Isometry Group of Lorentz Manifolds: A Coarse Perspective
aa r X i v : . [ m a t h . DG ] F e b ISOMETRY GROUP OF LORENTZ MANIFOLDS: A COARSEPERSPECTIVE by Charles Frances
Abstract . —
We prove a structure theorem for the isometry group Iso(
M, g ) of acompact Lorentz manifold, under the assumption that a closed subgroup has expo-nential growth. We don’t assume anything about the identity component of Iso(
M, g ),so that our results apply for discrete isometry groups. We infer a full classification oflattices that can act isometrically on compact Lorentz manifolds. Moreover, withoutany growth hypothesis, we prove a Tits alternative for discrete subgroups of Iso(
M, g ).
1. Introduction
In this article, we are interested in the question of which groups can appear as thegroup of isometries of a compact pseudo-Riemannian manifold (
M, g ). Although thisquestion makes sense, and has been considered, for more general classes of geometricstructures, it is striking to note that a complete answer is only known in a very smallnumber of cases. One of the most natural examples, for which we have a completepicture, is that of Riemannian structures (all the structures considered in this articleare assumed to be smooth, i.e of class C ∞ ). A famous theorem of Myers and Steenrod[ MS ], ensures that the group of isometries of a compact Riemannian manifold isa compact Lie group, the Lie topology coinciding moreover with the C topology.Conversely, for any compact Lie group G , one can construct a compact Riemannianmanifold ( M, g ) for which Iso(
M, g ) = G . This was proved independently in [ BD ],[ SZ ]. This settles completely the Riemannian case.Regarding compact Lorentzian manifolds, which are the subject of this article,things get substantially more complicated. Indeed, although it is always true thatthe group of isometries of a Lorentzian manifold is a Lie transformation group, anew phenomenon appears: even for compact manifolds ( M, g ), the group Iso(
M, g )may well not be compact. For instance, it is quite possible that this group is infinitediscrete.Seminal works are owed to R. Zimmer, which paved the way for the study of theisometry group of Lorentz manifolds. A first important contribution was [
Zim2 ],where it is shown that any connected, simple Lie group G , acting isometrically on CHARLES FRANCES a compact Lorentz manifold, must be locally isomorphic to SL(2 , R ). Several deepcontributions followed, [ Gr1 ], [ Kw ], before S. Adams, G. Stuck, and independentlyA. Zeghib obtained, about ten years later, a complete classification of all possible Liealgebras for the group Iso( M, g ). Their theorem can be stated as follows:
Theorem . — [
AS1 ] , [ AS2 ] , [ Z5 ] , [ Z4 ] . Let ( M, g ) be a compact Lorentz manifold.Then the Lie algebra Iso ( M, g ) is of the form s ⊕ a ⊕ k where:1. k is trivial, or the Lie algebra of a compact semisimple group.2. a is an abelian algebra (maybe trivial).3. s is trivial, sl (2 , R ) , a Heisenberg algebra heis (2 d + 1) , or an oscillator algebra. Altough this classification is only at the Lie algebra level, we have a fairly good pic-ture of the possibilities for the identity component Iso o ( M, g ) (see [ Z4 ] for a discussionabout this identity component).Few attemps were done to go beyhond the understanding of Iso o ( M, g ), and com-plete the picture for the full group of isometries of a compact Lorentz manifold(
M, g ). An intermediate situation was studied in [ ZP ], where the authors assume thatIso o ( M, g ) is compact (but still nontrivial), and the discrete part Iso(
M, g ) / Iso o ( M, g )is infinite.In the purely discrete case, the most advanced results were obtained by R. Zimmer,who proved in [
Zim2 , Theorem D] that no discrete, infinite, subgroup of Iso(
M, g )can have Kazhdan’s property ( T ). For instance, this rules out the possibility thatIso( M, g ) is isomorphic to a lattice in some higher rank simple Lie group, or in thegroup Sp(1 , n ), n ≥
2. Besides this result, and to the best of our knowledge, almostnothing was known when the group Iso(
M, g ) is infinite discrete. Our aim here, is tobegin filling this gap. Theorems A and B below are a step in this direction.
We begin ourgeneral study of the isometry group of a compact Lorentz manifold, by making agrowth assumption, namely there exists a closed, compactly generated subgroup ofIso(
M, g ) having exponential growth. In this case, we get a pretty clear picture ofthe situation: The group Iso(
M, g ) is either a compact extension of PSL(2 , R ), or acompact extension of a Kleinian group (a Kleinian group is any discrete subgroupof PO(1 , d ), d ≥ M, g ) is discrete, and contains afinitely generated subgroup of exponential growth, then Iso(
M, g ) must be virtuallyisomorphic to a Kleinian group, what reduces substantially the possibilities.
