as well as geometrical classification of closed 3-dimensional Lorentz manifolds admitting a noncompact isometry group.


Introduction
A celebrated theorem of Myers and Steenrod [MS], says that the isometry group of a n-dimensional Riemannian manifold is always a Lie transformation group of dimension at most n(n+1) 2 . More precisely, this group is closed in the group of homeomorphisms and the Lie topology coincides with the compact-open topology. This property of the isometry group, including the bound on the dimension, carries over to general pseudo-Riemannian structures (see for instance [No2] which deals with the more general case of affine connections). For closed Riemannian manifolds, Ascoli's theorem readily implies that the isometry group is a compact Lie group. It is very well known that this compactness property is specific to the Riemannian world, and fails for general closed pseudo-Riemannian manifolds. For instance the Lorentz torus R n /Z n , endowed with the metric induced by −dx 2 1 + dx 2 2 + . . . + dx 2 n , has isometry group O(1, n − 1) Z T n . This group is noncompact, since O(1, n − 1) Z is a lattice in O(1, n − 1).
The geometrical implications of those two antagonistic phenomena -noncompactness of the isometry group on the one hand, and compactness of the manifold on the other hand-were much studied in the Lorentzian case, on which we will focus here. A sample of significant results can be found in [Gr], [Z], [AS], [Z1], [Z3], [Z4], among a lot of other works.
One remarkable point is that the noncompactness of the isometry group is also expected to have strong topological consequences. This was first noticed by M. Gromov in [Gr] when the isometry group is "large", for instance when it contains a noncompact simple Lie group (see also further developments in [FZ]). Without any extra asumption on the acting group, let us mention the following striking result: Theorem 1.1. - [DA] Let (M, g) be a closed Lorentz manifold. We assume that M and g are real analytic, and M is simply connected. Then Iso (M, g) is a compact group.
The analyticity condition is crucial in the proof of Theorem 1.1, and when the dimension of the manifold is > 3, we actually don't know if the result holds in the smooth category.
The aim of this paper is to focus on the 3-dimensional situation, and to provide a thorough study of all closed 3-dimensional manifolds, which can be endowed with a Lorentz metric admitting a noncompact group of isometries.
1.1. Statement of results. -Let us recall a class of closed 3-manifolds which will play a prominent role in the sequel, namely the torus bundles over the circle (torus bundles for short). Let T 2 be a 2-torus R 2 /Z 2 , and let us consider the product [0, 1] × T 2 . We then make the identification (0, x) (1, Ax), where A is a given element of SL(2, Z). The resulting 3-manifold is denoted T 3 A . When A = id, we just get the 3-torus T 3 . If A ∈ SL(2, Z) is hyperbolic, namely is R-split with eigenvalues of modulus = 1, we say that T 3 A is a hyperbolic torus bundle. If A ∈ SL(2, Z) is parabolic, namely conjugated to a unipotent matrix (A = id), we say that T 3 A is a parabolic torus bundle. Finally, elliptic torus bundles are those for which A has finite order.
1.1.1. A topological classification. -Our first result is a topological classification of closed Lorentz 3-manifolds admitting a noncompact isometry group.
Theorem A. -Let (M, g) be a smooth, closed 3-dimensional Lorentz manifold. Assume that (M, g) is orientable and time-orientable, and that Iso(M, g) is noncompact. Then M is homeomorphic to one of the following spaces: 1. A quotient Γ\ PSL(2, R), where Γ ⊂ P SL(2, R) is any uniform lattice. 2. A 3-torus T 3 , or a torus bundle T 3 A , where A ∈ SL(2, Z) can be any hyperbolic or parabolic element. Conversely, any smooth compact 3-manifold homeomorphic to one of the examples above can be endowed with a smooth Lorentz metric with a noncompact isometry group.
We recall that a Lorentz manifold is said to be time-orientable whenever it admits a vector field X which is timelike everywhere, namely g(X, X) < 0 on M . The assumption about orientability and time-orientability of the manifold made in the theorem is not really relevant, and one could drop it (adding a few allowed topological types) with extra case-by-case arguments in our proofs. Notice that any closed 3manifold will have a covering of order at most four satisfying the assumptions of Theorem A. We thus see that a lot of 3-manifolds do not admit coverings appearing in the list of the theorem, where only four among the eight Thurston's geometries are represented. Hyperbolic manifolds are notably missing, and we can state: Corollary 1.2. -Let M be a smooth closed 3-dimensional manifold, which is homeomorphic to a complete hyperbolic manifold Γ\H 3 . Then for every smooth Lorentz metric g on M , the group Iso(M, g) is compact.
1.1.2. Continuous versus discrete isometries. -It is interesting to compare the conclusions of Theorem A to closely related results, and especially to the work [Z2], which was a great source of motivation for the present paper. In [Z2], A. Zeghib studies 3-dimensional closed manifolds admitting a non equicontinuous isometric flow. This hypothesis is actually equivalent to the noncompactness of the identity component Iso o (M, g). The classification can be briefly summarized as follows: 1. Up to a finite cover, the manifold M is homeomorphic either to a torus bundle T 3 A , with A ∈ SL(2, Z) hyperbolic, or to a quotient Γ\ PSL(2, R), for a uniform lattice Γ ⊂ PSL(2, R). 2. The manifold (M, g) is locally homogeneous. It is flat when M is a hyperbolic torus bundle, and locally modelled on a Lorentzian, non-Riemannian, leftinvariant metric on PSL(2, R) otherwise.
The definition of Lorentzian, non-Riemannian, left-invariant metrics on PSL(2, R) will be made precise in Section 2.1.
Does it make a big difference, putting the noncompactness assumption on Iso o (M, g) instead of Iso (M, g)? At the topological level, notice that 3-tori and parabolic torus bundles do not show up in Theorem 1.3. For Lorentz metrics on those manifolds, Iso o (M, g) is always compact, but we will see that there exist suitable metrics g, for which the full group Iso (M, g) is noncompact. It means that for those examples, the noncompactness comes from the discrete part Iso(M, g)/ Iso o (M, g). Actually, there are instances of 3-manifolds (see Section 2), where the isometry group is discrete, isomorphic to Z.
To put more emphasis on how the general case may differ from the conclusions of [Z2], let us state the following existence result: Theorem B. -Let M be a closed 3-dimensional manifold which is homeomorphic to a 3-torus T 3 , or a torus bundle T 3 A , with A ∈ SL(2, Z) hyperbolic or parabolic. Then it is possible to endow M with time-orientable Lorentz metrics g having the following properties: 1. The isometry group Iso (M, g) is noncompact, but the identity component There is no open subset of (M, g) which is locally homogeneous.
Observe that for any 3-dimensional closed Lorentz manifold (M, g) which is not locally homogeneous, Iso o (M, g) is automatically compact by Theorem 1.3 above.
The constructions leading to theorem B are rather flexible. In particular, on T 3 , or on any hyperbolic or parabolic torus bundle T 3 A , the moduli space of Lorentz metrics admitting a noncompact isometry group is by no means finite dimensional. This is again in sharp constrast with the second point of Theorem 1.3.
1.1.3. Geometrical results. -The topological classification given by Theorem A comes as a byproduct of a finer, geometrical understanding of closed Lorentz 3manifolds with noncompact isometry group. We actually get a quite complete geometrical description: Theorem C. -Let (M, g) be a smooth, closed 3-dimensional Lorentz manifold. Assume that (M, g) is orientable and time-orientable, and that Iso(M, g) is noncompact.
1.2. General strategy of the proof, and organization of the paper. -One aspect of the present work consists of existence results. This is the topic of Section 2, where we recollect well-known, and probably less known, examples of closed Lorentz 3manifolds having a noncompact isometry group. Examples are given, where Iso (M, g) is infinite discrete, or semi-discrete. This yields the existence part in Theorem A, and a proof of Theorem B.
The remaining part of the paper is then devoted to our classification results, namely Theorems A and C. The point of view we adopted, is that of Gromov's theory of rigid geometric structures [Gr].
Section 3 recall the main aspects of the theory, recast in the framework of Cartan geometry as in [M], [P]. The key result is the existence of a dense open subset M int ⊂ M , called the integrability locus, where Killing generators of finite order do integrate into genuine local Killing fields. Using the recurrence properties of the isometry group, this implies the crucial fact that the noncompactness of Iso (M, g) must produce a lot of local Killing fields (Proposition 3.5). Those continuous local symetries, arising from a potentially discrete Iso (M, g), will be of great help to understand the geometry of the connected components of M int , which can be roughly classified into three categories: constant curvature, hyperbolic, and parabolic (see Section 3.5). To unravel the global structure of M , we must understand how all the components of M int are patched together (notice that there can be infinitely many such components).
The first, and easiest case to study, is when all the components of M int are locally homogeneous. Results of [F2] show that (M, g) itself is then locally homogeneous, allowing to understand (M, g) completely. This is done in Section 4.
Section 5 studies the case where one component of M int is not locally homogeneous and hyperbolic. One then shows that M is a 3-torus or a hyperbolic torus bundle, and the geometry is that of examples 2. and 4. of Theorem C. This is summarized in Theorem 5.2. The key feature in this case is to show that (M, g) contains a Lorentz 2torus, on which an element h ∈ Iso (M, g) acts as an Anosov diffeomorphism (Lemma 5.4). We then show that it is possible to push this Anosov torus by a kind of normal flow, to recover the topological, as well as geometrical structure of (M, g).
The most tedious case to study is when (M, g) is not locally homogeneous, and there are no hyperbolic components at all. This is the purpose of Sections 6, 7 and 8. We show there that M is a 3-torus or a parabolic torus bundle, and the geometry is the one described in cases 3. and 4. of Theorem C. This is summarized in Theorem 8.1. The main observation here is that the manifold (M, g) is conformally flat (Section 6). We then get a developing map δ :M 3 → Ein 3 , which is a conformal immersion from the universal coverM 3 to a Lorentz model space Ein 3 , called Einstein's universe. After introducing relevant geometric aspects of Ein 3 in Section 7, we are in position to study in details the map δ :M 3 → Ein 3 in Section 8. We show that δ mapsM 3 in a one-to-one way onto an open subset of Ein 3 , which is conformally equivalent to Minkowski space. We are then reduced to the study of closed, flat, Lorentz 3-manifolds with noncompact isometry groups, which was already done in Section 4.
All those partial results are recollected in Section 9, where we see how they yield Theorem C and Corollary D.

A panorama of examples
The aim of this section is to construct a wide range of closed 3-dimensional Lorentz manifolds (M, g), with noncompact isometry group. Those examples will show that all topologies appearing in Theorem A do really occur. Moreover, Sections 2.4, 2.2 and 2.3.1 prove our Theorem B. Part of the examples presented here are well known, others like those described in Section 2.4.2, 2.2, 2.3.1 seem less classical, though elementary.
2.1. Examples on quotients Γ\ PSL(2, R). -The Lie group PSL(2, R), universal cover of PSL(2, R), admits a lot of interesting left-invariant Lorentzian metric. The most symmetric one is the anti-de Sitter metric g AdS . It is obtained by lefttranslating (a positive multiple of) the Killing form of the Lie algebra sl(2, R). The space ( PSL(2, R), g AdS ) is a complete Lorentz manifold with constant sectional curvature −1, called anti-de Sitter space AdS 3 . Because the Killing form is Ad-invariant, the metric g AdS is invariant by left and right multiplications of PSL(2, R) on itself. It follows that for any uniform lattice Γ ⊂ PSL(2, R), the metric g AdS induces a Lorentz metric g AdS on the quotient manifold Γ\ PSL(2, R), with a noncompact isometry group coming from the right action of PSL(2, R) on Γ\ PSL(2, R).
There are other metrics than g AdS on PSL(2, R), which allow the same kind of constructions. They are obtained as follows. Exponentiating the linear space spanned by the matrix 0 1 0 0 (resp. 1 0 0 −1 ), one gets a unipotent (resp. R-split) 1-parameter group {ũ t } (resp. {h t }) in PSL(2, R). The adjoint action of each of those flows, admits invariant Lorentz scalar products on sl(2, R), which are not equal to a multiple of the Killing form. One can left-translate those scalar products and get metrics g u and g h on PSL(2, R) which are respectively PSL(2, R) × {ũ t } and PSL(2, R) × {h t }-invariant. Actually there are families of such metrics g u and g h which are not pairwise isometric. Now, for each uniform lattice Γ ⊂ PSL(2, R), the quotient Γ\ PSL(2, R) can be endowed with induced metrics g u or g h carrying an isometric, noncompact action of R, coming from the right actions of, respectively, {ũ t } and {h t } on PSL(2, R).
In the sequel, the metric g AdS and metrics of the form g u or g h , will be refered to as Lorentzian, non-Riemannian, left-invariant metrics on PSL(2, R). Those are the only left-invariant metrics on PSL(2, R), the isometry group of which does not preserve a Riemannian metric.
2.2. Examples on hyperbolic torus bundles. -Let us start with the space R 3 endowed with coordinates (x 1 , x 2 , t) associated to a basis (e 1 , e 2 , e t ). We consider a hyperbolic matrix A in SL(2, Z). Hyperbolic means that A has two distinct real eigenvalues λ and λ −1 different from ±1.
Let us consider the group Γ generated by γ 1 = T e1 (the translation of vector e 1 ), γ 2 = T e2 and the affine transformation is discrete, acts freely properly and discontinuously on R 3 , giving a quotient manifold Γ\R 3 diffeomorphic to the hyperbolic torus bundle T 3 A . We see A as a linear transformation of Span(e 1 , e 2 ). This transformation is of the form (u, v) → (λu, λ −1 v) in suitable coordinates (u, v). For any smooth function a : R → (0, ∞), which is 1-periodic, the group Γ acts isometrically for the Lorentz metric g a = dt 2 + 2a(t)dudv on R 3 . Hence the metric g a induces a Lorentz metric g a on M = T 3 A . The flow of translations T t e3 acts on T 3 A as an Anosov flow. When a is a constant, the metric g a is flat, and up to finite index, the isometry group of (T 3 A , g a ) coincides with this flow. It is thus noncompact.
In the general case of a 1-periodic function a : R → (0, ∞), the linear transformation A 0 0 1 induces an isometry f of (T 3 A , g a ) which preserves individualy the Lorentz tori t = t 0 on T 3 A , and acts on them by an Anosov diffeomorphism. Interesting examples arise if one imposes a genericity condition on the function t → a(t).
Lemma 2.1. -Assume that the function a : R → (0, ∞) is 1-periodic, and that there is no sub-interval of R where it takes the form a(t) = Ae Bt , for some real numbers A and B. Then all Killing fields of g a are tangent to the hyperplanes t = t 0 . In particular, there is no nonempty open subset where the metric g a (resp. g a ) is locally homogeneous. Moreover, the isometry group Iso(T 3 A , g a ) virtually coincides with the subgroup < f > Z generated by f . It is thus infinite discrete.
Proof: Let us consider a local Killing field T for g a , defined on some open subset U ⊂ R 3 that we may assume to be a product of open intervals. We write T = α∂ t + β∂ u + γ∂ v , where α, β, γ are smooth functions defined on U . The Lie derivative L T g vanishes identically, what can be written: Equation (1) when the pair (i, j) is equal to (t, t), (u, u) and (v, v) respectively leads to: Equation (1) for the pair (u, v) yields: Finally, the pairs (t, u) and (t, v) lead to: Deriving (4) with respect to u and (5) with respect to v, we find ∂ 2 α ∂u 2 = 0 and ∂ 2 α ∂v 2 = 0. This leads to α(u, v) = α 1 u + α 2 v + α 3 , for some real numbers α 1 , α 2 , α 3 . We can now integrate equations (4) and (5). We find γ(t, v) = A 1 (t) + B(v) and β(t, u) = A 2 (t)+C(u) for some functions A 1 , A 2 , B and C. Plugging those expressions into (3), we end up with a(t) is constant on no sub-interval, this forces α to be identically zero. The Killing field T is tangent to the hyperplanes t = t 0 , as anounced.
Let us now determine Iso(T 3 A , g a ). The 1-periodic function a : R → (0, ∞) induces a smooth function a : S 1 → (0, ∞). The value a(t 0 ) has the following geometric meaning. If F t0 denotes the fiber of t 0 ∈ S 1 in the fibration T 3 A → S 1 , then a(t 0 ) = λ Γ vol(F t0 ), where vol(F t0 ) is the Lorentz volume of F t0 and λ Γ is a positive constant depending only on Γ. It follows that Iso(T 3 A , g a ) leaves invariant the fibers of a. Since a is not constant, there exists a finite fiber for a, so that a finite index subgroup of Iso(T 3 A , g a ) leaves invariant a Lorentz torus F t0 . We now make two extra observations. The first is that < f > has finite index in the isometry group of F t0 . The second is and that the subgroup of Iso(T 3 A , g a ) fixing F t0 pointwise is finite (this is just because a lorentz isometry is completely determines by its 1-jet at a point, and that the subgroup of O(1, n − 1) fixing pointwise a Lorentz hyperplane is finite). Those remarks show that < f > has finite index in Iso(T 3 A , g a ).
where r, s, z describe R. Observe that H is a subgroup isomorphic to the 3-dimensional Heisenberg group Heis. The action of H on R 3 is free and transitive. Observe also that H acts isometrically for the flat Lorentz metric h flat = dt 2 +2dudv. Let a : R → (0, ∞) be a smooth function, which is 1-periodic, and let us consider the metric When a is not constant, it is no longer true that h a is H-invariant. But it remains true that h a is invariant under the action of the discrete subgroup Γ ⊂ H, comprising where m, n, l describe Z. The gluing map between planes v = 0 and v = 1 is made by the matrix A = 1 1 0 1 . Thus the quotient Γ\R 3 is diffeomorphic to T 3 A , with A the unipotent matrix above. Torus bundles T 3 A are characterized up to homeomorphism by the conjugacy class in SL(2, Z) of the gluing matrix A. It follows that all parabolic torus bundles are obtained for gluing maps of the form A k = 1 k 0 1 , k ∈ N * , hence by considering finite index subgroups of Γ. The metric h a induces a Lorentz metric h a on the parabolic torus bundle T 3 A , and It is readily checked that this group does not have compact closure in Iso(T 3 A , h a ). We now make the following observation. Let X = X 1 ∂ ∂u + X 2 ∂ ∂t + X 3 ∂ ∂v be a local conformal Killing field for the flat metric h flat (namely the local flow of X preserves the conformal class of h flat ). Then L X h flat = α X h flat for a smooth function α X . Assume that X 3 is nonzero on a small open set, then X will be a Killing field for h a if and only if Because we are in dimension > 2, the set of local conformal Killing fields for h flat is finite dimensional, hence for a generic choice of smooth, 1-periodic a, the relation (6) will not be satisfied, whatever the conformal Killing field X we are considering. It follows that for such a generic set of functions, there will not be any open subset of T 3 A (resp. of T 3 ) where the metric h a will be locally homogeneous. These examples prove Theorem B for parabolic torus bundles.