Theorem A . —
Let ( M, g ) be a smooth, compact, ( n + 1) -dimensional Lorentz man-ifold, with n ≥ . Assume that the isometry group Iso(
M, g ) contains a closed, com-pactly generated subgroup with exponential growth. Then we are in exactly one of thefollowing cases:1. The group Iso(
M, g ) is virtually a Lie group extension of PSL(2 , R ) by a compactLie group.2. There exists a discrete subgroup Λ ⊂ PO(1 , d ) , for ≤ d ≤ n , such that Iso(
M, g ) is virtually a Lie group extension of Λ by a compact Lie group. SOMETRY GROUP OF LORENTZ MANIFOLDS By a
Lie group extension , we mean a group extension 1 → K → G → H → K, G, H is a Lie group, and each arrow a Lie grouphomomorphism.Let us make a few comments about the theorem. Although the hypotheses relate toa subgroup of Iso(
M, g ), the conclusion holds for the full isometry group. We do notmake directly a growth assumption about Iso(
M, g ), because even if the Lorentz man-ifold (
M, g ) is assumed to be compact, we don’t know if Iso(
M, g ) itself is compactlygenerated (making the notion of growth ill defined). The closedness assumption aboutthe subgroup of exponential growth is mandatory, to avoid for instance the “trivial”situation, where a non abelian free group is embedded in a compact subgroup ofIso(
M, g ).It is easy to prove that for a compact Lorentz surface, the isometry group is eithercompact, or a compact extension of Z , so that the assumptions of the theorem arenever satisfied in dimension 2, hence our hypothesis n + 1 ≥ M is 3-dimensional, Theorem A and its proof show that ( M, g ) hasto be of constant sectional curvature − , R ), and in the second one, ( M, g ) is a flat 3-torus. Thegroup Λ of Theorem A is then an arithmetic lattice in PO(1 , Fr1 ]). In higher dimension, the proof of TheoremA suggests that the geometry of (
M, g ) can be described completely. We will adressthis issue in later work.If we are in the first case of Theorem A, then Iso(
M, g ) has finitely many con-nected components. The identity component Iso o ( M, g ) is finitely covered by a prod-uct K × PSL(2 , R ) ( m ) , where K is a connected compact Lie group, and PSL(2 , R ) ( m ) stands for the m -fold cover of PSL(2 , R ). Such a situation appears through the follow-ing well known construction. On the Lie group PSL(2 , R ), let us consider the Killingmetric, namely the metric obtained by pushing the Killing form on s l (2 , R ) by lefttranslations. This is a Lorentzian metric of constant sectional curvature −
1, which isactually bi-invariant. If one considers a uniform lattice Γ ⊂ PSL(2 , R ), the quotientPSL(2 , R ) / Γ is a 3-dimensional Lorentz manifold, and the left action of PSL(2 , R ) isisometric. Actually the isometry group of PSL(2 , R ) / Γ is virtually PSL(2 , R ). Tak-ing products with compact Riemannian manifolds, provides examples of compactLorentzian manifolds, having isometry group virtually isomorphic to K × PSL(2 , R ),for K any compact Lie group.Let us now investigate the second case of Theorem A. The situation is then reversed:The identity component Iso o ( M, g ) and the discrete part Iso(
M, g ) / Iso o ( M, g ) is infi-nite, isomorphic to a (finite extension of a) Kleinian group. Here is a classical construc-tion illustrating this case. On R n +1 , n ≥
2, let us consider a Lorentzian quadratic form q . It defines a Lorentzian metric g on R n +1 which is flat and translation-invariant.If Γ ≃ Z n +1 is the discrete subgroup of translations with integer coordinates, weget a flat Lorentzian metric g on the torus T n +1 = R n +1 / Γ. The isometry groupof ( T n +1 , g ) is easily seen to coincide with O( q, Z ) ⋉ T n +1 . If q is chosed to be arational form (namely q has rational coefficients), then due to a theorem of A. Borel CHARLES FRANCES and Harish-Chandra, O( q, Z ) is a lattice in O( q ) ≃ O(1 , n ) (in particular, it is finitelygenerated of exponential growth). In section 2.4.2, we will elaborate on this con-struction, and provide examples of compact Lorentzian manifolds (
M, g ) for whichthe group Iso(
M, g ) is discrete and isomorphic to O( q, Z ) as above. The question ofwhich Kleinian groups Λ may appear in point (2) of Theorem A is interesting wouldcertainly deserve further investigations.Observe finally, that the statements of Theorem A also contains known, but non-trivial facts, about isometric actions of connected Lie groups. For instance it is shownin [ AS2 ], [ Z4 ] that if the affine group of the line Aff( R ) acts faithfully and isometri-cally on a compact Lorentz manifold, then the action extends to an isometric action ofa group G locally isomorphic to PSL(2 , R ). It is plain that if Aff( R ) acts isometrically,then we must be in the first case of Theorem A, so that the theorem provides directlya subgroup locally isomorphic to PSL(2 , R ) in Iso( M, g ). Actually, a generalizationof [
AS2 ], [ Z4 ] will be needed during the proof of Theorem A (see Theorem 7.7). We now investigate whatcan be said if we remove the growth assumption made in Theorem A. It is then stillpossible to describe the discrete, finitely generated subgroups of Iso(
M, g ). This isthe content of our second main result.