2.3.2.
Examples modelled on Lorentz-Heisenberg geometry. -We denote by heis the 3-dimensional Heisenberg Lie algebra, and Heis the connected, simply connected, associated Lie group. Recall that heis admits a basis X, Y, Z, for which the only nontrivial bracket relation is [X, Y ] = Z. Let B ∈ SL(2, Z) be a hyperbolic matrix, and consider the automorphism ϕ of heis, which in the basis X, Y, Z writes B 0 0 1 .
It defines an automorphism Φ of the Lie group Heis. The matrix B is diagonal in some basis X , Y of Span(X, Y ), with eigenvalues λ, λ −1 . The Lorentz scalar product defined by < X , Y >= 1, < Z, Z >= 1, and all other products are zero, can be left-translated on Heis to give an homogeneous Lorentz metric g LH called the Lorentz-Heisenberg metric on Heis. The reader will find more details and further references about this geometry in [DZ,Section 4.1]. One can actually show that different choices of the hyperbolic matrix B will produce isometric spaces.
By construction, Φ acts isometrically on (Heis, g LH ), and so do left translations. It is explained in [DZ,Section 4.1] that the identity component Iso o (Heis, g LH ) is 4-dimensional isomorphic to R Heis. The R-factor corresponds a 1-parameter group of automorphisms of heis containing Φ.
We now consider the following lattice in Heis: We call O(1, 2) the group of linear transformations preserving g flat , and we introduce Γ the discrete subgroup generated by the translations T u , T v , T w of vectors u, v, w. The quotient Γ\R 3 inherits an induced metric g flat from g flat , and the isometry group of (T 3 , g 0 ) is O(1, 2) Z T 3 . Because the quadratic form −u 2 + v 2 + w 2 has rationnal coefficients, a theorem of Borel and Harish-Chandra ensures that O(1, 2) Z is a lattice in O(1, 2). In particular, O(1, 2) Z T 3 is noncompact. The identity component of the isometry group is however compact in this case.

Non locally homogeneous examples.
-These examples are built in the same way as those of Sections 2.2 and 2.3.1, so that we will be rather sketchy in our description.
We consider the metric g a , introduced in Section 2.2, for a : R → (0, ∞) a smooth 1-periodic function.
The metric g a is invariant by the discrete group Γ generated by the translations of vectors e 1 , e 2 and e t . Hence g a induces a metric g a on T 3 . As in Section 2.2, for generic choices of the function a : R → (0, ∞), there is no open set on which g a is locally homogeneous. The isometry group is then Z T 2 (the Z-factor comes from the transformation A 0 0 1 , as in 2.2).
We can also consider the metric h a introduced in Section 2.3.1, and take for Γ the discrete subgroup generated by the translations of vectors (e u , e t , e v ). This yields a metric h a on T 3 = Γ\R 3 . For generic choices of the 1-periodic function a : R → (0, ∞), there is no open set on which h a is locally homogeneous, and the isometry group is noncompact, isomorphic to Z T 2 .
These examples prove Theorem B for 3-dimensional tori.
3. Curvature, recurrence, and local Killing fields 3.1. Generalized curvature map and integrability locus. -Let us consider (M, g), a smooth Lorentz manifold of dimension n ≥ 2. All the material presented below holds actually in the much wider framework of Cartan geometries, but we will not need such a generality.
3.1.1. Cartan connection associated to the metric. -Let π :M → M denote the bundle of orthonormal frames onM . This is a principal O(1, n − 1)-bundle over M , and it is classical (see [KN][Chap. IV.2 ]) that the Levi-Civita connection associated to g can be interpreted as an Ehresmann connection α onM , with values in the Lie algebra o(1, n − 1). For the reader's convenience, we briefly recall the link between the two points of view. The kernel of the form α determines a distribution H onM , which is transverse to the fibers and O(1, n − 1)-invariant. Let us consider a curve γ : [0, 1] → M , and a frame at x = γ(0), that we see as a pointx ∈M . There is a unique liftγ of γ toM , which starts atx and is tangent to H. This curve t →γ(t) describes the family of frames obtained by parallel transportingx along γ, for the Levi-Civita connection. Let now θ be the soldering form onM , namely the R n -valued 1-form onM , which to every ξ ∈ TxM associates the coordinates of the vector π * (ξ) ∈ T x M in the framê x. The sum α + θ is a 1-form ω : TM → o(1, n − 1) R n called the canonical Cartan connection associated to (M, g) (see [Sh,Chap. 6] for a nice introduction to Cartan geometries).
In the following, we will denote by g the Lie algebra o(1, n − 1) R n . The Cartan connection ω satisfies the two crucial properties: -For everyx ∈M , ωx : TxM → g is an isomorphism of vector spaces.
3.1.2. Generalized curvature map. -The curvature of the Cartan connection ω is a 2-form K onM , with values in g. IfX andŶ are two tangent vectors at a same point ofM , it is given by the relation: Because ωx establishes an isomorphism between TxM and g at each pointx ofM , ω provides a trivialization of the tangent bundle TM =M × g. It follows that any field of k-linear forms onM , with values in some vector space W, can be seen as a map fromM to Hom(⊗ k g, W). This remark applies in particular for the curvature form, yielding a curvature map κ :M → W 0 , where the vector space W 0 is Hom(∧ 2 (R n ); g) (the curvature is antisymmetric and vanishes when one argument is tangent to the fibers ofM ).
We can now differentiate κ, getting a map Dκ : TM → W 0 . Our previous remark allows us to see Dκ as a map Dκ :M → W 1 , with W 1 = Hom(g, W 0 ). Applying this procedure r times, we define inductively the r-derivative of the curvature D r κ :M → Hom(⊗ r g, W r ) (with W r defined inductively by W r = Hom(g, W r−1 )).
Let us now set m = dim O(1, n − 1) = n(n−1) 2 . The generalized curvature map of (M, g) is the map Dκ = (κ, Dκ, . . . , D m κ). The O(1, n − 1)-module W 1 ⊕ . . . ⊕ W m will be simply denoted W in the following. 3.2. Integrability theorem and structure of Is loc -orbits. -For x ∈ M , the Is loc -orbit of x is the set of points y ∈ M such that y = f (x) for some local isometry f : U ⊂ M → V ⊂ M . The kill loc -orbit of x is the set of points y ∈ M that can be reached by flowing along (finitely many) successive local Killing fields.
Local flows of isometries on M clearly induce local flows on the bundle of orthonormal frames, which moreover preserve ω. It follows that any local Killing field X on U ⊂ M lifts to a vector fieldX onM , satisfying LX ω = 0. Conversely, local vector fields ofM such that LX ω = 0, that we will henceforth call ω-Killing fields, commute with the right O(1, n−1)-action onM . Hence, they induce local vector fields X on M , which are Killing because their local flow maps orthonormal frames to orthonormal frames. It is easily checked that a ω-vector field which is everywhere tangent to the fibers of the bundleM → M must be trivial. As a consequence, there is a one-to-one correspondence between local ω-Killing fields onM and local Killing fields on M . We will use this correspondence all along the paper. The same remark holds for local isometries.
Observe finally that ifX is a ω-Killing field onM (namely LX ω = 0), then the local flow ofX preserves Dκ, henceX belongs to Ker(DxDκ) at each point. The integrability theorem below says that the converse is true onM int .
1. For everyx ∈M int , and every ξ ∈ Ker(DxDκ), there exists a local ω-Killing fieldX aroundx such thatX(x) = ξ. 2. The Is loc -orbits in M int are submanifolds of M int , the connected components of which are kill loc -orbits.
The deepest, and most difficult part, of the theorem is the first point. Such an integrability result as well as the structure of Is loc -orbits first appeared in [Gr]. The results were recast in the framework of Cartan geometry by K. Melnick in the analytic case (see [M]). The reference [P] gives an alterative approach for smooth Cartan geometries, leading to the statement of Theorem 3.1. A proof that the integrability property actually holds on the set where the rank of Dκ is locally constant (first point of the theorem) can be found in Annex A of [F2].
Let us recall how the second point of Theorem 3.1 easily follows from the first one (see also [P,Sec. 4.3.2]). The generalized curvature map Dκ :M → W is invariant under all local isometries. It follows thatM int is invariant as well. Givenx ∈M int , and w = Dκ(x), the Is loc -orbit Is loc (x) is contained in Dκ −1 (w) ∩M int . Now since Dκ has locally constant rank onM int , Dκ −1 (w) ∩M int is a submanifold ofM int , and the first point of Theorem 3.1 exactly means that the kill loc -orbit ofx coincides with the connected component of Dκ −1 (w) ∩M int containingx, hence is a submanifold on M int . The set Is loc (x) is a union of such connected components, hence a submanifold too. The point we have to check is that this property remains true when one projects everything on M . Observe first that the projection of Dκ −1 (w)∩M int on M coincides with that of Dκ −1 (O.w) ∩M int , where O.w stands for the O(1, n − 1)-orbit of w in W. Now, using the constancy of rank(Dκ) on Dκ −1 (O.w) ∩M int , the O(1, n − 1)equivariance of Dκ, and the fact that O(1, n − 1)-orbits in W are locally closed, one shows that Dκ −1 (O.w) ∩M int is a submanifold ofM int . By O(1, n − 1)-invariance of this set, its projection on M int is a submanifold too.
3.3. Components of the integrability locus and kill loc -algebra. -Let us recall here classical facts about the behaviour of Killing fields on the integrability locus M int (see [DaG,Section 5.15] for a general discussion in the framework of rigid geometric structures).
For each x ∈ M , let us consider a sequence U i of nested connected open neighbourhoods of x, such that {x} = i∈N U i . For each i ∈ N, denote by kill(U i ) the Lie algebra of Killing fields defined on U i . A Killing field of U i vanishing on an open subset must be identically zero, so that the restriction maps yield a sequence of Lie algebra embeddings kill(U i ) → kill(U i+1 ). The dimension of each kill(U i ) is finite (bounded by n(n+1) 2 ) and is nondecreasing with i, hence it stabilizes for i ≥ i 0 . In other words, all Killing fields defined on U i for i ≥ i 0 are restrictions of Killing fields of U i0 . We can thus state: As a consequence, there is a good notion of Lie algebra of local Killing fields at x that we denote by kill loc (x): It coincides with kill(U (x)), where the open set U (x) is given by the previous fact. Observe that for every y ∈ U (x) there is a natural embedding of Lie algebras kill loc (x) → kill loc (y). It is obtained by restricting Killing fields on U (x) to a small neighbourhood of y.
3.3.1. Components and analytic continuation. -The integrability locus M int splits into a union of connected components M i . The M i s will be just called components in the sequel. If x belongs to a component M, then the open set U (x) given by Fact 3.2 will be chosen to be included in M. We are going to see that Killing fields on such components behave as if the structure was analytic.
Let x ∈ M int and denote by M be the component containing x. It follows from Theorem 3.1 that the dimension of the Lie algebra kill loc (x) is equal to dimM −rk(Dκ). Hence the dimension of kill loc (x) does not depend on the point x ∈ M. We already oberved that for y ∈ U (x), there is a Lie algebra embedding kill loc (x) → kill loc (y). Since the two dimensions coincide this embedding is actually an isomorphism. In other words, any Killing field defined on a connected open subset V ⊂ U (x) can be extended to a Killing field defined on U (x) (recall that U (x) ⊂ M). This extension result is similar to the one obtained by Amores in [Am]. It allows to perform anaytic continuation of local Killing fields along paths contained in M int . As in [Am], this leads to the: Observe that by the discussion above, the isomorphism type of kill loc (x) does not depend on x ∈ M, and we will sometimes write kill loc (M) instead of kill loc (x).

Isotropy algebra.
-For x ∈ M , we consider Is(x), the isotropy algebra at x, namely the Lie algebra of local Killing fields defined in a neighbourhood of x and vanishing at x. Proof: Let us considerx ∈M in the fiber of x. Every local Killing field X around x which vanishes at x, lifts to a local ω-Killing field aroundx, denotedX, which is vertical (namely tangent to the fiber) atx. We call evx the mapX → ω(X(x)). The relation ϕ tX .x =x.e tevx(X) , available for t in a neighbourhood of 0, together with the invariance of Dκ under ω-Killing flows, shows that evx is a linear embedding from Is(x) to the Lie algebra of the stabilizer of Dκ(x) in O(1, n−1) (this map is one-to-one because a local ω-Killing field onM vanishing at a point must be identically zero). Cartan's formula LX = ιX •d+d•ιX shows that wheneverX andŶ are two ω-Killing fields aroundx, the relation ω([X,Ŷ ]) = K(X,Ŷ ) − [ω(X), ω(Ŷ )] holds. WhenX orŶ is vertical, K(X,Ŷ ) = 0, proving that evx is an anti-morphism of Lie algebras. To see that evx is onto, let us consider {e tξ } t∈R , a 1-parameter group of O(1, n − 1) fixing Dκ(x). Clearly, ξ belongs to Ker(DxDκ), so that by Theorem 3.1, ω −1 (ξ) is the evaluation atx of a local ω-Killing field. This ω-Killing field being vertical atx, its projection yields a local Killing field of Is(x). ♦ 3.4. Nontrivial recurrence provides nontrivial Killing fields. -We still deal here with (M n , g) a closed n-dimensional Lorentz manifold (n ≥ 2). There is, as in Riemannian geometry, a notion of Lorentzian volume, which provides a smooth, Iso(M n , g)-invariant measure on M . This measure is finite under our asumption that M is closed. When the group Iso(M n , g) is noncompact, Poincaré's recurrence theorem applies and almost every point of M is recurrent for the action of Iso (M n , g). Recall that a point x is said to be recurrent when there exists a sequence of isometries (f k ) leaving every compact subset of Iso (M n , g), and such that f k (x) converges to x.
We are going to see that such a recurrence phenomenon is responsible for the existence of nontrivial continuous local symetries. The precise statement is: Proposition 3.5. -Let (M n , g) be a closed, n-dimensional Lorentz manifold, and assume that Iso (M n  We summarize here the main arguments for the reader's convenience. Let x be a recurrent point for Iso (M n , g) and choosex ∈M in the fiber of x. The recurrence hypothesis means that there exists (f k ) tending to infinity in Iso (M n , g), and (p k ) a sequence of O(1, n − 1) such that f k (x).p −1 k tends tox. By equivariance of the generalized curvature map Dκ :M → W, we also have Observe that (p k ) tends to infinity in O(1, n − 1), because Iso(M n , g) acts properly onM .
The O(1, n − 1)-orbits on W are locally closed, because the action of O(1, n − 1) is linear, hence algebraic. As a consequence, there exists a sequence ( k ) in O(1, n − 1) with k → id and k .p k .Dκ(x) = Dκ(x). Since (p k ) tends to infinity by properness of the action of Iso(M n , g) onM , so does ( k .p k ), proving that the stabilizer Ix of Dκ(x) in O(1, n − 1) is noncompact. This group is algebraic, hence the identity component To prove the second point, we start with x ∈ M int , and consider a connected, simply connected neighbourhood U ⊂ M int of x. By Fact 3.3, every algebra Is(y), y ∈ U , is realized as a Lie algebra of Killing fields defined on U . The first point of proposition 3.5 says that there exists X a nontrivial Killing field on U , such that X(x) = 0. The zero locus of X is a nowhere dense set in U (actually a submanifold of codimension ≥ 1). We can thus pick y ∈ U satisfying X(y) = 0, and apply again the first point of the proof at y. We get a second nontrivial Killing field Y defined on U , and vanishing at y. Again, the zero locus of Y is a submanifold of codimension ≥ 1. The set of vectors in T y M which are transverse to this submanifold is thus open in T y M , so that we can pick a vector u ∈ T y M such that g y (u, X(y)) = 0, and u is transverse to the zero locus of Y . Let t → γ u (t) be the geodesic passing through y at t = 0 and such thatγ u (0) = u. Because X is a Killing field, the operator ∇X is antisymmetric for g.
This leads to Clairault's equation: d dt g(γ u (t), X(γ(t))) = 0. In particular g(γ u , X) is constant on γ u , hence does not vanish on γ u by our choice of u.
On the other hand, by the same Clairault's equation, g γu (γ u , Y ) = 0 along γ u . This implies that X(γ(t)) and Y (γ u (t)) are linearly independent as soon as Y (γ u (t)) = 0, a property satisfied for small and nonzero values of t. We obtain an open subset of U where the orbits of the local Killing algebra have dimension ≥ 2, while for every point z ∈ U , the dimension of Is(z) is ≥ 1. The minoration dim kill loc (U ) ≥ 3 follows. ♦ Remark 3.6. -The proof above does not use the fact that the metric g is Lorentzian, and Proposition 3.5 actually holds for any pseudo-Riemannian (non Riemannian) metric.
3.5. Components of the integrability locus, and their classification. -We now stick to dimension 3, and we consider a closed Lorentz manifold (M, g). We assume that the isometry group Iso(M, g) is noncompact.
On each component M ⊂ M int , the discussion of Section 3.2 shows that there is a well-defined Lie algebra kill loc (M) of local Killing fields (beware that some monodromy phenomena may occur), which by Proposition 3.5 is at least 3 dimensional. The first point of Proposition 3.5 ensures that the dimension of Is(x) is at least 1. It can not be 2, because Fact 3.4 would then imply that a vector in W has a stabilizer of dimension 2 under the linear action of O(1, 2) on W. But since O(1, 2) is locally isomorphic to SL(2, R), all finite dimensional linear representation of O(1, 2) are easily described, and on checks that no vector can have a stabilizer of dimension exactly 2. We conclude that the dimension of Is(x) is 1 or 3 for every x ∈ M, and the dimension of kill loc (M) is thus 6, 4 or 3.
-When this dimension is 6, the component has constant sectional curvature. Indeed, at each point the 3-dimensional isotropy acts transitively on the Grassmannian of Lorentzian (resp. Riemannian) 2-planes.
-When the dimension of kill loc (M) is 4, it is not hard to check that M is locally homogeneous (see for instance [DM,Lemma 4]). The dimension of Is(x) is then 1 at each point x ∈ M.
-When the dimension of kill loc (M) is 3, then the dimension of Is(x) is 1 or 3 at each point x ∈ M. The kill loc -orbits have dimension 0 or 2, and the component is nowhere locally homogeneous.
Proposition 3.5 ensures that whenever the dimension of Is(x) is 1, then this algebra generates a hyperbolic or a parabolic flow in O(T x M ) O(1, 2). In the first case, we say that x is a hyperbolic point, and in the second one we call x a parabolic point.

Definition 3.7 (Hyperbolic and parabolic components)
A component M of M int which is not of constant curvature is said to be hyperbolic when it contains a hyperbolic point. Otherwise, it is called parabolic.
Observe that this defintion allows a priori a hyperbolic component to contain parabolic points (it will turn out later that this does not occur).
To summarize, components of M int split into three (rough) categories. a) The first category comprises all components having constant sectional curvature. b) The second category comprises hyperbolic components. Those in turn split into two subgategories: i) The locally homogeneous ones, for which the dimension of kill loc (M) is 4.
ii) The non locally homogeneous ones, for which the dimension of kill loc (M) is 3. c) The remaining components are parabolic. They can also be splitted into: i) The locally homogeneous ones for which dim(kill loc (M)) = 4.
ii) The non locally homogeneous ones for which dim(kill loc (M)) = 3. Let us notice that components M with constant curvature are those for which kill loc (M) has dimension 6. Theorem 3.1 says that on M, dim kill loc (M) = dimM − rk(Dκ). We infer that points belonging to components of constant curvature are those for which the rank of Dκ is locally equal to 0. In the same way, points belonging to locally homogeneous components are those for which the rank of Dκ is locally equal to 2. Finally, we prove: Lemma 3.8. -Points x ∈ M belonging to a component of M int which is not locally homogeneous are exactly those at which the rank of Dκ is 3.
Proof: Recall the generalized curvature map Dκ :M → W introduced in Section 3.2. We saw in Proposition 3.5 that for every x ∈ M int , kill loc (x) has dimension ≥ 3. Because Dκ is invariant along kill loc -orbits inM , the corank of Dκ is a least 3 on M int , and becauseM has dimension 6, the rank of Dκ is at most 3 on the dense set M int , hence onM . The rank can only increase locally, hence points where the rank of Dκ is 3 actually stay in M int . ♦

Locally homogeneous Lorentz manifolds with noncompact isometry group
In this section, we prove Theorem A in the case where all the components of M int are locally homogeneous, implying that (M, g) is locally homogeneous on a dense open set. Our study will also settle the locally homogeneous case of Theorem C.
We observed in the previous section that if all components of M int are locally homogeneous, and under our standing assumption that Iso(M, g) is noncompact, the Lie algebra of local Killing vector fields has dimension ≥ 4 on each component. We can apply the results of [F2], saying that we must then have M int = M , and the manifold (M, g) is locally homogeneous. There are a lot of homogeneous 3-dimensional models for Lorentz manifolds. Fortunately, very few of them can appear as the local geometry of a closed manifold with a noncompact isometry group. We have indeed: The Theorem applies in our situation since Proposition 3.5 ensures that Is(x) generates a noncompact subgroup of O(T x M ) for almost every (hence every) point x.
Lorentzian, non-Riemannian, left-invariant metrics g AdS , g u and g h on PSL(2, R) were introduced in Section 2.1, while Lorentz-Heisenberg geometry was described in Section 2.3.2. It remains to explain what is the Lorentz-Sol geometry.
The Lie algebra sol is the 3-dimensional Lie algebra with basis T, X, Y and nontrivial bracket relations [T, Z] = Z, [T, X] = −X. The corresponding connected, simply connected Lie group is denoted SOL. On sol, we can consider the Lorentz scalar product such that < T, Z >=< X, X >= 1, and all other products are 0. After left-translating this scalar product on SOL, we get a Lorentz metric g sol on SOL which is called the Lorentz-Sol metric. The isometry group of (SOL, g sol ) contains SOL (acting by left-translations), but it is actually 4-dimensional. The Lie algebra of Killing fields is obtained by adding Y to T, X, Z, with bracket relations [T, Y ] = 2Y and [X, Y ] = Z (see [DZ,Section 4.2] for further details). The key property for proving Proposition 4.3 is a Bieberbach rigidity theorem for closed manifolds modelled on Lorentz-Sol geometry.
Moreover, the intersection of Γ with Iso o (SOL, g sol ) is a lattice Γ 0 ⊂ SOL acting by left translations.
We know that (M, g) is isometric to some quotient Γ\ SOL, by Theorem 4.4. Let us denote by L SOL the subgroup of Iso(SOL, g sol ) comprising all left-translations by elements of SOL. The group Γ 0 = Γ ∩ Iso o (SOL, g sol ) is Zariski-dense in L SOL by Theorem 4.4. It follows that Nor(Γ), the normalizer of Γ in Iso(SOL, g sol ), must normalize L SOL . But the description of Iso(SOL, g sol ) made above shows that L SOL has finite index in its normalizer. We infer that Nor(Γ)/Γ is compact, which proves Proposition 4.3.

4.2.
Minkowski and Lorentz-Heisenberg geometries with noncompact isometry group. -We now focus on closed Lorentz manifolds locally modelled on Minkowski space, or on Lorentz-Heisenberg geometry. In both cases, one has a Bieberbach type theorem. This is very well known in the flat case, thanks to the works [FG] and [GK], and the completeness result of Carrière for closed flat Lorentz manifolds [Ca]. For manifolds modelled on Lorentz-Heisenberg geometry, this is proved in [DZ,Proposition 8.1]. The precise statement is the following: Theorem 4.5 (Bieberbach's theorem for flat and Lorentz-Heisenberg manifolds) is isometric to the quotient Γ\R 1,2 . Moreover, there exists a connected 3dimensional Lie group G ⊂ Iso(R 1,2 ), which is isomorphic to R 3 , Heis or SOL, and which acts simply transitively on R 1,2 , satisfying that Γ 0 = G ∩ Γ has finite index in Γ and is a uniform lattice in G.

If (M, g) is locally modelled on Lorentz-Heisenberg geometry, then it is isometric
to the quotient of (Heis, g Heis ) by a discrete subgroup Γ ⊂ Iso(Heis, g Heis ). Moreover, there exists a finite index subgroup Γ 0 ⊂ Γ which is a lattice Γ 0 ⊂ Heis acting by left translations.
Notice that in [DZ], the authors make use of the classification result obtained in [Z2], namely Theorem 1.3, to prove this Bieberbach's theorem for Lorentz-Heisenberg manifolds. We will explain at the end of the paper (Section 11) how to adapt the proof of [DZ,Proposition 8.1] in order to avoid the use of [Z2]. Hence, there is no vicious circle in our arguments, and our main results are genuinely independent of Theorem 1.3.
Theorem 4.5 says that up to finite cover, a closed Lorentz manifold (M, g) modelled on Minkowski, or Lorentz-Heisenberg geometry, is homeomorphic to T 3 or to T 3 A for A ∈ SL(2, Z) hyperbolic or parabolic. Noncompactness of the isometry group allows to be more precise.
Proposition 4.6. -Let (M, g) be a closed, 3-dimensional Lorentz manifold, such that Iso(M, g) is noncompact. We assume that (M, g) is orientable and time- Proof: The situation provided by Theorem 4.5 is the following (both in the flat and Lorentz-Heisenberg case). We have a 3-dimensional Lie group G, which is either R 3 , Heis or SOL, as well as a left-invariant metric µ on G, and the manifold (M, g) is isometric to a quotient of (G, µ) by a discrete subgroup Γ ⊂ Iso(G, µ). Moreover, if we denote by L G the group of left-translations by elements of G, the intersection Γ 0 = Γ ∩ L G has finite index in Γ, and is a uniform lattice in L G . Observe that if Nor(Γ) denotes the normalizer of Γ in Iso(G, µ), then the isometry group Iso (M, g) coincides with the quotient group Γ\ Nor(Γ). An important remark for the following is that the group Nor(Γ) normalizes L G . It is obvious in the case of Lorentz-Heisenberg geometry (G, µ) = (Heis, g Heis ). In this case, the identity component Iso o (G, µ) is of the form R L G (see [DZ,Section 4.1]), and L G is thus normalized by the full isometry group Iso(G, µ).
In the case of Minkowski geometry, one has to remember that the group G (more accurately L G ) is the identity component of the crystallographic hull of Γ (see [FG,Section 1.4]). The last part of [FG,Theorem 1.4] ensures that in the case of Minkowski geometry, the crystallographic hull is unique. It follows that Nor(Γ) must normalize this crystallographic hull, as well as its identity component L G .
As a consequence, elements of Nor(Γ) (in particular elements of Γ) belong to the group Aut(G) µ L G , where Aut(G) µ denotes the automorphisms of G preserving the metric µ. Let us denote by Γ l the projection of Γ on Aut(G) µ . If this projection is trivial, we get that Γ ⊂ L G . The manifold M is obtained as a quotient of R 3 , SOL or Heis by a uniform lattice, and we are done.
If this projection is nontrivial, we are going to get a contradiction. Indeed, Γ l must be a finite subgroup of Aut(G) µ , because Γ 0 ⊂ L G has finite index in Γ. The isotropy representation ρ of Aut(G) µ at e identifies Γ l with a nontrivial finite sub- We thus see that Nor o (Γ)/Γ 0 is compact, implying the compactness of Nor(Γ)/Γ. This in turns implies Iso(M, g) compact: Contradiction. ♦ 4.3. Anti-de Sitter structures with noncompact isometry group. -It remains to study closed Lorentz manifolds (M, g) modelled on a Lorentzian, non-Riemannian, left-invariant metric on PSL(2, R). We observe that the identity components Iso o ( PSL(2, R), g u ) and Iso o ( PSL(2, R), g h ) are actually contained in Iso o ( PSL(2, R), g AdS ). Thus, if (M, g) is a closed, orientable and time-orientable, Lorentz manifold modelled on ( PSL(2, R), g u ) or ( PSL(2, R), g h ), there exists an anti-de Sitter metric g on M which is preserved by a finite index subgroup of Iso (M, g). Hence, it will be enough for us to focus on the topology of closed anti-de Sitter manifolds with noncompact isometry group. In the sequel, we will denote AdS 3 the space ( PSL(2, R), g AdS ) Proposition 4.7. -Let (M, g) be a closed, orientable and time-orientable, anti-de Sitter manifold. If Iso(M, g) is noncompact, then M is homeomorphic to a quotient Γ\ PSL(2, R), for a uniform lattice Γ ⊂ PSL(2, R).
It is worth noticing that all closed (orientable and time-orientable) 3-dimensional anti-de Sitter manifolds are Seifert fiber bundles over hyperbolic orbifolds (a short proof of this fact can be found in [Tho,Corollary 4.3.6]). Conversely, any Seifert fiber bundle over a hyperbolic orbifold, with nonzero Euler number, can be endowed with an anti-de Sitter metric (see [Sco]). The assumption that Iso(M, g) is noncompact reduces the possibilities for the allowed Seifert bundles. For instance, all nontrivial circle bundles over a closed orientable surface of genus g ≥ 2 admit anti-de Sitter metrics, but only those for which the Euler number divides 2g − 2 do occur in Proposition 4.7.
It was shown in [Kl] that closed anti-de Sitter manifolds are complete. It follows that (M, g) as in Proposition 4.7 is a quotient of AdS 3 by a discrete subgroup Γ ⊂ Iso( AdS 3 ). ActuallyΓ ⊂ Iso o ( AdS 3 ) because (M, g) is orientable and timeorientable. The center of PSL(2, R) is infinite cyclic, generated by an element ξ. The group PSL(2, R)× PSL(2, R) acts on AdS 3 by left and right translations: coincides with the identity component of the isometries of PSL(2, R) endowed with its anti-de Sitter metric.
An important result, known as finiteness of level, says thatΓ ∩ Z = id. This was first stated in [KR]. A detailed proof can be found in [Sa2,Theorem 3.3.2.3]. Geometrically, this theorem ensures that there exists a finite group of isometries Λ ⊂ Iso(M, g), which acts freely and centralizes a finite index subgroup of Iso (M, g), such that the quotient manifold of (M, g) by Λ is a quotient of PSL(2, R) by a discrete group Γ ⊂ PSL(2, R)×PSL(2, R). Let us denote by (M 3 , g) this new Lorentz manifold, and observe that Iso (M 3 , g) is still noncompact. Observe also that the projection π mapsΓ onto Γ. The structure of the group Γ is well understood. Up to conjugacy, there exists Γ 0 a uniform lattice in PSL(2, R), and a representation ρ : Γ 0 → PSL(2, R) such that This was established in [KR,Theorem 5.2] when Γ is torsion-free. For a group with torsion, the adapted proof can be found in [Tho,Lemma 4 we infer that the leaves are of the form {gAG | g ∈ PSL(2, R)}, and ρ(Γ 0 ) normalizes AG, namely ρ(Γ 0 ) ⊂ AG.
We now consider the normalizer H of Γ in PSL(2, R) × PSL(2, R). The first projection π 1 (H) must normalize Γ 0 . Since uniform lattices in PSL(2, R) are of finite index in their normalizer, we can replace H by a finite index subgroup and assume . As a consequence, ρ(Γ) can not be Zariski dense in AG. Otherwise, h −1 2 ρ(h 1 ) should be trivial implying that h = (h 1 , h 2 ) actually belongs to Γ. We thus would get that Iso (M 3 , g) is finite, a contradiction.
As a result, ρ(Γ) is included in a 1-parameter subgroup of AG. This implies that the groupΓ is contained in a product PSL(2, R) × R, where PSL(2, R) acts by left translations, and R ⊂ PSL(2, R) is a R-split or unipotent 1-parameter group acting on the right. We consider the projection π 1 :Γ → PSL(2, R) on the left-factor. The group Γ := π 1 (Γ) projects surjectively on Γ by π, hence is a uniform lattice in PSL(2, R). Moreover, the kernel of π 1 must be trivial, otherwise some nontrivial element ofΓ would belong to {id}×R, and Nor(Γ)/Γ would be compact, contradicting the hypothesis Iso(M, g) noncompact. It follows thatΓ is isomorphic to Γ.

Conclusions. -
The previous results show that closed, orientable and timeorientable, Lorentz 3-dimensional manifolds which are locally homogeneous are homeomorphic to T 3 , a torus bundle T 3 A for A hyperbolic or parabolic, or a quotient Γ\ PSL(2, R). This proves Theorem A for manifolds such that all components of M int are locally homogeneous. Our analysis shows moreover that when M is homeomorphic to a hyperbolic torus bundle or to a 3-torus, the metric must be flat. When M is homeomorphic to a parabolic torus bundle, the metric is either flat, or locally modelled on Lorentz-Heisenberg geometry. When M is homeomorphic to Γ\ PSL(2, R), the geometry is locally anti-de Sitter or locally modelled on a Lorentzian, non-Riemannian, left-invariant metric on PSL(2, R). This is in accordance to points 2, 3, 4 of Theorem C.