Theorem B (Tits alternative for discrete subgroups)
Let ( M n +1 , g ) be a smooth compact ( n + 1) -dimensional Lorentz manifold, with n ≥ .Then every discrete, finitely generated, subgroup of Iso(
M, g ) either contains a freesubgroup in two generators, in which case it is virtually isomorphic to a discretesubgroup of PO(1 , d ) , or is virtually nilpotent of growth degree ≤ n − . In the previous statement, we say that a group G is virtually isomorphic to a group G , if there exists a finite index subgroup G ′ ⊂ G , and a finite normal subgroup G ′′ ⊂ G ′ , such that G ′ /G ′′ is isomorphic to G .Theorem B is obtained by combining Theorem A, together with the recent resultsabout coarse embeddings of amenable groups, obtained by R. Tessera in [ Te ] (seeSection 2.5).Actually, a generalization of [ DKLMT , Theorem 3], from the setting of finitelygenerated, to that of compactly generated, unimodular amenable groups, would al-low to prove a Tits alternative – in its classical formulation – for all finitely gener-ated subgroups of Iso(
M, g ) (without the discreteness assumption). Also, it wouldyield a statement similar to that of Theorem B for finitely generated subgroups ofIso(
M, g ) / Iso o ( M, g ). We defer those developpements to subsequent works, whensuch generalizations of [
DKLMT , Theorem 3] will be available.
Another interesting corollary ofTheorem A, is a complete desciption of lattices (in noncompact simple Lie groups),which can appear as discrete subgroups of Iso(
M, g ), where (
M, g ) is a compactLorentz manifold. We will indeed obtain:
SOMETRY GROUP OF LORENTZ MANIFOLDS Corollary C . —
Let ( M n +1 , g ) be a compact ( n + 1) -dimensional Lorentz manifold.Assume that a discrete subgroup Λ ⊂ Iso(
M, g ) is isomorphic to a lattice in a non-compact simple Lie group G .1. Then G is locally isomorphic to PO(1 , k ) , for ≤ k ≤ n .2. If equality k = n holds, then ( M n +1 , g ) is either a -dimensional anti-de Sittermanifold, or a flat Lorentzian torus, or a two-fold cover of a flat Lorentziantorus. We recover this way, but by other methods, Zimmer’s results ([
Zim2 , TheoremD]) saying that lattices having property ( T ) do not appear as discrete subgroups ofIso( M, g ). The novelty in Corollary C is to cover the case of lattices in O(1 , k ) orSU(1 , k ), which was, to the best of our knowledge, not known.