Manifolds admitting a hyperbolic component
Our aim in this section is to prove Theorem A under the assumption that our Lorentz manifold (M, g) admits at least one hyperbolic component M. Actually Theorem A will be implied by a more precise description provided by Theorem 5.2 to be stated below. Let us assume that there exists x ∈ ∂M. We are going to see that x ∈ M int , which will yield a contradiction. We pick x 0 ∈ M. In the sequel, we use the notation Dκ(z) for the O(1, 2)-orbit of Dκ(ẑ) in W (whereẑ is any point in the fiber of z). By assumption, the rank of Dκ is constant equal to 2 on M. Moreover, by local homogeneity, Dκ(M) is a single O(1, 2)-orbit in W. This orbit is 2-dimensional because M is locally homogeneous, but does not have constant curvature. The stabilizer of points in Dκ(x 0 ) are hyperbolic 1-parameter subgroups of O(1, 2) by assumption that M is hyperbolic. In every finite-dimensional representation of O(1, 2), such hyperbolic orbits are closed. This is a standard fact, the proof of which is, for instance, detailed in [F2,Annex B]. It follows that Dκ(x) = Dκ(x 0 ). Hyperbolic 1-parameter groups are open in the set of 1-parameter groups of O(1, 2). It follows that there is a sufficiently small neighbourhood U of x in M such that the rank of Dκ on U is ≥ 2, and for every y ∈ U , the orbit Dκ(y) is 2-dimensional with hyperbolic 1-parameter groups as stabilizers of points (notice that Dκ(y) can not be 3-dimensional because of the first point of Proposition 3.5). If at some point y ∈ U , the rank of Dκ is 3, then y belongs to a component of M int which is not locally homogeneous, by Lemma 3.8. This component must be hyperbolic because stabilizers in Dκ(y) are hyperbolic. Since we assumed that there are no such components, it follows that the rank of Dκ is constant equal to 2 on U . But then x ∈ M int leading to the desired contradiction.
♦ The case of locally homogeneous manifolds was already settled in Section 4, so that we will assume in all the remaining part of this section that M int contains a component which is hyperbolic but not locally homogeneous. We will call M this component. We are going to show that under these circumstances, M is diffeomorphic to a hyperbolic torus bundle. More precisely, the geometry of (M, g) can be described as follows: Theorem 5.2. -Assume that (M, g) is a closed, orientable and time-orientable 3dimensional Lorentz manifold, such that Iso(M, g) is noncompact. Assume that (M, g) admits a hyperbolic component which is not locally homogeneous. Then 1. The manifold M is diffeomorphic to a 3 torus T 3 , or a torus bundle T 3 A where A ∈ SL(2, Z) is a hyperbolic matrix. 2. The universal cover (M ,g) is isometric to R 3 endowed with the metric dt 2 + 2a(t)dudv for some positive nonvanishing, periodic, smooth function a : R → (0, +∞). 3. There is an isometric action of the Lie group SOL on (M ,g).
The proof of Theorem 5.2, will be the aim of Sections 5.2 to 5.6 below. Proof: Closed Lorentz manifolds with constant curvature are geodesically complete ( [Ca], [Kl]). It follows that a closed, flat Lorentz surface (Σ, g) is a quotient of the Minkowski plane R 1,1 , by a discrete subgroup Γ ⊂ O(1, 1) R 2 , which acts freely and properly on R 1,1 . Observe that nontrivial elements of O(1, 1) are of two kinds. Either they are hyperbolic (namely have two real eigenvalues of modulus = 1), or they have order 2 (an orthogonal symmetry with respect to a spacelike (resp. timelike) line). It is readily checked that Γ is either a lattice in R 2 , or admits a subgroup of index 2 which is such a lattice. Assume we are in the first case, and let H ⊂ Iso(Σ, g) be a subgroup which does not have compact closure. We lift H toH ⊂ O(1, 1) R 2 . The groupH has a nontrivial projection on SO(1, 1), otherwise H would be compact. ThusH contains a conjugate of a hyperbolic element of O(1, 1), which acts as an Anosov diffeomorphism on Σ.
In the second case, where the projection of Γ on O(1, 1) is an order 2 subgroup, the normalizer N or(Γ) must have trivial projection on SO(1, 1), which implies that Γ is cocompact in N or(Γ). This shows that flat Klein bottles have compact isometry group. ♦ 5.2.2. Closed kill loc -orbits. -Our next aim is to exhibit some kill loc -orbits which are closed surfaces.
Lemma 5.4. -In every hyperbolic component M, there exists a kill loc -orbit Σ 0 which is a flat Lorentz 2-torus, and such that there exists h ∈ Iso(M, g) leaving Σ 0 invariant, and acting on Σ 0 as a linear hyperbolic automorphism.
Let us prove this lemma. We consider our distinguished component M that, we recall, is not locally homogeneous, and hyperbolic. We pick x ∈ M a hyperbolic point, and we choosex ∈M in the fiber of x. We already observed in the proof of Lemma 3.8 that the rank of Dκ is at most 3 on M . It is exactly 3 atx, still by Lemma 3.8, hence remains constant equal to 3 in a neighbourhood ofx. Hence, if U ⊂M is a small open set aroundx, Dκ(U ) is a 3-dimensional submanifold of W. If U is chosen small enough, the O(1, 2)-orbit of every point in Dκ(U ) will be 2-dimensional and will have hyperbolic 1-parameter groups as stabilizers of points. Let us now callΛ the closed subset ofM where the rank of Dκ is ≤ 2. By Sard's theorem, the 3-dimensional Hausdorff measure of Dκ(Λ) is zero. We infer the existence of w ∈ Dκ(U ) \ Dκ(Λ). Movingx inside U , we assume that w = Dκ(x), and we denote by O(w) the O(1, 2)-orbit of w in W. By O(1, 2)-equivariance of Dκ, the inverse image Dκ −1 (O(w)) avoidsΛ, hence the rank of Dκ is constant equal to 3 on Dκ −1 (O(w)). Lemma 3.8 then leads to the inclusion Dκ −1 (O(w)) ⊂M int . By the discussion right after Theorem 3.1, the projection of Dκ −1 (O(w)) on M is a submanifold N of M . The stabilizer of w in O(1, 2) is hyperbolic, thus as mentioned in the proof of Lemma 3.8, the orbit O(w) is closed in W. It follows that N is closed in M , hence compact. By (the proof of ) Theorem 3.1, the Is loc -orbit of x is a union of connected components of N , and the connected component of x in N , denoted Σ 0 , coincides with the kill locorbit of x. It is a connected compact surface in M . Let us show that this surface has Lorentz signature. The Lie algebra Is(x) is generated by a local Killing field X around x, vanishing at x, and such that the flow is a hyperbolic 1-parameter group. Linearizing X around x thanks to the exponential map, we see there are two distinct lightlike directions u and v in T x M such that the two geodesics γ u : s → exp(x, su) and γ v : s → exp(x, sv) are left invariant by φ t X . In particular, for s = 0 close to 0,γ u (s) andγ v (s) are colinear to X, hence tangent to O(γ u (s)) and O(γ v (s)) respectively. By continuity, this property must still hold for s = 0. We infer that T x (O(x)) contains the two distinct lightlike directions u and v, hence has Lorentz signature. By local homogeneity of the kill loc -orbit Σ 0 , we get that Σ 0 is Lorentz, and moreover has constant Gauss curvature. The only closed Lorentz surfaces of constant curvature are flat tori or Klein bottles. Now Iso(M, g) sends Σ 0 to components of the Is loc -orbit of x, and there are finitely many such components by compactness of N . As a consequence the subgroup H ⊂ Iso(M, g) leaving Σ 0 invariant is noncompact. Observe that if g 0 is the metric induced by g on Σ 0 , then the injection H → Iso(Σ 0 , g 0 ) is proper (see for instance [Z2,Prop. 3.6]). It follows that Iso(Σ 0 , g 0 ) is a noncompact group. Lemma 5.3 ensures that (Σ 0 , g 0 ) is a flat Lorentz torus, and there exists h ∈ Iso(M, g) acting on Σ 0 by a hyperbolic linear automorphism. 5.3. Pushing Anosov tori along the normal flow. -From the 2-torus Σ 0 and the diffeomorphism h ∈ Iso(M, g) given by Lemma 5.4, we are going to recover the topology of the whole manifold M , as well as its geometry.
5.3.1. Preliminary definitions. -On the torus Σ 0 , h acts as an Anosov diffeomorphism. It means that there are two 1-dimensional subbundles E s ans E u , inducing a h-invariant splitting T (T 2 ) = E s ⊕ E u , so that vectors in E s (resp. in E u ) are exponentially contracted under Df n as n → +∞ (resp. n → −∞). This property and the fact that h acts isometrically for the metric g imply that the bundles E s and E u are lightlike. We choose a frame field (E − , E + ) with the property that E − and E + are future lightlike, satisfy g(E − , E + ) = 1, and generate the E u and E s respectively. Because M is assumed to be orientable, this defines a smooth normal field ν : Σ 0 → T Σ ⊥ 0 with the property that (E − , E + , ν) is a direct frame of T z M at each point z ∈ Σ 0 , and g(ν, ν) = +1.
In all the rest of the section, we pick once for all z 0 ∈ Σ 0 a periodic point of h (recall that the set of periodic points is dense in Σ 0 ). This point has period m 0 , and replacing h by h 2m0 if necessary, we will assume henceforth that h(z 0 ) = z 0 and h * ν = ν.
For every z ∈ Σ 0 , we will call γ z the oriented geodesic arc through z, with tangent ν(z) at z. Observe that γ z0 is a closed spacelike geodesic. Indeed, since h is a Lorentz isometry, the fixed points set Fix(h) is a closed, totally geodesic submanifold of M . The matrix of the differential D z0 h, expressed in the basis ( Linearizing h around z 0 thanks to the exponential map, we see that the component of Fix(h) containing z 0 , is precisely γ z0 .
5.3.2. The normal flow, and an auxiliary pseudo-Riemannian manifold. -It will be usefull in the sequel to consider the manifold N = R × Σ 0 . On this manifold, we have the vector field ∂ ∂t . Pushing the vector fields E − , E + on {0} × Σ 0 by the flow of ∂ ∂t , we get two more vector fieldsẼ − ,Ẽ + on N . The frame field (Ẽ − ,Ẽ + , ∂ ∂t ) provides N with an orientation.
Let us consider the map f : (t, z) → exp(z, tν(z)). It is well-defined and smooth on some maximal open subset U max ⊂ N . An easy application of the inverse mapping theorem shows that f : (− , ) × Σ 0 → M is a one-to-one immersion for small > 0.
A key property of the map f is its equivariance with respect to the action of h, namely : which is available for (t, z) ∈ U max (observe that f (U max ) is left invariant by h). Relation (7) just follows from the fact that h is an isometry preserving the normal field ν.
In the following, we are going to introduce It will be sometimes more suggestive to restrict f to {0}×Σ 0 , and consider the normal flow of Σ 0 , φ t : Σ 0 → M defined by φ t (z) := exp(z, tν (z)). By what we said before, φ t is at least defined on (0, τ m ), and for all t ∈ (0, τ m ), φ t : Σ 0 → M is a proper embedding, with image Σ t ⊂ M . Equivariance relation (7) shows that h preserves Σ t and acts on it as an Anosov diffeomorphism. In particular, the stable and unstable bundles must be lightlike for the metric g, showing that Σ t is a Lorentz torus. We denote by g t the restriction of g to Σ t . The map t → φ t (z 0 ) provides a (cyclic) parametrization of the closed geodesic γ z0 at speed +1. Hence for every t ∈ R, the map z → φ t (z) is defined and smooth on some small neighbourhood U t ⊂ Σ 0 containing z 0 . We call E ± (t) := D z0 φ t (E ± (z 0 )), and the formula a(t) := g γz 0 (t) (E − (t), E + (t)) defines a smooth function a : R → R. This in turns defines on the manifold N a symmetric (2, 0)-tensorg = dt 2 + a(t)g 0 (here g 0 is the metric induced by g on Σ 0 ). Our main task in the following will be to show that a(t) does not vanish. Doing this, we will prove thatg is a genuine Lorentz metric.
5.4. First geometric properties of the normal flow. -We detail here the main geometric properties of the normal flow φ t which, we recall, is defined on (0, τ m ).
The second point follows easily. Indeed, we already noticed that a(t) > 0 for t ∈ (0, τ m ), which ensures thatg is Lorentzian on (0, τ m )×Σ 0 . By the first point, f will be isometric if we prove that T γz(t) Σ t is orthogonal to γ z (t) for all t ∈ (0, τ m ). Now, observe that for a linear Lorentz transformation L = the only Lorentz plane invariant by L is the one generated by the two first basis vectors, namely the orthogonal to the line of fixed point of L. This remark shows that if z ∈ Σ 0 is a periodic point for h, T γz(t) Σ t ⊥ γ z (t) holds. By density of periodic points of h on Σ 0 , the property actually holds for all z ∈ Σ 0 . Finally f is orientation preserving because (E − , E + , ν) is positively oriented, and we defined the orientation of N to make (Ẽ − ,Ẽ + , ∂ ∂t ) positively oriented. ♦ Let us also say a few words about the surfaces Σ t . We observe that generally, Σ t are not totally geodesic submanifolds of M . However, they enjoy the weaker condition: Fact 5.6. -For every t ∈ (0, τ m ), the parametrized lightlike geodesics of Σ t for the metric g t are parametrized geodesics for the metric g.
This can be checked directly by computation for the metricg on (0, τ m ) × Σ 0 . From Fact 5.6, we infer the following relation, available for all z ∈ Σ 0 , t ∈ (0, τ m ) and s ∈ R : 5.5. Completeness of the normal flow. -Thanks to the previous section, we understand pretty well the behaviour of the normal flow for t ∈ (0, τ m ). Our next step is to show that the flow can be extended for t ≥ τ m . Proof: We just have to show that a(τ m ) = 0. Recall the point z 0 ∈ Σ 0 we introduced at the begining of Section 5.3.1. This point is fixed by h, and we already observed that f (t, z 0 ) exists for all t ∈ R. Hence, on U ⊂ Σ 0 , a small convex neighbourhood of z 0 (convex relatively to the metric g 0 ), we have an extended flow φ t : U → M defined for t ∈ (0, τ m + δ), δ > 0. Saturating U by the action of (h m ) m∈Z , we get a dense open set V on which φ t is defined for t ∈ (0, τ m + δ). Observe that V contains the stable and unstable manifolds of h at z 0 , namely W ± := {exp(z 0 , sE ± ) | s ∈ R}.
Assume for a contradiction that there exists some sequence (t k ) in (0, τ m ), such that t k → τ m and λ − t k → 0. Relation (8) says that for s ∈ R, and k ∈ N ). This implies φ τm (exp(z 0 , sE − (z 0 ))) = φ τm (z 0 ) for all s ∈ R. In particular, because the unstable manifold W − is dense in V, we get that φ τm (z) = φ τm (z 0 ) for every z ∈ V. Let us choose a h-periodic point z 1 ∈ V, z 1 = z 0 , of period q ∈ N * (such a point exists by density of h-periodic points in Σ 0 ). By what we just said, γ z0 (τ m ) = γ z1 (τ m ). We observe that γ z0 (τ m ) and γ z1 (τ m ) can not be linearly independent, otherwise D γz 0 (τm) h q would fix pointwise a 2-dimensional space in T γz 0 (τm) M , implying that at γ z0 (τ m ), Dh q is trivial or has order 2. A Lorentz isometry being completely determined by its first jet at a given point, this situation would lead to h 2q = id, a contradiction. We infer that γ z0 (τ m ) = −γ z1 (τ m ), and z 1 = γ z0 (2τ m ). Applying the same argument to a periodic point z 2 ∈ V different from z 0 and z 1 , we get a contradiction. Interverting the role of W − and W + , the same argument holds if λ + t k → 0 for some sequence t k → τ m , and the lemma follows.
♦ We have shown that the Lorentz metricg on (0, τ m ) × Σ 0 extends to a Lorentz metric on (− , τ m + ) × Σ 0 . Our next goal is to extend our isometric embedding f to a map f : [0, τ m ] × Σ 0 . This will be done thanks to the following general extension result, which is of independent interest. In the previous proposition, smooth isometric immersion means that f : Ω → M admits a well defined differential D z f : T z L → T f (z) M for every z ∈ Ω, which is isometric with respect tog and g, and varies smoothly with z.
Proof: The main part of the proof is to show the following:

Proof:
We consider at x, a unit spacelike vector ν which is normal to T x (∂Ω) and points toward Ω. We consider γ a small geodesic segment starting from x and satisfying γ (0) = ν, as well as a sequence (x k ) of points of γ ∩ Ω converging to x. Since M is compact, we may assume that f (x k ) converges to a point y ∈ M . In small neighbourhoods U and V of x and y, we choose two orthonormal frame fields, which yield at each points z, z of U and V respectively, isometric identifications i z : R 1,n−1 → (T z L,g), i z : R 1,n−1 → (T z M, g) (here R 1,n−1 stands for n-dimensional Minkowski space). Obviously, one can choose our orthonormal frame fields such that i −1 γ(t) (γ (t)) is a constant vector ξ ∈ R 1,n−1 . Also, there are U, V neighbourhoods of the origin in R 1,n−1 (depending only of our initial choice of U and V ) such that u → exp(z, i z (u)), u ∈ U, and v → exp(z , i z (v)), v ∈ V, make sense and are diffeomorphisms on their images for every z ∈ U and z ∈ V . In the trivialization given by the frame fields, the sequence of differentials (D x k f ) becomes a sequence of matrices (A k ) in O(1, n − 1). Since f is an isometry, we have the relation for every u ∈ U.
We can prove the lemma if we show that the sequence (A k ) is contained in a compact set of O(1, n − 1). For if it is the case, we may assume A k → A ∞ , and shrinking maybe U, we will have A k (U) ⊂ V for all k ∈ N. Then, we choose C ⊂ R 1,n−1 an open cone with vertex 0, containing −aξ (for some small a > 0) and contained in U. For k 0 large enough, U x = exp(x k0 , i x k 0 (C)) contains x, and if C is chosen connected and narrow enough around −aξ, U x ∩ Ω and U x ∩ ∂Ω are connected. The map f x : U x → M given by f x (exp(x k0 , i x k 0 (u))) = exp(y k0 , i y k 0 (u)), u ∈ C is a one-to-one immersion which coincides with f on U x ∩ Ω.
It remains to explain why the sequence (A k ) must be bounded. If not, we apply the KAK decomposition of O(1, n − 1) to the sequence (A k ), and after considering a subsequence we can write A k as a product M k We see that there exists a lightlike hyperplane H ⊂ R 1,n−1 (namely the image by N −1 ∞ of Span(e 2 , . . . , e n )) with the following dynamical property: For every u ∈ H, there exists u k → u such that after extracting a subsequence, Because H is lightlike while T x (∂Ω) has Lorentz signature, one can find a nonzero u ∈ H ∩ U such that i x (u) ∈ T x (∂Ω) and i x (u) points toward Ω. We choose a sequence (u k ) in U converging to u such that A k (u k ) tends to u ∞ (after extraction). We can also pick some v ∈ U \ H such that i x (v) points toward Ω. Then we can find (s k ) a sequence of real numbers tending to 0 such that A k (s k v) converges to v ∞ = 0, and (u k ). Observe that exp(x k , u k ) and exp(x k , u k + s k v) belong to Ω for k large. Now, f (exp(x k , u k )) tends to f (exp(x, u)), and relation (9) shows that f (exp(x, u)) = exp(y, u ∞ ). On the other hand, f (exp(x k , u k + s k v)) should also converge to f (exp(x, u)), because s k → 0. But relation (9) says that this sequence actually converges to exp(y, u ∞ + v ∞ ). Since v ∞ = 0, and because we can rescale u and v so that u ∞ and u ∞ + v ∞ belong to V, we have exp(y, u ∞ + v ∞ ) = exp(y, u ∞ ), and we get a contradiction. ♦ Lemma 5.9 easily provides a smooth extension of f , f : Observe that the relation f * g =g which is available on U x ∩ Ω must still hold on U x ∩ Ω. This proves that f is an isometric immersion. ♦ 5.5.2. The normal flow at τ m realizes the first return map. -We apply Proposition 5.8 choosing for L the Lorentz manifold ((− , τ m + ) × Σ 0 ,g) and for Ω the product (0, τ m )×Σ 0 . We get a smooth, orientation preserving, extension f : We recall the first return set Ω r ⊂ Σ 0 and the first return map θ r : Ω r → Σ 0 introduced at the end of Section 5.3.3.
-The extension f maps {τ m } × Σ 0 diffeomorphically and isometrically onto Σ 0 . In other words, the first return time τ r coincides with τ m , the first return set Ω r coincides with Σ 0 , and and φ τm : Σ 0 → Σ 0 realizes the first return map.
Once we know that f is one-to-one in restriction toΣ τm , we get that f (Σ τm ) is a Lorentz surface of M , to which we can again apply the normal flow. This results into an extension of f to a smooth immersion defined on a domain (0, τ m + ) × Σ 0 . If f (Σ τm ) does not meet Σ 0 , it is easily checked that for > 0 small enough, f is one-to-one on (0, τ m + ) × Σ 0 , contradicting the definition of τ m .
We infer that there exist z 1 and z 2 in Σ 0 such that f (τ m , z 1 ) = f (0, z 2 ). Observe that D (τm,z1) f (TΣ τm ) and D (0,z2) f (TΣ 0 ) can not intersect transversely, otherwise f would not be one-to-one on (0, τ m ) × Σ 0 . We can recast this property saying that γ z1 (τ m ) = z 2 , and γ z1 (τ m ) is orthogonal to T z2 Σ 0 . In other words, z 1 belongs to the first return set Ω r , and the return time is τ r = τ m . It means in particular that for all z ∈ Ω r , f (τ m , z) ∈ Σ 0 . By density of Ω r in Σ 0 , we finally get that f mapsΣ τm isometrically and diffeomorphically onto Σ 0 . ♦ 5.6. End of proof of Theorem 5.2. -Let us just recollect what we did so far. First, showing that the normal flow φ t is defined on (− , τ m + ) with φ τm (Σ 0 ) = Σ 0 immediately implies that φ t is defined for every t ∈ R. Equivalently, the map f is defined on all of N = R × Σ 0 . Next, Proposition 5.5 implies that (φ τm ) * g 0 = a(τ m )g 0 , with a(τ m ) > 0. Because the global Lorentz volume of Σ 0 must be preserved, we get a(τ m ) = 1. The transformation φ τm is a Lorentz isometry of (Σ 0 , g 0 ) commuting with h: It must be either ±id or a linear hyperbolic transformation. The possibility φ τm = −id is ruled out by the assumption that (M, g) is time-orientable.
In the following, we denote by A the transformation φ τm . We just showed that t → a(t) is τ m -periodic, and thanks to Propositions 5.5 and 5.8, we get that f : (N,g) → (M, g) is an isometric immersion. Let us call ϕ : N → N the transformation ϕ(t, x) = (t + 1, A −1 x). Then ϕ acts isometrically forg, and f • ϕ = f . Calling Γ the cyclic group generated by ϕ, we finally see that f induces an isometry between Γ\N (endowed with the metric induced byg) and (M, g). This shows the topological part of Theorem 5.2.
Since Σ 0 is a flat torus, the universal cover (M ,g) is isometric to R 3 endowed with the metric dt 2 + 2a(t)dudv. Affine transformations preserving the planes t = t 0 and acting by Lorentz isometries on the Minkowski (u, v)-plane, provide an isometric action of SOL on (M ,g). This shows points 2) and 3) of Theorem 5.2.