Let us explain roughly what is the general strategy to prove our mainTheorem A, and its corollaries.The starting point of our study, and an essential tool throughout this article, is thenotion of coarse embedding , introduced by M. Gromov in [
Gr1 ]. Thus, Section 2 isdevoted to recalling what this important notion is. Gromov’s fundamental observationwas that the isometry group of a compact ( n + 1)-dimensional Lorentzian manifoldadmits a coarse embedding into the real hyperbolic space H n . This mere remarkalready provides important restrictions on the group Iso( M, g ), through basic coarseinvariants such as the asymptotic dimension, or some growth properties. To givethe reader a flavour of how those basic tools can be used in the context of isometricactions, we prove a particular case of Corollary C in Section 2.4. Then, we explainhow to link our main Theorem A to the results of [ Te ], in order to obtain TheoremB. Section 3 is devoted to the notion of limit set associated to a coarse embeddinginto H n , and the related limit set for the isometry group Iso( M, g ). It is explainedin Sections 3.4 and 3.5, how our assumtion of a compactly generated subgroup ofexponential growth forces this limit set to be infinite. The crucial step here, is to provethat the growth assumption we made on a subgroup of Iso(
M, g ), implies exponentialgrowth of derivatives for the action of Iso(
M, g ).This property is exploited further in Section 4. Using Gromov’s theory of rigidgeometric structures, one establishes a link between the limit set of the isometrygroup on the one hand, and local Killing fields on the manifold (
M, g ) on the otherhand. The general picture is that when Iso(
M, g ) has a big (infinite) limit set, thenthe manifold (
M, g ) must have a lot of local Killing fields. This statement is madeprecise in Theorem 4.1.The abundance of local Killing fields proved in Section 4, allows us to exhibit,in Section 5.2, a compact Lorentz submanifold Σ ⊂ M , which is locally homo-geneous (with semisimple isotropy) and left invariant by a finite index subgroupIso ′ ( M, g ) ⊂ Iso(
M, g ). This is the content of Theorem 5.1. The restriction mor-phism ρ : Iso ′ ( M, g ) → Iso(Σ , g ) is proper (it has closed image and compact kernel),
CHARLES FRANCES what essentially reduces the proof of Theorem A to the setting of locally homogeneousmanifolds.In Section 6, we thus tackle the problem of describing all compact Lorentz mani-folds, which are locally homogeneous with semisimple isotropy, and have an isometrygroup of exponential growth. To this aim, we have to prove a completeness theoremfor this class of manifolds, akin to the one obtained by Y. Carri`ere and B. Klingler forcompact Lorentz manifolds of constant sectional curvature. This is done in Theorem6.7.This completeness result opens the way for the final proof of Theorem A, which isachieved in the last Section 7.
2. Coarse embedding associated to an isometric action2.1. Coarse embeddings between metric spaces and groups. —
Let (
X, d X )and ( Y, d Y ) be two metric spaces. A map α : X → Y is called a coarse embedding ifthere exist two nondecreasing functions φ − : R + → R and φ + : R + → R + satisfyinglim r → + ∞ φ ± ( r ) = + ∞ , such that for any x, y in X , one has:(1) φ − ( d X ( x, y )) ≤ d Y ( α ( x ) , α ( y )) ≤ φ + ( d X ( x, y ))Equivalently, for any pair of sequences ( x k ) , ( y k ) in X , one has d Y ( α ( x k ) , α ( y k )) → + ∞ if and only if d X ( x k , y k ) → + ∞ .The function φ + (resp. φ − ) is called the upper control (resp. the lower control).In the case where φ + and φ − are affine functions, we recover the more popular notionof quasi-isometric embedding. The spaces ( X, d X ) and ( Y, d Y ) are said to be coarselyequivalent whenever there exists a second coarse embedding β : ( Y, d Y ) → ( X, d X )such that β ◦ α is a bounded distance from identity.It seems that this notion was considered for the first time by M. Gromov in [ Gr1 ,Section 4], under the name of placement . . — We will often speak inthis paper of coarse embeddings between groups, or from a group to a metric space.To make this notion precise, say that a distance d on a topological group G is called an adapted metric , when it is right invariant , proper (the balls are relatively compactsets), and when compact subsets have finite diameter for d . Observe that we don’trequire d to be