The local geometry of manifolds with no hyperbolic component
We keep going in our study of closed 3-dimensional Lorentz manifolds (M, g), such that Iso(M, g) is not compact. Thanks to sections 4 and 5, we can prove Theorem A when all the components of the integrability locus M int are locally homogeneous, or when there exists at least one hyperbolic component. Looking at the posibilities for the different components listed in Section 3.5, it only remains to investigate the case where all the components are either parabolic or of constant curvature, and there is at least one non locally homogeneous component. This section is devoted to a careful geometric study of such manifolds, and our aim is to prove the Recall that (M, g) is said to be conformally flat if each sufficiently small open neighbourhood of M is conformally diffeomorphic to an open subset of Minkowski space. Hence, Theorem 6.1 tells us that at the conformal level, our structure (M, [g]) is locally homogeneous. This local information will be decisive to recover the global properties of (M, g), both topologically and geometrically, a task that will be carried over in Section 8.
6.1. More on the geometry of parabolic components. -Parabolic components split into two categories, the locally homogeneous ones for wich the Killing algebra is 4-dimensional, and the others for which it is 3-dimensional.
6.1.1. Locally homogeneous parabolic components. -The study of locally homogeneous parabolic components was made in [F2], and can be summarized as follows: Actually, the statement of [F2] is slightly more precise since it describes which semidirect products R heis can occur. However, we will not need this extra information here. For the sequel, it will be important to notice that in the first case of Proposition 6.2, the Lie subalgebra heis contains the isotropy algebra at each points, hence acts with 2-dimensional pseudo-orbits (this follows from the computations done in [F2,Section 4.3.3]).
6.1.2. Parabolic components which are not locally homogeneous. -We investigate now the geometry of parabolic components which are not locally homogeneous. The proof of Proposition 6.3 involves quite a bit of computations, that we defer to Annex A, at the end of the text. Let us mention an important corollary which will be crucial later on. Proof: We already know (Theorem 3.1) that the Is loc -orbit of x is a submanifold Σ of M int . We have to show that Σ is closed in M int . We thus consider a sequence (x k ) of Σ converging to a point x ∞ ∈ M int . Letx be a lift of x inM . We recall the generalized curvature map Dκ :M → W (see Section 3.1). Let us call w = Dκ(x), and O.w the orbit of w under the action of O(1, 2) on W. Since M is a parabolic component, O.w is 2-dimensional and the isotropy at w is a 1-parameter unipotent subgroup of O(1, 2). Let (x k ) be a sequence ofM lifting (x k ), such thatx k →x ∞ . Since We infer that Dκ(x ∞ ) ∈ O.w. Hence, replacing the sequencex k byx k .p k for a bounded sequence (p k ) of P , we may assume that Dκ(x k ) = w for all k. By the discussion following Theorem 3.1, Dκ −1 (w) ∩M int is a submanifold, the connected component of which are kill loc -orbits. We conclude that for k large enough,x ∞ and x k are in the same kill loc -orbit. The same is thus true for x ∞ and x k , and the corollary is proved. ♦ 6.2. Conformal flatness. -Under the standing assumptions stated at the begining of Section 6, the only non locally homogeneous components in M are parabolic. It follows from Proposition 6.3 that the scalar curvature of g is constant on each component, and equal to zero on the non locally homogeneous ones. As a consequence, the scalar curvature vanishes identically on M , which implies that components of constant sectional curvature are actually flat, hence conformally flat. It thus remains to show that all parabolic components (locally homogeneous or not) are conformally flat. Observe that conformal flatness is given by a tensorial condition, namely the vanishing of the Cotton-York tensor in dimension 3, so that (M, g) will be conformally flat as soon as a dense open subset of M is. Observe also that the vanishing of the scalar curvature says that locally homogeneous parabolic components are exactly those described by the first point of Proposition 6.2. This fact, together with Proposition 6.3 and the remark after Proposition 6.2 reduces the proof of Theorem 6.1 to the following general observation: Proposition 6.5. -Let (N, h) be a 3-dimensional Lorentz manifold. Assume that there exists on N a Lie algebra n of Killing fields which is isomorphic to heis(3), and whose pseudo-orbits have dimension ≤ 2. Then all pseudo-orbits are 2-dimensional and lightlike, and (N, h) is conformally flat.
Proof: Our hypothesis that all pseudo-orbits have dimension ≤ 2 implies that the isotropy algebra at each point x ∈ N is a nontrivial subalgebra of n. This isotropy algebra is thus isomorphic to R, R 2 or heis(3). Because there is no subalgebra of o(1, 2) isomorphic to R 2 or heis(3), the isotropy algebra of n is 1-dimensional at each point, and all pseudo-orbits of n have dimension 2. Let us consider X, Y, Z three Killing fields generating n, and satisfying the relations [X, Y ] = −Z and [X, Z] = [Y, Z] = 0. We are going to look at the subalgebra a spanned by Y and Z. Because no subalgebra of o(1, 2) is isomorphic to R 2 , pseudo-orbits of a have dimension 1 or 2. We claim that the open subset Ω where the pseudo-orbits of a are 2-dimensional is dense in N . To see this, let us consider ∆ a 1-dimensional pseudo-orbit of a, and let x ∈ ∆. The isotropy, in a, of the point x is spanned by an element U = aY + bZ. There is another vector vector field V = cY + dZ such that v := V (x) = 0. Since U and V commute, U actually vanishes at each point of ∆. Let t → γ(t) be a geodesic for the metric h, satisfying γ(0) = x, h(γ (0), v) = 0, and γ (0) ∈ R.v. Clairault's equation ensures that for t > 0 small enough, h(γ (t), U (γ(t))) = 0 and h(γ (t), V (γ(t))) = 0. Observe that U (γ(t)) = 0, because locally, the zero set of a nontrivial Killing field on a 3-dimensional manifold is a submanifold of dimension ≤ 1. We thus get that γ(t) ∈ Ω for t > 0 small, ensuring the density of Ω. This density property shows that we will be done if we show that Ω is conformally flat. To this aim, we consider a point x 0 ∈ Ω. Since, [Y, Z] = 0 and Y, Z span a 2dimensional space at each point of Ω, there exist local coordinates (x 1 , x 2 , x 3 ) around (0, 0, 0) such that Z = ∂ ∂x1 and Y = ∂ ∂x2 . Because the orbits of n are 2-dimensional, X is of the form λ ∂ ∂x1 + µ ∂ ∂x2 for some functions λ and µ. The bracket relations [X, Z] = 0 and [X, Y ] = −Z lead to 0 = ∂λ ∂x1 = ∂µ ∂x1 = ∂µ ∂x2 and ∂λ ∂x2 = 1. Hence we can write Observe that replacing X by X − a(0)Z − b(0)Y will not affect the bracket relations between X, Y and Z, so that we will assume in the following that a(0) = b(0) = 0. Let us consider a point p = (p 1 , p 2 , p 3 ). The vector field U = X − (p 2 + a(p 3 ))Z − b(p 3 )Y is nonzero and vanishes at p. We compute that at p: Since U belongs to n, hence is Killing for the metric h, we infer that the matrix which is the matrix of ∇U (p), must be antisymmetric for the Lorentz scalar product h p . It is readily checked that a rank 1 nilpotent matrix never has this property (basically because exp(tA) would be a nontrivial 1-parameter group in (a conjugate of) O(1, 2) fixing pointwise a 2-plane, which is impossible). We thus infer that the derivative b is nowhere 0. In particular, there exist a smooth map ψ defined around 0, such that ψ(0) = 0 and b(ψ(x 3 )) = x 3 . The transformation then yields a local diffeomorphism fixing the origin. Applying ϕ * to X, Y, Z, we get three vector fields , and Z = ∂ ∂x 1 which are Killing for the metric h = ϕ * h.
Let again p = (p 1 , p 2 , p 3 ) be a point in our coordinate chart. The vector field U = X − p 2 Z − p 3 Y vanishes at p, and is a Killing field for h . A straigthforward computation yields everywhere. It follows that the matrix   0 1 0 0 0 1 0 0 0   must be antisymmetric with respect to h p . This allows us to see that the matrix of h p in the frame ( ∂ ∂x1 , ∂ ∂x2 , ∂ ∂x3 ) is of the form Killing fields for h , we see that β and γ only depend on the variable x 3 , and we conclude that the metric h writes as : shows that h is locally isomorphic to −2β(x 3 )dx 1 dx 3 +β(x 3 )dx 2 2 , hence is conformally flat. ♦

Geometry on Einstein's universe
Lorentz conformally flat structures in dimension n = 3 are examples of (G, X)structures in the sense of Thurston. In particular, there is a universal space among those structures, called Einstein's universe Ein 3 , such that if (M, g) is Lorentz and conformally flat, there exists a conformal immersion δ :M → Ein 3 , which is equivariant under a representation of π 1 (M ) into Conf(Ein 3 ) (see Section 7.1.3 below). The proof of Theorem A for manifolds (M, g) satisfying hypotheses of Theorem 6.1 and with a noncompact isometry group, will rely in a crucial way on the study of this developing map δ. This study will be carried over in the next section 8, and it will require a deeper knowledge of the geometry of Ein 3 . That's why we dedicate the present section to studying Ein 3 in more details. The reader eager to learn more about the geometry of Ein 3 is refered to [F1] or [BCDGM].
7.1. Basics on Einstein's universe. -Einstein's universe is the Lorentz analogue of the Riemannian conformal sphere. We recall its construction, sticking to dimension 3, which is the relevant one for our purpose.
Let R 2,3 be the space R 5 endowed with the quadratic form We consider the null cone and denote by N 2,3 the cone N 2,3 with the origin removed. The projectivization P ( N 2,3 ) is a smooth submanifold of RP 4 , and inherits from the pseudo-Riemannian structure of R 2,3 a Lorentz conformal class (more details can be found in [F1], [BCDGM]). We call the 3-dimensional Einstein universe, denoted Ein 3 this compact manifold P( N 2,3 ) with this conformal structure. One can check that a 2-fold cover of Ein 3 is conformally diffeomorphic to the product (S 1 × S 2 , −g S 1 ⊕ g S 2 ). The orthogonal group of Q 2,3 , isomorphic to O(2, 3), acts naturally on the 4dimensional projective space, preserving Ein 3 and its conformal structure. It turns out (see Theorem 7.1 below) that PO(2, 3) is the full conformal group of Ein 3 . Observe that Ein 3 is homogeneous under the action of PO(2, 3). 7.1.1. Photons and lightcones. -It is a remarkable fact of Lorentz geometry that all the metrics of a given conformal class have the same lightlike geodesics (as sets but not as parametrized curves). In the case of Einstein's universe, the lightlike geodesics are the projections on Ein 3 of totally isotropic 2-planes P ⊂ R 2,3 (namely planes P on which Q 2,3 vanishes identically). We will rather use the term photon for the lightlike geodesics of Einstein's universe. Observe that all photons of Ein 3 are simple closed curves.
Given a point p in Ein 3 , the lightcone with vertex p, denoted by C(p), is the union of all photons containing p. If p ∈ Ein 3 is the projection of u ∈ N 2,3 , the lightcone C(p) is just P(u ⊥ ∩ N 2,3 ). The lightcone C(p) is singular (from the differentiable viewpoint) at its vertex p, and C(p)\{p} is topologically a cylinder. The entire cone C(p) has the topology of a 2-torus pinched at p. 7.1.2. Stereographic projection. -There is for Ein 3 a generalized notion of stereographic projection, which shows that Ein 3 is a conformal compactification of the Minkowski space.
Let us call R 1,2 the space R 3 endowed with the quadratic form Q 1,2 (x, x) = 2x 1 x 3 + x 2 2 . Consider ϕ : R 1,2 → Ein 3 given in projective coordinates of P(R 2,3 ) by Then ϕ is a conformal embedding of R 1,2 into Ein 3 , called the inverse stereographic projection with respect to p 0 := [e 0 ]. The image ϕ(R 1,2 ) is a dense open set of Ein 3 with boundary the lightcone C(p 0 ). Observe that this proves the fact (rather hard to visualize): The complement of a lightcone C(p) in Ein 3 is connected.
7.1.3. Developing conformally flat structures into Einstein's universe. -It is a standard fact that Einstein's universe satisfies an analogue of the classical Liouville's theorem on the sphere (see for instance [Sh,Chap. 7,Coro. 3.5] for a proof which generalizes to all signatures (p, q)). Namely: The existence of the stereographic projection (10), and the transitivity of the action of PO(2, 3) on Ein 3 shows that Ein 3 is conformally flat. Liouville's theorem 7.1 shows that any 3-dimensional, conformally flat Lorentz structure (M, g) is actually a (PO(2, 3), Ein 3 )-structure, in the sense of Thurston.
As a consequence, for every conformally flat Lorentz structure (M, g), there exists a conformal immersion δ : (M ,g) → Ein 3 called the developing map of the structure. Here,M is the universal cover of the manifold M , andg is the lifted metric. This developing map comes with a holonomy morphism ρ : Conf(M ,g) → PO(2, 3) satisfying the equivariance relation: (M ,g). Notice that the terminology holonomy morphism is usally used for the restriction of ρ above to the group π 1 (M ), seen as the group of deck transformations ofM . However, we will really need the extension of ρ to Conf (M ,g) in the following.
7.2. More geometry on Ein 3 . -7.2.1. The foliation F ∆ . -We refer here to the notations introduced in Section 7.1. Let P be the plane in R 2,3 spanned by the vectors e 0 and e 1 . The form Q 2,3 vanishes identically on P , hence the projection of P on Ein 3 defines a photon that we will denote by ∆. The open subset obtained by removing ∆ to Ein 3 will be called Ω ∆ . Given a point p ∈ ∆, we consider the lightcone C(p) with vertex p. Since ∆ is a photon, we have ∆ ⊂ C(p). Now, the intersection of C(p) with Ω ∆ , namely C(p) \ ∆ is a lightlike hypersurface of Ω ∆ , diffeomorphic to a plane. We call it F ∆ (p). We now make the observation that in Ein 3 , there is no nontrivial lightlike triangle, namely if two photons ∆ 1 and ∆ 2 intersect ∆ transversely at two distinct points, then ∆ 1 ∩ ∆ 2 = ∅. This is the geometric counterpart of the following algebraic fact: In R 2,3 , there are no 3-dimensional spaces on which Q 2,3 vanish identically. It follows that if p = p are points of ∆, C(p)∩C(p ) = ∆, or in other words F ∆ (p)∩F ∆ (p ) = ∅. This shows that {F ∆ (p)} p∈∆ are the leaves of a codimension 1 lightlike foliation of Ω ∆ , that we will call F ∆ . Actually, there is a smooth submersion π ∆ : Ω ∆ → ∆, which to a point x ∈ Ω ∆ associates p = π ∆ (x) ∈ ∆ such that x ∈ C(p). The fibers of π ∆ are precisely the leaves of F ∆ , and the space of leaves is naturally identified with ∆. For x ∈ Ω ∆ , we will adopt the notation F ∆ (x) for the leaf of F ∆ containing x. 7.2.2. Symetries of the foliation F ∆ . -Let us call G ∆ the stabilizer of ∆ in PO(2, 3). Obviously, G ∆ preserves Ω ∆ and the foliation F ∆ .
It is readily checked that this group is a semi-direct product where the group N is isomorphic to the 3-dimensional Heisenberg group Heis(3), and given in PO(2, 3) by the matrices: The factor PGL(2, R) is the subgroup of PO(2, 3) corresponding to matrices: Observe that ∆ being obtained as the projectivization of a null plane of R 2,3 , it is naturally identified with RP 1 . The action of G ∆ on the space of leaves of F ∆ corresponds to the projective action of the factor PGL(2, R) on ∆. The subgroup S ∆ ⊂ G ∆ which preserves individually all the leaves of F ∆ is a semi-direct product where the factor R * + corresponds to matrices: Let us end this algebraic parenthesis by giving more details about the action of the group N . Obviously, N fixes the point p 0 = [e 0 ] ∈ Ein 3 , hence if we perform a stereographic projection given by formula (10), the group N becomes a subgroup of conformal transformations of R 1,2 . These transformations are affine, given by In PO(2, 3), such transformations take the matricial form: From this matricial representation, it is straigthforward to check the following Fact 7.2. -1. The set of fixed points for the action of the group N (resp. T ) on Ein 3 is exactly ∆. 2. For every x ∈ Ω ∆ , the N -orbit of x is the leaf F ∆ (x) 3. The action of T is free on Ω ∆ \F ∆ (p 0 ), and orbits of T on this open set coincide with leaves of F ∆ . 4. On F ∆ (p 0 ), orbits of T are 1-dimensional and coincide with the photons of C(p 0 ), with p 0 removed.
In the rest of the paper, we will adopt the notations g ∆ , s ∆ , n, t for the Lie subalgebras of o(2, 3) corresponding to the groups G ∆ , S ∆ , N, T . o(2, 3). -The Lie group N admits a Lie algebra n ⊂ o(2, 3) that will be called the standard Heisenberg algebra of o(2, 3).