continuous, so that the last condition is nontrivial. It is a classical result(see [ Str ]) that any locally compact, second countable, topological group admits anadapted metric (actually one can even find adapted metrics generating the topologyof G ).An important example of adapted metric will be the word metric on a secondcountable, locally compact, compactly generated group. Those are groups G , forwhich there exists a compact subset S , that we may assume symmetric, namely S = S − , such that G = S n ∈ N S n . Here S n = S.S. . . . .S denotes the set of g ∈ G that can be written as a product g = w ....w n with w k ∈ S (and S = { G } ). One candefine ℓ S ( g ) = min { n ∈ N , g ∈ S n } . Then, defining d S ( g, h ) = ℓ S ( gh − ), we obtain SOMETRY GROUP OF LORENTZ MANIFOLDS an adpated metric on G , called the word metric (associated to the generating set S ).Observe that generally, this metric is not continuous.By a coarse embedding α : G → G between two (locally compact, second count-able) topological groups, we will mean in what follows that α is coarse between ( G , d )and ( G , d ), with d , d adapted metrics. The notion does not depend on the choiceof such metrics, because of the following easy topological characterization: Fact 2.1 . —
A map α : G → G between two topological groups, endowed withadapted metrics, is a coarse embedding, when for each pair of sequences ( f k ) and ( g k ) in G , f k g − k stays in a compact subset of G if and only if α ( f k ) α ( g k ) − stays in acompact subset of G . Iso(
M, g ). — Let ( M n +1 , g ) be a compact ( n + 1)-dimensionalLorentz manifold. We recall briefly how one makes Iso( M, g ) into a Lie transfor-mation group. Let us call ˆ M the bundle of orthogonal frames on M , which is aO(1 , n )-principal bundle over M . Notice that every f ∈ Iso(
M, g ) induces naturallya diffeomorphism ˆ f : ˆ M → ˆ M , which moreover preserves a parallelism on ˆ M , comingfrom the Levi-Civita connection of g . One then shows that Iso( M, g ) acts freely onˆ M , and orbits of Iso( M, g ) on ˆ M are closed submanifolds of ˆ M (closed but of coursegenerally not compact). This identification of Iso( M, g ) as a submanifold of ˆ M isthe way one defines the Lie group topology on Iso( M, g ) (see [ St , Cor VII.4.2] for amore detailed account). In particular Iso( M, g ) is secound countable and locally com-pact, hence admits adapted metrics. Observe that for a compact Lorentz manifold( M n , g ), it is not clear (and maybe false, eventhough we don’t have any example)that the group Iso( M, g ) is compactly generated. Using the fact that the exponentialmap of g locally linearizes isometries, it is not very hard to check that this Lie topol-ogy is actually the C -topology on Iso( M, g ), making Iso(
M, g ) a closed subgroupof Homeo( M ). The very definition of the Lie topology implies that Iso( M, g ) actsproperly on ˆ M . Iso(
M, g ) into O(1 , n ). — We fix once for all a section σ : M → ˆ M such that the image σ ( ˆ M ) has compact closure in ˆ M . Such sectionsexist since M is compact. In all the paper, we will always deal with such boundedsections.
Having fixed a bounded section σ : M → ˆ M as above, we get for every x ∈ M amap D x : Iso( M, g ) → O(1 , n )defined by the following relation: σ ( f ( x )) = ˆ f ( σ ( x )) . ( D x ( f )) − . The element D x ( f ) is nothing but the matrix of the tangent map D x f : T x M → T f ( x ) M , if we put the frames σ ( x ) and σ ( f ( x )) on T x M and T f ( x ) M respectively.In [ Gr1 ], M. Gromov made the following crucial observation:
CHARLES FRANCES
Lemma 2.2 (see [ Gr1 ] Sections 4.1.C and 4.1.D) . —
Let ( M n +1 , g ) be a com-pact Lorentz manifold. Then for every x ∈ M , the derivative cocycle D x : Iso( M, g ) → O(1 , n ) is a coarse embedding.Proof . — The proof follows easily from the properness of the action of Iso( M, g ) onˆ M . Indeed, given two sequences ( f k ) and ( g k ) of Iso( M, g ), f k g − k stays in a compactsubset of Iso( M, g ) if and only if ˆ f k ˆ g − k σ ( g k x ) stays in a compact set of ˆ M (because σ ( g k x ) is contained in a compact subset of ˆ M and Iso( M, g ) acts properly on ˆ M ).But from the relation: ˆ f k ˆ g − k σ ( g k x ) D x ( g k ) D x ( f k ) − = σ ( f k x ) , and the properness of the right action of O(1 , n ) on ˆ M , we see that our initial assertionis equivalent to D x ( g k ) D x ( f k ) − (hence