Standard Heisenberg algebras in
It is not true that all subalgebras of o(2, 3) which are isomorphic to heis(3) are conjugated to the standard algebra n. There is however the following useful characterization: Proof: As any solvable Lie subalgebra of o(2, 3), h must leave invariant a line R.v or a 2-plane P in R 2,3 . Such a vector v can not be timelike or spacelike, otherwise the decomposition R 2,3 = R.v ⊕ v ⊥ would lead to an embedding of h in one of the Lie algebras R ⊕ o(1, 3) or R ⊕ o(2, 2) R ⊕ sl(2, R) ⊕ sl(2, R). But none of those algebras contains a subalgebra isomorphic to heis(3). Similarly, P can not be of signature (+, +), (+, −) or (−, −), otherwise the decomposition R 2,3 = P ⊕ P ⊥ would lead to an embedding of h into o(2) ⊕ o(2, 1) (3). One checks as above that this is not possible. The only possibilities are then: a) The vector v is lightlike or P has signature (0, +) (resp. (0, −)). This means that H has a global fixed point in Ein 3 , that we can assume to be p 0 after conjugating within PO(2, 3). b) The form Q 2,3 vanishes identically on P , in which case H has an invariant photon that we can assume to be ∆. We first deal with case a). After considering a stereographic projection of pole p 0 , h becomes a subalgebra of Conf(R 1,2 ) (R ⊕ o(1, 2)) R 3 . Here the normal subalgebra R 3 integrates into the subgroup of translations. Let us consider the projection π : (R ⊕ o(1, 2)) R 3 → o(1, 2). Since o(1, 2) does not have any subalgebra isomorphic to heis(3) or R 2 , the rank of π |h is 0 or 1. Because R R 3 (with R acting by homothetic transformations on R 3 ) does not contain a copy of heis(3), this rank is actually 1, hence the kernel of π |h , denoted a, has dimension 2 in h, hence is abelian. The only subalgebras isomorphic to R 2 in R R 3 are actually contained in R 3 .
Our hypothesis on the orbits of the group H implies that the translation vectors in a span a lightlike plane, hence after conjugating within Conf(R 1,2 ), we can assume a = t, where t was introduced at the end of Section 7.2.2.
The first point of Fact 7.2 implies that since H centralizes t, H ⊂ G ∆ . The hypothesis on the orbits of H says that on some open set, H-orbits and T -orbits coincide. Points 3 and 4 of Fact 7.2 imply that the action of H on ∆ is trivial on some nonempty open set, hence trivial. This yields H ⊂ S ∆ . Because the normalizer of t in S ∆ is N , we finally get H = N , and the proof is completed in this case.
Consider now case b). Because H leaves ∆ invariant, H is a subgroup of G ∆ . As above, we can look at the morphism π : g ∆ (R ⊕ sl(2, R)) n → sl(2, R).
The same arguments as above show that the kernel of π |h is a 2-dimensional abelian Lie subalgebra a ⊂ h. Observe that a ⊂ s ∆ , and the only 2-dimensional abelian subalgebras of s ∆ are contained in n. After conjugating within G ∆ , we can ensure a = t. We then finish the proof as in the first case. ♦

The global geometry of manifolds without hyperbolic components
This section is devoted to establishing Theorem A in the only remaining case to be studied, namely that of closed 3-dimensional Lorentz manifolds (M, g) which are not locally homogeneous, such that M int does not admit any hyperbolic component, and with a noncompact isometry group Iso (M, g). By Theorem 6.1, those manifolds are conformally flat.
What we will really show in this section is: Theorem 8.1. -Let (M, g) be a closed, orientable and time-orientable, 3dimensional Lorentz manifold, such that Iso(M, g) is noncompact. We assume that (M, g) is not locally homogeneous, and that M int does not admit any hyperbolic component. Then: 1. The manifold M is homeomorphic to a 3-torus, or a parabolic torus bundle T 3 A . 2. There exists a metric g = e 2σ g in the conformal class of g which is flat, and which is preserved by Iso(M, g). 3. There exists a smooth, positive, periodic function a : R → (0, ∞) such that the universal cover (M ,g) is isometric to R 3 endowed with the metric g = a(v)(dt 2 + 2dudv).
4. There is an isometric action of Heis on (M ,g).
This result clearly implies Theorem A in the case under study. Its proof will be the aim of Sections 8.1 to 8.6 below. In all those sections, (M, g) satisfies the asumptions of Theorem 8.1.
8.1. Approximately stable foliation on M . -So far, we saw that (M, g) is an agregate of (possibly infinitely many) components, the local geometry of which we understand fairly well. But we need a global object which allows one to understand how those components fit together. This global object turns out to be a foliation provided by the noncompactness of Iso(M, g) as follows.
Consider a sequence (f n ) in Iso (M, g) which tends to infinity, and call AS(f n ) the subset of T M comprising all vectors v ∈ T M for which there exists a sequence (v n ) in T M converging to v, such that |Df n (v n )| is bounded (where |.| is the norm associated to an auxiliary Riemannian metric on M ). In [Z3], A. Zeghib proved the following result : Theorem 8.2. -[Z3, Theorem 1.2] Let (M, g) be a closed Lorentz manifold, and (f n ) a sequence of Iso (M, g) tending to infinity. Replacing if necessary (f n ) by a subsequence, the set AS(f n ) is a codimension 1, lightlike, Lipschitz distribution in T M , which integrates into a codimension 1, totally geodesic, lightlike foliation.
The foliation given by Theorem 8.2 is called the approximately stable foliation of (f n ).
In the particular case of a 3-dimensional manifold, codimension 1, totally geodesic, lightlike foliations have very nice properties that were studied by A. Zeghib in [Z5]. He proved in particular: Theorem 11] Let (M, g) be a 3-dimensional closed Lorentz manifold. Let F be a C 0 , codimension 1, totally geodesic, lightlike foliation of M . Then: 1. A leaf of F is homeomorphic to a plane, a cylinder or a torus. 2. The foliation F has no vanishing cycles.
A consequence of the non-existence of vanishing cycles is that loops of a leaf F representing a nontrivial element in π 1 (F ) also represent a nontrivial element in π 1 (M ).
We now choose a sequence (f n ) tending to infinity in Iso (M, g), and after considering a suitable subsequence, we denote the approximatively stable foliation of (f n ) by F. By Theorem 8.3, the leaves of F are planes, cylinders or tori. Our main aim, and a decisive step to prove Theorem 8.1 will be to show that all leaves of F are tori, yielding the torus bundle structure of M . It will be convenient in the sequel to consider the lift of F to the universal coverM . We will callF this lifted foliation.  (2, 3). We pick x 0 ∈M and U 0 a 1-connected neighbourhood of x 0 on which the developing map δ is injective. If U 0 is chosen small enough, the Lie algebra kill(U 0 ) of Killing fields on U 0 coincides with kill loc (x 0 ). Einstein's universe Ein 3 satisfies a generalization of Liouville's theorem (Theorem 7.1): Any conformal Killing field defined on some connected open set of Ein 3 is the restriction of a global one. Thus the algebra δ * (kill(U 0 )) is a subalgebra of o(2, 3) isomorphic to the 3-dimensional Heisenberg algebra. The pseudo-orbits of δ * (kill(U 0 )) on δ(U 0 ) are 2-dimensional and lightlike by the second point of Proposition 6.3. Lemma 7.3 applies and says that post-composing δ by an element of PO(2, 3), we may assume δ * (kill(U 0 )) = n. We will now work with a developing map δ having this property, and say that δ is adapted to M. We will consider the associated holonomy morphism ρ : Observe that by this proposition, the trace of the foliation F on parabolic components which are not locally homogeneous, actually does not depend on the sequence (f n ).
8.3.1. The pullback foliationF ∆ and its geometric properties. -We consider the developing map δ :M → Ein 3 , which we recall is adapted to M, and take the pullback by δ of the foliation F ∆ defined in Section 7.2.1. We get in this way a (singular) foliationF ∆ onM . Actually,F ∆ is a genuine foliation by lightlike hypersurfaces on the open setΩ ∆ = δ −1 (Ω ∆ ). Singularities occur on the complement ofΩ ∆ inM , namely∆ := δ −1 (∆). This singular set is either empty (in which caseF ∆ is a regular foliation onM ), or a 1-dimensional lightlike manifold.
Let us emphasize the fact that a priori, we don't have any invariance property forF ∆ under the action of the fundamental group π 1 (M ). In particular, there is no reason forF ∆ to define any foliation on M .
In the following, we will identify o(2, 3) with the Lie algebra of conformal Killing fields of Ein 3 (see Theorem 7.1). We can pull back the vector fields of the Lie algebra n by the developing map δ :M → Ein 3 , getting a Lie algebrañ of conformal Killing fields on (M ,g). By Fact 7.2, the pseudo-orbits ofñ coincide with the leaves ofF ∆ . 8.3.2. FoliationF ∆ and kill loc -orbits. -A first important feature of the foliationF ∆ is its relation to the kill loc -orbits inM.
Lemma 8.5. -The restriction toM of any vector field ofñ is a Killing field forg. Conversely, any local Killing field defined on some open set U ⊂M is the restriction of a vector field inñ.
Proof: Recall the point x 0 ∈M and the 1-connected open subset U 0 introduced in Section 8.2. By the fact that our developing map δ is adapted to M, any Killing field on U 0 is the restriction of a vector field ofñ. Sinceñ and kill(U 0 ) have same dimension, the restriction to U 0 of any vector field ofñ must be Killing. Let X be a vector field of n, and let us call Y = X |U0 . Let us pick an arbitrary y ∈M, and draw a simple curve γ joining y to x 0 insideM. Let us consider V a 1-connected open neighbourhood of γ contained inM and containing U 0 . Because the dimension of kill loc (z) is constant onM, the vector field Y can be extended by analytic continuation to a Killing field forg (still denoted Y ) defined on V . But now, Y and X |V are two conformal Killing fields on V , which coincide on U 0 . They must then coincide on V , showing that X is Killing forg in a neighbourhood of y. We have thus proved that the restriction of X toM is Killing. A dimentional argument as above shows that conversely, a Killing field defined on some connected open subet ofM is the restriction of a field inñ. ♦ Corollary 8.6. -The componentM is contained inΩ ∆ .
Proof: Points of∆ are singularities for the vector fields ofñ. Hence if a point x ∈∆ belongs toM, Lemma 8.5 will provide a Lie subalgebra of Killing fields vanishing at x and isomorphic to heis(3). The isotropy representation then yields an embedding of Lie algebras heis(3) → o(1, 2). This is impossible. ♦ We conclude this paragraph with the following important lemma.
-Let x be a point ofM, andF ∆ (x) the leaf ofF ∆ through x. Theñ F ∆ (x) is contained inM, and coincides with the kill loc -orbit of x.
Proof: Let us consider a leafF ∆ having a nonempty intersection withM. Assume for a contradiction that V =F ∆ ∩M is not all ofF ∆ . It means that V is an open subset ofF ∆ having a nontrivial boundary ∂V insideF ∆ . Of course, ∂V ⊂ ∂M (this last boundary is taken inM ). SinceF ∆ is a pseudo orbit ofñ, it is easy to show that there exists y ∈ ∂V , a vector field X ∈ñ and a point x ∈ V such that the local orbit t → φ t X .
x is defined on [0, 1], φ t X .x belongs to V for t ∈ [0, 1/2) but φ 1/2 X .x ∈ ∂V . We denote byR the bundle of frames onM , and exceptionally in this proof, we adopt the notationM for the bundle of orthonormal frames ofM (and not of M ). The local action of φ t X lifts naturally toR. We pickx ∈M in the fiber of x, and look at the orbit t → φ t X .x inR. Because X is Killing onM (Lemma 8.5), this orbit is contained inM for t ∈ [0, 1/2), and the same is true for t ∈ [0, 1/2] becauseM is closed inR. We now look at the generalized curvature map Dκ :M → W, and its derivative that we see as a map DDκ :M → Hom(g, W). The map t → DDκ(φ t X .x) makes sense for t ∈ [0, 1/2], and is constant on this interval because X is Killing oñ M. In particular, the kernel of DDκ(φ t X .x) is the same for all t ∈ [0, 1/2], hence the rank of Dκ is the same atx and at φ 1/2 X .x. We get that the rank of Dκ at φ 1/2 X .x is 3, but we already observed in the proof of Lemma 3.8, that all points where Dκ has rank 3 are contained inM int . We infer φ 1/2 The last part of the lemma follows easily. Lemma 8.5, together with Corollary 8.6 ensures that for every x ∈M, the kill loc -orbit of x coincides withF ∆ (x) ∩M. But F ∆ (x) ∩M =F ∆ (x) by the first part of the proof. We also lift the foliation F to a foliationF on the universal coverM . For each x ∈M , we denote byF (x) the leaf ofF containing x.
Thanks to Lemma 8.7, Proposition 8.4 will be a simple consequence of: This shows that the foliationF which is a priori only transversally Lipschitz, is transversally smooth in restriction toM.
Proof: We work onM, and we consider the two 1-dimensional lightlike distributions D ∆ = TF ∆ ⊥ andD = TF ⊥ . Our aim is to show that those distributions coincide oñ

M.
For every x ∈M, let us introduce the set C(x), comprising all lightlike directions u ∈ P(T xM ) such that there exists a lightlike totally geodesic hypersurface Σ through x, with T x Σ ⊥ = u. Let us recall a key observation made in [Z4]: Lemma 8.9. -[Z4, Proposition 2.4] If the set C(x) spans T xM , then the sectional curvature at x is constant.
If at some point x ofM, the directionsD ∆ andD do not coincide, then Lemma 8.9 implies that they must both be fixed by the local flow generated by the isotropy algebra Is(x). But a nontrivial parabolic 1-parameter flow in O(1, 2) has only one invariant direction: Contradiction.
We are thus led toF ∆ (x) ⊂F (x) for every x ∈M. This inclusion can not be proper, otherwiseF ∆ could be extended in a smooth way to points of∆. ♦ Remark 8.10. -The previous proof shows actually that onM, any lightlike, totally geodesic, codimension 1 foliation has to coincide withF. Lemma 8.11. -Let x ∈Ω ∆ , and assume that the leavesF (x) andF ∆ (x) coincide. Then δ is injective in restriction toF (x).
Proof: Considering if necessary a finite cover of M (which will not changeM ), there exists W a vector field on M , tangent to F and satisfying g(W, W ) = 1. We lift W to a vector fieldW onM , which is tangent toF. Notice thatW is complete. By assumption,W is tangent toF (x). The proof follows now closely the arguments of [Z2,Proposition 6.5]. Let us callD the 1-dimensional foliation integratingF ⊥ . A fundamental remark made in [Z5,Proposition 2] is that becauseF (x) is totally geodesic, any vector field U tangent toD acts as a Killing field on the degenerate surface (F (x),g). It follows easily that if γ and η are two curves onF (x) parametrized by [0, 1], such that γ(0) and η(0) belong to the same leaf ofD, γ (0) and η (0) point to the same side, and if γ and η have the same length with respect tog, then γ(1) and η(1) also belong to a same leaf ofD. Applying this remark to the integral curves of the flow {ψ t } generated byW , we obtain that ψ t maps leaves ofD to leaves of D. GivenD 0 a leaf ofD inF (x), the union U(D 0 ) = t∈R ψ t (D 0 ) is open inF (x), and two such open sets either coincide, or are disjoint, so thatF (x) = U(D 0 ). By hypothesis, the leavesF ∆ (x) andF (x) coincide, so that the developing map δ sends F (x) to F ∆ = F ∆ (δ(x)) ⊂ Ein 3 . Let γ : I →F (x) be an injective parametrization of the leaf ofD through x. We observe that for every s ∈ I, δ is injective on the curve t → ψ t (γ(s)), because in F ∆ , there is no closed curve transverse to photons of F ∆ . Also, for every t ∈ R, δ is injective in restriction to s → ψ t (γ(s)), because no photon in F ∆ is closed. The injectivity of δ onF (x) follows. ♦ 8.4.2. The group Iso(M, g) is not a torsion group. -Our noncompactness hypothesis on the group Iso(M, g) does not prevent a priori Iso(M, g) from being a torsion group. In particular, we still don't know if there exists a single element h ∈ Iso(M, g) such that {h k } is infinite discrete. The aim of this paragraph is to show it is indeed the case, and to prove the stronger statement: Proposition 8.12. -Let F ⊂ M be a leaf of F containing at least one recurrent point. Let S F be the stabilizer of F in Iso (M, g). There exists h ∈ S F such that the group {h k } is not relatively compact in Iso (M, g).
Notice that there are examples of noncompact Lorentz surfaces admitting a noncompact isometry group which is a torsion group (see [MoB,Remarque 4.12]).
We recall (Proposition 8.4) that M is saturated by the leaves of F. The proof of Proposition 8.12 will require the intermediate lemmas 8.13 and 8.14 below. We lift F to a leafF ⊂M and call SF the stabilizer ofF in Iso (M ,g). Observe that SF projects surjectively on S F under the epimorphism Iso(M ,g) → Iso (M, g).
Lemma 8.13. -For every leaf F ⊂ M containing recurrent points, the groups S F and SF are closed, noncompact subgroups of Iso (M, g) and Iso(M ,g) respectively.

Proof:
We first prove that S F is closed in Iso (M, g). If (f k ) is a sequence of S F which converges to f ∞ ∈ Iso (M, g), then for k very large, f −1 k f ∞ belongs to the identity component Iso o (M, g). Because F coincides with a kill loc -orbit of M (Lemma 8.8), we thus have Let us now check that S F is noncompact. By assumption on F , there is a recurrent point x in F . It means that there exists a sequence (f k ) tending to infinity in Iso (M, g) such that f k (x) → x. Because the Is loc -orbit of x is a 2-dimensional submanifold, the connected components of which are kill loc -orbits (see Theorem 3.1), we get that f k (x) ∈ F for k large enough. In particular, S F is a noncompact subgroup of Iso (M, g).
The corresponding assertions on SF are then straigthforward. ♦ Lemma 8.14.
-Let F ⊂ M be a leaf of F, andF a lift of F toM .
1. The holonomy morphism ρ maps the group SF into the group S ∆ . In particular, any element of SF leaves invariant the leaves ofF which are sufficiently close toF . 2. The morphism ρ : SF → S ∆ is injective and proper.

Proof:
We heavily use the notations introduced in Section 7.2. We choose a transversal I ⊂M to the foliationF, that cutsF at x. We assume that I is small enough, so that δ sends I injectively on a transversal J of F ∆ . Shrinking I if necessary, J meets each leaf of F ∆ at most once, so that by Lemma 8.11, I meets each leaf of F at most once. We call V the open subset obtained by saturating I by leaves of F. We recall the submersion π ∆ : Ω ∆ → ∆ introduced in Section 7.2.1. By what we just said, I is the space of leaves of V, and the map ϕ := π ∆ • δ : I → ∆ gives an identification of I with J = π ∆ (J), the space of leaves of the foliation induced by F ∆ on δ(V). Under this identification, the point x is sent to a point p ∈ J .
Leth be an element of SF . Its action on the space of leaves ofF yields a germ h of diffeomorphism of I fixing x. The equivariance relation δ •h = ρ(h) • δ shows that ρ(h) permutes leaves of F ∆ near δ(F ). In particular, ρ(h) maps J to an interval of ∆ containing p. We infer that ρ(h) preserve ∆, what yields ρ(h) ∈ G ∆ . Moreover, denoting l : G ∆ PGL(2, R) N → PGL(2, R), we get the equivariance relation ϕ • h = l(ρ(h)) • ϕ. Now, l(ρ(h)) acts as an element of PGL(2, R) on ∆, admitting p as fixed point. We know the local dynamics of a Möbius transformation around one of its fixed points: If l(ρ(h)) is nontrivial, we can choose q ∈ J , q = p, such that l(ρ(h k ))(q) belongs to J for all k ≥ 0, and lim k→∞ l(ρ(h k ))(q) = p. This means that ifF is a leaf corresponding to ϕ −1 (q), the iteratesh k (F ) will accumulate onF . ButF is a kill loc -orbit by Proposition 8.4, and closeness of the Is loc -orbit ofF iñ M int (Corollary 6.4) says thatF and all the h k (F ), k ∈ N, belong to the same Is locorbit. This accumulation phenomenon then contradicts the fact that Is loc -orbits are submanifolds inM int (see Theorem 3.1). We conclude that l(ρ(h)) is trivial, which implies that ρ(h) ∈ S ∆ . Moreover, h is trivial, which means that all leaves ofF close toF are left invariant byh.
We now prove the second point of the Lemma. Leth ∈ SF such that ρ(h) = id. Equivariance relation δ •h = ρ(h) • δ, together with Lemma 8.11, shows that the action ofh onF . The following fact then impliesh = id. Proof: The proof relies on the fact that the map, which to an isometry associates its 1-jet at a given point, is injective and proper, and that restricting elements of O(1, n − 1) to a lightlike hyperplane is also injective and proper. Details can be found in [Z2,Prop 3.6], for instance. ♦ Properness of the map ρ : SF → S ∆ follows the same lines. If (h k ) is a sequence of SF such that ρ(h k ) is relatively compact in S ∆ . Then ρ(h k ) |δ(F ) is relatively compact, hence the retriction ofh k toF is relatively compact by Lemma 8.11. Fact 8.15 yields that (h k ) is relatively compact in SF .
♦ We can now proceed to the proof of Proposition 8.12. We know from Theorem 8.3, that the leaves of F are discs, cylinders or tori, and there are no vanishing cycles. It means (see the comment right after Theorem 8.3) that the leafF is a disc, and the stabilizer Γ F ofF in π 1 (M ) is either trivial, or a discrete subgroup isomorphic to Z or Z 2 . On the other hand, we also know thatS F /Γ F is noncompact, because of Lemma 8.13. a) Case where F is a disk. We choose a nontrivialh ∈ SF . It restricts to a nontrivial transformation ofF (Fact 8.15), hence ρ(h) is a nontrivial element of S ∆ , by Lemma 8.14. Every nontrivial element of S ∆ generates an infinite discrete group.
In particular, {ρ(h) k } k∈Z is not relatively compact in S ∆ . Second point of Lemma 8.14 says that {h k } is not relatively compact in Iso (M ,g). Fact 8.15 thus implies that {h k |F } is not relatively compact in Homeo(F ), hence the same is true for {h k |F }, because the projection π :F → F is a diffeomorphism in the case we are considering. Finally, {h k } is not relatively compact in Iso (M, g). b) Case where F is a cylinder. Because F does not have vanishing cycles, Γ F := SF ∩ Γ is nontrivial, generated by a single element γ. The automorphism group of Γ F is {±1} and Γ F is normalized by SF . Hence after considering an index 2 subgroup of SF , we may assume that γ is centralized by all elements of SF . We observe that ρ(γ) is nontrivial by Lemma 8.14, and consider its centralizer in S ∆ . Two cases can then occur: -The group ρ(SF ) is contained in a 1-paramater subgroup of S ∆ . In this case, ρ(SF )/ < ρ(γ) > is relatively compact. This implies that (S F ) |F is relatively compact, hence S F is relatively compact in Iso (M,g) (again Fact 8.15). This is ruled out by Lemma 8.13.
-If we are not in the previous case, we can findh ∈ SF such that the group generated by ρ(h) and ρ(γ) is discrete isomorphic to Z 2 . As above, applying second point of Lemma 8.14 and Fact 8.15, one gets thath projects to h ∈ S F , such that {h k } is infinite discrete in Iso (M, g). c) Case where F is a torus. This time, Γ F is isomorphic to Z 2 and generated by γ 1 and γ 2 . Lemma 8.14 ensures that τ 1 := ρ(γ 1 ) and τ 2 := ρ(γ 2 ) generate a discrete subgroup of S ∆ isomorphic to Z 2 . Such a subgroup must be contained in N , and after conjugating ρ within G ∆ (what amounts to post-compose δ by some element of G ∆ ), we have that < τ 1 , τ 2 >⊂ T . We must have ρ(SF ) ⊂ N because SF normalizes Γ F , and because S F is noncompact, ρ(SF ) ⊂ T . Pickingh ∈ SF such that ρ(h) ∈ T , we get an element h ∈ S F which, by similar arguments as above, generates an infinite discrete group {h k } ⊂ Iso(M, g).
8.4.3. Existence of a toral leaf. -We now consider an element h ∈ Iso(M, g) given by Proposition 8.12, namely {h k } is not relatively compact in Iso (M, g). Theorem 8.2 provides an approximately stable foliation F h associated to a subsequence of {h k }, and since all what we did before did not assume anything special on F, we can decide that now F = F h .
Proposition 8.16. -Every leaf F of F which is contained in M is a torus.
Let F be a leaf of F contained in M, such that almost every point of F is recurrent for {h k }. We lift F toF ⊂M, and we will also assume that δ(F ) is not contained in the leaf F ∆ (p 0 ) (see Fact 7.2). Observe that since M is closed, Poincaré recurrence ensures that almost every point of (M, g) is recurrent for {h k }. It follows that for almost every leaf of F, almost every point is recurrent (leaves of M coincide with kill loc -orbits hence are locally closed and transversally smooth on M. There is thus nothing tricky in desintegrating the volume form in M along those leaves). Hence almost every leaf of F in M is an F with the properties above. We claim that F is a torus. To see this, we lift h to an elementh ∈ SF , and our assumption is that the set of points inF , which are recurrent under the group <h, Γ F > have full measure inF . By Lemma 8.11, almost all points of δ(F ) must be recurrent for the group ρ(<h, Γ F >). We now go back to the analysis made at the end of Section 8.4.2. If F is not a torus, we are in cases a) or b) of this discussion. In case a), Γ F is trivial. The action of ρ(h) on δ(F ) is conjugated to that of an affine transformation on (an open subset of) the plane (see Section 7.2 and formula (15)). The set of recurrent points of ρ(h) has thus zero measure on δ(F ). A contradiction.
In case b), we saw that the group ρ(<h, Γ F >) is conjugated to a lattice in the closed subgroup T . Since by assumption, δ(F ) is not contained in the leaf F ∆ (p 0 ), point 3) of Fact 7.2 shows that T has no recurrent point on δ(F ). We reach a new contradiction.
The arguments above show that almost all leaves F ⊂ M are tori. Now, for a codimension 1 foliation on a closed manifold, the union of all compact leaves is itself compact (see [God,Chap II,Corollary 3.10]). Proposition 8.16 follows.
8.5. All leaves of F are tori. -We keep the notations of the last section. We still consider h ∈ Iso (M, g) such that {h k } is not relatively compact. We consider the approximately stable foliation F associated to some sequence (h n k ), with n k → ∞. Proposition 8.16 and its proof provide us with a leaf F 0 ⊂ M which is an h-invariant torus. We lift F 0 toF 0 ⊂M , and h toh ∈ Iso(M ,g) preservingF 0 . The group Γ 0 = SF 0 ∩ π 1 (M ) is discrete and isomorphic to Z 2 , generated by two elements γ 1 and γ 2 . Lemma 8.14 ensures that τ 1 := ρ(γ 1 ) and τ 2 := ρ(γ 2 ) generate a discrete subgroup of S ∆ isomorphic to Z 2 . After conjugating by an element of G ∆ , we may assume that τ 1 and τ 2 are elements of T .
We call in the sequelH the subgroup generated byh, γ 1 and γ 2 . Lemma 8.14 ensures that ρ(h) ∈ S ∆ = R * + N (see Section 7.2.2). Elements of S ∆ which are not in N act on N with nontrivial dilation. They can not preserve any lattice in T . This says that becauseh normalizes Γ 0 , we must have ρ(H) ⊂ N . There are thus three elements X, Y, Z in the Lie algebra n such that τ 1 = e X , τ 2 = e Z and ρ(h) = e Y . The center of n is contained in Span(X, Z), and we pick Z 0 = 0 in this center.
We now pull back the four vector fields X, Y, Z, Z 0 of Ein 3 by the developing map δ :M → Ein 3 . This way, we get vector fieldsX,Ỹ ,Z,Z 0 onM . Observe that τ * onM . After introducing those notations, we can state what will be the last technical step of our study: Proposition 8.17. -For every x ∈M , we have: iii) The restriction of the 3 vector fieldsX,Ỹ ,Z are complete onF ∆ (x), and the The proposition will show thatF coincideF ∆ onM , and Γ 0 -invariance of the leaves ofF will easily imply that leaves of F are all tori. In the next section 8.6 we will derive more consequences from this equality E =M , and prove Theorem 8.1.
We are going to consider the set E ⊂M , comprising all points x ∈M satisfying the three conditions of Proposition 8.17, and show that E is nonempty, open and closed inM , yielding E =M . Lemma 8.18. -Let x ∈Ω ∆ such thatF (x) =F ∆ (x). Assume moreover that F ∆ (x) is invariant by a subgroup Λ ⊂ π 1 (M ), isomorphic to Z 2 and such that ρ(Λ) ⊂ T . Then the map δ is a diffeomorphism fromF ∆ (x) to F ∆ (δ(x)). Moreover F ∆ (δ(x)) = F ∆ (p 0 ).
Proof: Lemma 8.11 ensures that δ is a diffeomorphism fromF ∆ (x) to an open subset U ⊂ F ∆ (δ(x)). The group Λ is isomorphic to Z 2 and acts properly discontinuously on the diskF (x) =F ∆ (x). By a cohomological dimension argument, the action must be cocompact. The group ρ(Λ) is thus a lattice in T , and must act properly and cocompactly on U . Last point of Fact 7.2 says that the action of ρ(Λ) can not be proper on any open subset of F ∆ (p 0 ). We thus infer that F ∆ (δ(x)) = F ∆ (p 0 ). In particular, again by Fact 7.2, the action of ρ(Λ) is proper and cocompact on F ∆ (δ(x)). We then must have U = F ∆ (δ(x)). ♦ The completeness ofX,Ỹ andZ onF 0 follows from Lemma 8.18, applied for Λ = Γ 0 , because X, Y, Z are complete on leaves of F ∆ . The relations τ 1 = e X , τ 2 = e Z and ρ(h) = e Y imply that the relations φ 1X = γ 1 , φ 1 Z = γ 2 , φ 1 Y =h hold oñ F 0 . We infer thatF 0 ⊂ E. 8.5.2. The set E is open. -We begin by stating a lemma that we will use repeatedly in the sequel.
Lemma 8.19. -Let U ⊂Ω ∆ be a connected open set. Let f, g : U →M two conformal immersions. Assume that for some x ∈Ω ∆ ,F ∆ (x) ∩ U = ∅, and that f and g coincide onF ∆ (x) ∩ U , then f and g coincide on U .
Proof: Shrinking U if necessary and looking at δ(U ) ⊂ Ein 3 , we are reduced to the situation of two transformations g 1 and g 2 of PO(2, 3) which coincide on some open subset of a lightcone in Ein 3 . At level of linear algebra, it means that those two transformations of PO(2, 3) must coincide on a lightlike hyperplane of R 2,3 . This easily implies g 1 = g 2 . ♦ Let us start with x ∈ E. Vector fieldsX,Ỹ andZ are complete in restriction tõ F ∆ (x) =F (x), so given > 0, we can choose U ⊂Ω ∆ a small neighbourhood of x such that φ tX .y is defined on [− , 1 + ] for every y ∈ U , and for every t ∈ [− , 1 + ] and every y ∈ U , φ s Z .φ tX .y is defined for every s ∈ [− , 1 + ]. Lemma 8.19 says that identity φ 1X .y = γ 1 .y holds on U , because it holds on U ∩F ∆ (x). It follows easily from the property γ * 1X =X that for every y ∈ U , φ tX .y is defined for t ∈ R. Relation γ * 1Z =Z now implies that φ s Z .φ tX .y makes sense for every t ∈ R, s ∈ [− , 1 + ], and y ∈ U . Let us call U = {φ tX .y | t ∈ R, y ∈ U }. This is an open set on which φ 1 Z is defined. Relation φ 1 Z = γ 2 holds on U ∩F ∆ (x), hence on U by Lemma 8.19. Together with the property γ * 2Z =Z, this implies that φ s Z .φ tX .y makes sense for every y ∈ U , and s, t ∈ R. Now, Lemma 8.18 says that F ∆ (δ(x)) = F ∆ (p 0 ). If U was chosen small enough, δ(y) ∈ F ∆ (p 0 ) for every y ∈ U . It follows that (t, s) → e sZ .e tX .δ(y) is a diffeomorphic parametrization of F ∆ (δ(y)). In other words, for every y ∈ U , {φ s Z .φ tX .y | (s, t) ∈ R 2 } coincides with the leafF ∆ (y), and the developing map δ :F ∆ (y) → F ∆ (δ(y)) is a diffeomorphism. Completeness of vector fieldsX,Ỹ ,Z onF ∆ (y) follows, because vector fields X, Y, Z are complete on leaves of F ∆ .
To conclude that U ⊂ E, it remains to check thatF ∆ (y) coincides withF (y) for every y ∈ U . A first observation is thatF ∆ (y) is diffeomorphic to F ∆ (δ(y)), hence is a disk. It follows from a cohomological dimension argument that because Γ 0 Z 2 , the quotient Σ = Γ 0 \F ∆ (y) is a torus in M . Recall from (16) the relatioñ Remember also thatZ 0 is a linear combination of X andZ, hence tangent toF ∆ (y). On the torus Σ,Z 0 thus induces a vector field Z 0 which is h-invariant. Notice that Z 0 is lightlike because Z 0 is lightlike on Ein 3 , and Z 0 is nonsingular because the singularities of Z 0 are exactly the points of ∆, andF ∆ (y) ⊂Ω ∆ . Hence, for every z ∈ Σ, Z 0 (z) ⊥ = T z Σ. On the other hand, equality D z h n k (Z 0 (z)) = Z 0 (h n k (z)) shows that Z 0 (z) belongs to the aproximately stable distribution of h n k (see the definition of this distribution in Section 8.1). The aproximately stable distribution has codimension 1 and is lightlike, so that it coincides with Z 0 (z) ⊥ = T z Σ for all z ∈ Σ. We conclude that Σ is a leaf of F, which proves F (y) =F ∆ (y). 8.5.3. The set E is closed. -We consider a sequence (x k ) of E converging to x ∞ ∈ M . The leafF (x ∞ ) is accumulated by the sequence of leavesF (x k ) =F ∆ (x k ). In particular, the vector fieldsX,Ỹ ,Z being tangent toF ∆ (x k ) for all k, they are also tangent toF (x ∞ ). Point 2) of Fact 7.2 then says thatF (x ∞ ) \∆ is a union of leaves ofF ∆ . If the setF (x ∞ ) ∩∆ is not empty, those leaves ofF ∆ might be prolongated smoothly accros the singular set∆, a contradiction. We infer thatF (x ∞ ) ⊂Ω ∆ , and The union of the compact leaves of F is a compact subset of M (see [God, Chap. II, Corollary 3.10]). Since F has no vanishing cycles,F (x ∞ ) is left invariant by a discrete subgroup Λ 1 ⊂ π 1 (M ) which is isomorphic to Z 2 . We choose I ⊂ M , a small transversal to the foliation F containing the point π(x ∞ ). Following the loops of F (π(x ∞ )) defining Λ 1 in the neighboring leaves, we get a corresponding holonomy morphism for the leaf F (π(x ∞ )): hol : Λ 1 → Diff loc π(x∞) (I).
Here Diff loc π(x∞) (I) denotes the pseudo-group of local diffeomorphisms of I fixing π(x ∞ ). We refer to [God, Chap. II] for more details on holonomy of a foliation. If γ ∈ Λ 1 and z ∈ I is close enough to π(x ∞ ), then hol(γ 2 ).z makes sense. If we have hol(γ 2 ).z = z, then the pseudo-orbit hol(γ n ).z is defined for all n ≥ 0 or all n ≤ 0 and is infinite. In this case the leaf F (z) cannot be closed. Because F (π(x k )) is a torus for each k ∈ N, it follows that replacing Λ 1 by some index 2 subgroup, we may assume that Λ 1 leaves invariantF (x k ) for k large enough.
This property also shows that ρ(Λ 1 ), which is a priori not a subgroup of G ∆ , leaves invariant infinitely many leaves of F ∆ . Those leaves are traces on Ω ∆ of lightcones in Ein 3 with vertex on ∆. We infer that ρ(Λ 1 ) fixes infinitely many points on ∆. It follows that ρ(Λ 1 ) leaves ∆ invariant: ρ(Λ 1 ) ⊂ G ∆ . Since a Möbius transformation fixing infinitely many points on the circle must be trivial, we actually have ρ(Λ 1 ) ⊂ S ∆ .
Let y ∞ ∈F (x ∞ ), and let U ⊂Ω ∆ be a small neighbourhood of y ∞ . By completeness ofX,Ỹ andZ in restriction toF ∆ (x ∞ ), and shrinking U if necessary, the local diffeomorphisms φ 1X , φ 1 Y and φ 1 Z are defined on U . For k large, U ∩F ∆ (x k ) = ∅, and identities φ 1X = γ 1 , φ 1 Y =h and φ 1 Z = γ 2 hold on U ∩F ∆ (x k ). Lemma 8.19 says that those identities hold on U . Finally y ∞ was arbitrary inF ∆ (x ∞ ) so that these identities hold onF ∆ (x ∞ ). This proves x ∞ ∈ E. 8.6. Proof of Theorem 8.1. -Let us draw further conclusions from Proposition 8.17. The coincidence of the foliationsF andF ∆ implies thatF ∆ is π 1 (M )-invariant. Moreover, the Γ 0 -invariance of each leafF ∆ , together with Lemma 8.18 implies that δ is injective on each leafF ∆ , and that δ(M ) ⊂ Ω ∆ \ F ∆ (p 0 ). Also, it follows from Proposition 8.17 that Γ 0 is exactly the subgroup of π 1 (M ) leaving each leaf ofF invariant. It follows that Γ 0 is normal in π 1 (M ). We claim that Γ 0 is also normalized by N π1 , the normalizer of π 1 (M ) in Iso (M ,g). Indeed, if f ∈ N π1 , then f Γ 0 f −1 leaves each leaf of f (F) invariant. Now f (F) is a lightlike, totally geodesic, codimension 1 foliation. Remark 8.10 ensures that on any non locally homogeneous componentM, f (F) coincides withF. In particular, The group ρ(N π1 ) normalizes ρ(Γ 0 ), hence T since ρ(Γ 0 ) is Zariski-dense in T . By fact 7.2, the lightcone C(p 0 ) can be characterized as the set of points where the orbits of ρ(Γ 0 ) are contained in a photon of Ein 3 . It follows that C(p 0 ) is left invariant by Nor(T ), the normalizer of T in PO(2, 3). Applying the stereographic projection ϕ of pole p 0 (see Section 7.1.2) we can see ρ(N π1 ) as a subgroup of Conf(R 1,2 ). We then show: Lemma 8.20. -Seen in Conf(R 1,2 ), the elements of ρ(N π1 ) are contained in the group In particular, we have the inclusion ρ(N π1 ) ⊂ Iso(R 1,2 ).
Proof: After performing the stereographic projection ϕ, the foliation F ∆ restricted to Ein 3 \C(p 0 ) becomes a foliation of R 1,2 . Formula (10) for ϕ readily shows that this is the foliation by affine planes of direction Span(e 1 , e 2 ). Recall (see Section 7.2.2) that the group T corresponds to the group of translations of vectors v ∈ Span(e 1 , e 2 ). Since Nor(T ), hence ρ(N π1 ), must preserve this foliation (this is a consequence of Fact 7.2), we see that elements of ρ(N π1 ) belong to the subgroup G ⊂ Conf(R 1,2 ) comprising all elements of the form:  normalizes a lattice in Span(e 1 , e 2 ), then the determinant of its restriction to Span(e 1 , e 2 ) is ±1. It follows that µ = ± 1 √ |λ| .
We saw that ρ(h) belongs to the group N , hence has the form: In particular, becauseh normalizes Γ 0 , if τ ∈ ρ(Γ 0 ), and if we see τ as a translation of vector v ∈ Span(e 1 , e 2 ), then ρ(Γ 0 ) will also contain v = v − A.v, where A = 1 y 0 1 . In other words ρ(Γ 0 ) contains a translation of vector αe 1 , α = 0. The fact that ρ(N π1 ) normalizes the discrete group ρ(Γ 0 ), leads to the relation λµ = ±1 in (17). Together with the relation µ = ± 1 √ |λ| , this leads to |µ| = |λ| = 1, and the Lemma follows. ♦ Lemma 8.20 says that our (Ein 3 , PO(2, 3))-structure is actually a (R 1,2 , Iso(R 1,2 ))structure. We conclude that there exists g in the conformal class of g which is flat, and which is preserved by Iso (M, g). We can thus apply the results of Section 4.2. Theorem 4.5 and Proposition 4.6 say that (M, g ) is the quotient of R 3 , Heis or SOL by a lattice. But ρ(π 1 (M )) ⊂ G by Lemma 8.20, and G does not contain any subgroup isomorphic to SOL. We thus get that M is homeomorphic to T 3 or to a torus bundle T 3 A with A ⊂ SL(2, Z) parabolic. This proves points 1) and 2) of Theorem 8.1. Finally, Carrière's completeness result [Ca] says that δ :M → R 1,2 is a conformal diffeomorphism. It follows that if the coordinates associated to (e 1 , e 2 , e 3 ) in R 1,2 are (u, t, v), the metricg is of the form a(u, t, v)(dt 2 + 2dudv).
It remains to check that the function a depends only on v. First, the foliation by planes with direction Span(e 1 , e 2 ) is totally geodesic. If∇ denotes the Levi-Civita connection ofg, we thus have: Identifyingh and ρ(h), we saw that It follows that ρ(h) acts on each hyperplane v = v 0 by the affine transformation: where z(v 0 ) = z 0 − y 2 2 v 0 and x(v 0 ) = x 0 + yv 0 . The group Γ 0 is generated by two translations τ 1 , τ 2 of (linearly independant) vectors a b and c d respectively. The w-coordinate ofh k • τ m 1 • τ n 2 u t is t + kx(v 0 ) + mb + nd. Becauseh k • τ m 1 • τ n 2 acts isometrically forg this leads to a(t, v) = a(t + kx(v) + mb + nd, v) for every (k, m, n) ∈ Z 3 . Since b and d can not be both zero (let say b = 0), and because x(v) and b are rationally independant for almost every value of w (because y = 0), we get that for almost every w, t → a(t, v) is constant. As a consequence, a = a(v), and the fact that it is a periodic function follows easily from the compactness of M .
Finally the group N which comprises transformations of the form is isomorphic to Heis and acts isometrically on (M ,g). This concludes the proof of Theorem 8.1.

Conclusions
The study made in Section 4, as well as Theorems 5.2 and 8.1 provide all possible topologies for a closed 3-dimensional, orientable and time-orientable, Lorentz manifold with a noncompact isometry group. Those are the 3-dimensional torus, hyperbolic or parabolic torus bundles, and compact quotients Γ\ PSL(2, R). Together with the examples provided in Section 2, this yields Theorem A.
Let us now look at the geometries which can occur on those manifolds, and prove Theorem C. The manifolds Γ\ PSL(2, R) occur only in Proposition 4.7. Hence the only metrics on such manifolds which admit a noncompact isometry group are covered by PSL(2, R), endowed with a Lorentzian, non-Riemannian, left-invariant metric. In particular those manifolds (M, g) are locally homogeneous and (M ,g) admits an isometric action of PSL(2, R).
Parabolic torus bundles appear in Proposition 4.6 and Theorem 8.1. We saw there that the universal cover is isometric to R 3 endowed with a metric a(v)(dt 2 + 2dudv), with a smooth and periodic. This universal cover admits an isometric action of Heis. If the manifold (M, g) is locally homogeneous, Proposition 4.6 ensures that g is flat or locally isometric to the Lorentz-Heisenberg metric.
Hyperbolic torus bundles appear only in Proposition 4.6 Theorem 5.2. We saw that the universal cover is isometric with R 3 endowed with a metric dt 2 + 2a(t)dudv, with a smooth and periodic. There is an isometric action of SOL on this universal cover. The manifold (M, g) is locally homogeneous if and only if it is flat.
Finally, 3-tori appear in Proposition 4.6 Theorem 5.2 and Theorem 8.1. The metric on the universal coverM is provided by those two last theorems, and there is always an isometric action of Heis or SOL on (M ,g). Finally, (M, g) is locally homogeneous if and only if it is flat.
Those results alltogether prove Theorem C and Corollary D.
The other irreducible submodule E 5 of the curvature module is 5-dimensional, spanned by the matrices: We call κ 0 the element of Hom(∧ 2 (R 3 ), o(1, 2)) corresponding to the identity matrix, namely κ 0 maps e ∧ h to E, e ∧ f to H and h ∧ f to F . We also call κ 1 the element of Hom(∧ 2 (R 3 ), o(1, 2)) corresponding to the matrix The two dimensional vector space spanned by κ 0 and κ 1 is the set of fixed points of the action of {e tE } t∈R on the curvature module.
10.0.2. Identification of the kill loc -algebra. -We consider a parabolic component M which is not locally homogeneous. In such a component, the points are either parabolic, or points where the isotropy algebra is 3-dimensional and the sectional curvature is constant. Constant curvature on a nonempty open subset U ⊂ M would imply that the algebra of Killing fields is 6-dimensional on U , hence on M. As it contradicts our assumption that M is a parabolic component, we conclude that the set of parabolic points is a dense open set Ω ⊂ M. Observe that at a parabolic point x ∈ Ω, if X a local Killing field around x, generating the isotropy Is(x), the 1-parameter group D x ϕ t X is unipotent in O(T x M ). In a suitable basis (u 1 , u 2 , u 3 ) of T x M satisfying g(u 1 , u 3 ) = 1 = g(u 2 , u 2 ) and all the other products are 0, the matrix of D x ϕ t X reads   1 t −t 2 /2 0 1 −t 0 0 1   We quickly check that the only 2-plane stable by D x ϕ t X is spanned by u 1 and u 2 , so that on Ω, the kill loc -orbits must be lightlike surfaces.
Let us now fix a point x ∈ Ω. We work in the fiber bundleM (and lift all local Killing fields on M to local ω-Killing fields onM ). After multiplying X by a suitable constant, we can findx ∈M in the fiber of x such that ω(X(x)) = E. We now choose Z and Y two local Killing fields around x such that Z(x) = u 1 and Y (x) = u 2 . After adding to Z and Y a suitable multiple of X, we can write, atx: ω(Ẑ) = e + βH + γF and ω(Ŷ ) = h + αH + νF. The curvature κ(x) is Ad(e tE )-invariant, hence is of the form κ = σκ 0 + bκ 1 . In particular, the following identities hold atx: (18) κ(e ∧ h) = σE, κ(e ∧ f ) = σH, κ(h ∧ f ) = bE + σF.
Notice that σ, b, α,β,γ,ν depend on x andx, but since those points are fixed, there will be considered as constant in the sequel. Cartan's formula LÛ ω = ιÛ dω+d(ιÛ ω) shows that wheneverÛ ,V are two ω-Killing fields onM , the following relation holds: Here K is the curvature of ω, as defined in Section 3.1.2. We recall that it is linked to the curvature function κ by the relation K(Û ,V ) = κ(ω(Û ), ω(V )).
In the sequel, we will call H the span of ω(Ẑ), ω(X), ω(Ŷ ) atx, and we are going to write Equation (19) atx, using identities (18) The fact that kill loc (x) is a Lie algebra, together with the property H ∈ H forces γ to vanish. Next, two Killing fields which coincide atx must be equal (by freeness of the action of isometries on the orthonormal frames), which implies [Ẑ,X] = −βX.  Notice that establishing (20) and (21), we have actually shown that ad(X) is a nilpotent endomorphism of kill loc (x). This property did not use anything special on x, so that we actually have: Fact 10.1. -At each z ∈ Ω, if U is a local Killing field around z generating the isotropy at z, then ad(U ) is a nilpotent endomorphism of kill loc (z). At x, Z(x) is lightlike and nonzero and Y (x) is spacelike, orthogonal to Z(x). The orthogonal to Y (x) at x is a Lorentzian plane spanned by Z(x) and another vector w ∈ T x M . Let us call t → γ(t) the geodesic through x satisfyingγ(0) = w. Clairault's equation ensures that the quantities g(γ(t), Z(γ(t))), g(γ(t), X(γ(t))) and g(γ(t), Y (γ(t))) do not depend on t. In particular, for t > 0, both Y (γ(t)) and X(γ(t)) are orthogonal toγ(t) while Z(γ(t)) is not. For t > 0 small enough, Y (γ(t)) is still spacelike, hence nonzero, and γ(t) belongs to Ω. In particular, the kill loc -orbit at γ(t) is 2-dimensional, so that Y and X must be colinear at γ(t). One then has X(γ(t)) = λ t Y (γ(t)), for some real λ t . Observe finally that w is not fixed by D x φ t X , hence is transverse to the set where X vanishes. In particular, for t ≥ 0 small, X(γ(t)) = 0 only for t = 0, and thus λ t = 0 if t = 0.
For t ≥ 0 small, X, Y, Z generate kill loc (γ(t)), hence −2λ t α = 0 because of Fact 10.1. Since λ t = 0 if t = 0, we get α = 0. Injecting this data in equation (22) and (20), we find that the matrix of ad(λ t Y − X) in the basis Y, Z, X is: The characteristic polynomial of ad(λ t Y − X) is Hence, the nilpotency of ad(λ t Y − X) (Fact 10.1) implies (23) λ t σ + 2ν = 0, If σ = 0, we get that t → λ t is constant, which is not the case since we observed that λ 0 = 0 but λ t = 0 for t > 0 small. We end up with the equality σ = ν = 0. The vector fields Z, X, Y then satisfy the bracket relations: showing that Lie algebra kill loc (x) is isomorphic to heis. We also proved that σ, the scalar curvature at x, vanishes, but since x was arbitrary in the open set Ω, we finally get the vanishing of the scalar curvature on Ω, and then on M by density.
10.0.3. Description of the kill loc -orbits. -The fact that the local Killing algebra is isomorphic to heis shows that no point in M has a 3-dimensional isotropy algebra. Indeed, the isotropy representation at those points would yield an embedding heis → o(1, 2), what is impossible. We thus get Ω = M, and all the kill loc -orbits on M are 2-dimenional and lightlike.
On the other hand, since the isotropy algebra Is(x) generates a parabolic 1parameter subgroup of O(1, 2) at each x, there is a totally geodesic lightlike hypersurface F (x), whose tangent space is left invariant by the isotropy (see [DZ,Lemma 3.5] and its proof). We already observed that at x ∈ M, the local isotropy preserves only one 2-plane of T x M . This implies that the kill loc -orbits are everywhere tangent to a leaf of a totally geodesic foliation of M, hence the kill loc -orbits are themselves totally geodesic. This concludes the proof of Proposition 6.5.

Annex B: About the completeness of closed Lorentz-Heisenberg manifolds
Our aim here is to explain how to adapt the proof of [DZ,Prop. 8.1], and get Theorem 4.5 for closed Lorentz-Heisenberg manifolds without using Theorem 1.3.
We call heis the 3-dimensional Heisenberg Lie algebra, namely the Lie algebra generated by Z, Y, X, with relation [Y, X] = Z. The Lorentz-Heisenberg metric g LH on the Lie group Heis is the left-invariant Lorentz metric, which is given on heis by < X, Y >= 1, < Z, Z >= 1, and all other products are zero. It is explained in [DZ,Section 4.1], that the Lie algebra of Killing fields on (Heis, g LH ) is 4-dimensional, and is generated by X, Y, Z as well as a fourth element T satisfying the braket relations [T, Y ] = Y , [T, X] = −X and [T, Z] = 0. The identity component G of Iso(Heis, g LH ) is thus isomorphic to a semi-direct product R heis, where the R-factor integrates into a group of hyperbolic automorphisms of heis.
If (M, g) is a closed Lorentz manifold locally modelled on (Heis, g LH ), we consider a developing map δ :M → Heis, and the corresponding holonomy morphism ρ : π 1 (M ) → Iso(Heis, g LH ). Since G has finite index in Iso(Heis, g LH ), we can replace (M, g) by finite cover and assume that Γ := ρ(π 1 (M )) is contained in G. Since Z is central in G, the vector field δ * (Z) projects on (M, g) to a Killing vector field, the flow of which will be called the characteristic flow on (M, g), denoted ϕ t Z . The main part set U containingx, such that δ : U → gB is a diffeomorphism. One easily deduces that δ :M → Heis has the path-lifting property, hence is a diffeomorphism. ♦ Lemma 11.3 ensures that Γ := ρ(π 1 (M )) is a discrete subgroup of G R Heis that acts freely properly and cocompactly on Heis. We call Γ := Γ ∩ Heis.
If Γ = {1}, then Γ projects injectively on the R-factor of G R Heis. It is thus abelian. It is easily checked that nontrivial connected abelian subgroups of G have dimension 1 or 2. Looking at the identity component of the Zariski closure of Γ in G, we get that Γ is isomorphic to Z or Z 2 . A cohomological dimension argument shows that the action of Γ can not be cocompact on Heis.
Discrete subgroups of Heis which are not cyclic must intersect nontrivially the center of Heis. On the other hand, Γ intersects the center of Heis trivially, otherwise ϕ t Z would be a periodic flow, contradicting our assumption that it is not relatively compact. We conclude that Γ is infinite cyclic, and normalized by Γ. Considering a finite index subgroup, we get that Γ centralizes Γ . Because Γ is not contained in the center of Heis, its centralizer in G is 1-dimensional. We conclude that Γ Z, contradicting again that its action on Heis is cocompact. ♦ Aknowledgments. -We warmly thank the referee(s) of this article for providing extremely valuable remarks on the initial draft.