D x ( f k ) D x ( g k ) − ) staying in a compact subsetof O(1 , n ), and we conclude by Fact 2.1. H n . — Let H n denote the real hyperbolic space, andlet o ∈ H n be a base point. The group O(1 , n ) acts isometrically on H n . If x ∈ M n +1 is given, we can derive from D x a coarse embedding D x : Iso( M, g ) → ( H n , d hyp ),defined by D x ( f ) = D x ( f ) − .o . The embedding D x is coarse because so is D x , andthe action of O(1 , n ) on H n is proper. . — To get useful in-variants under coarse equivalence of metric spaces, it is better to focus on spaceswhich are quasi-geodesic, and of bounded geometry. We recall that a metric space( X, d X ) is quasi-geodesic if there exist a, b > x, y )in X can be joigned by a ( a, b )-quasi-geodesic. In other words, there exists, for anysuch pair ( x, y ) an interval [0 , L ], as well as a map γ : [0 , L ] → X with γ (0) = x and γ ( L ) = y , and for every t, t ′ ∈ [0 , L ]:1 a | t ′ − t | − b ≤ d X ( γ ( t ′ ) , γ ( t )) ≤ a | t ′ − t | + b. A metric space (
X, d X ) has bounded geometry, if it is quasi-isometric to some( X ′ , d X ′ ) satisfying:1. ( X ′ , d X ′ ) is uniformly discrete, namely there exists C > x = x ′ d X ′ ( x, x ′ ) ≥ C .2. For every r >
0, there exists n r such that every ball of radius r has at most n r elements.For us, the basic examples of quasi-geodesic metric spaces of bounded geometrywill be homogeneous Riemannian manifolds, for instance Euclidean space R n or realhyperbolic space H n . Also very important is the case of a second countable, locallycompact and compactly generated group G , endowed with a word metric d S . Thevery definition of the word metric makes ( G, d S ) a quasi-geodesic space. It may notbe locally finite, but we can always consider Λ ⊂ X be a C -metric lattice in G . Itmeans that d ( x, x ′ ) ≥ C if x = x ′ in Λ, and also that there is a constant D > SOMETRY GROUP OF LORENTZ MANIFOLDS such that any x ∈ X is at distance at most D from Λ. Such lattices exist by Zornlemma, and for C > , d S ) is locally finite, uniformly discrete, andquasi-isometric to ( G, d S ). In particular (Λ , d S ) is quasi-geodesic.One important feature of the quasi-geodesic assumption is the following: Lemma 2.3 . —
Let ( X, d X ) and ( Y, d Y ) be two metric spaces, with ( X, d X ) quasi-geodesic. Then for any coarse embedding α : X → Y , the upper control function φ + can be chosen affine.Proof . — By assumption, there exist a, b > x and y , there exists a ( a, b )-quasi-geodesic. Since α is a coarse embedding, thereexists c > x, y ∈ X , d X ( x, y ) ≤ a + b implies d Y ( α ( x ) , α ( y ) ≤ c .Let x, y ∈ X , and γ : [0 , L ] → X be a ( a, b )-quasi-geodesic between x and y . Then d Y ( α ( x ) , α ( y )) ≤ Σ E ( L ) i =0 d Y ( α ( γ ( i )) , α ( γ ( i + 1))) ≤ Lc . But since γ is a ( a, b )-quasi-geodesic, we have a L − b ≤ d X ( x, y ), so that d Y ( α ( x ) , α ( y )) ≤ ac d X ( x, y ) + abc . . — Let ( X, d X ) be a metric space of boundedgeometry, that we assume first to be uniformly discrete. For any x ∈ X , and r > β X ( x , r ) = | B ( x , r ) | (the number of points in B ( x , r )).Given two functions f : R + → R + and g : R + → R + , one says that f (cid:22) g whenthere exist constants λ, µ and c such that f ( r ) ≤ λg ( µr + c ), and f ≈ g when f (cid:22) g and g (cid:22) f . It is pretty clear that changing x into y , one gets β X ( x , ˙) ≈ β X ( y , ˙).Recall that ( X, d X ) is said to have polynomial growth if there exists a constant C >
0, and d ∈ N , such that β X ( x , r ) ≤ Cr d for r big enough. The minimal d forwhich it holds is then called the growth degree of ( X, d X ).One says that ( X, d X ) has exponential growth, when there exists some a > β X ( x , r ) ≥ e ar for r big enough.The property β X ( x , ˙) ≈ β X ( y , ˙) ensures that this definition is independent of thechoice of x .If ( X, d X ) is a space with bounded geometry, which is not locally finite, we takesome ( X ′ , d X ′ ) which is quasi-isometric to ( X, d X ) and locally finite, and define β X = β X ′ (equality makes sense up to the relation ≈ ). Lemma 2.4 . —
Let ( X, d X ) and ( Y, d Y ) be two quasi-geodesic metric spaces ofbounded geometry. Then the growth function of Y dominates that of X , namely β X (cid:22) β Y . In particular, if ( Y, d Y ) has polynomial growth of degree d , then ( X, d X ) has polynomial growth of degree d ′ ≤ d , and if ( X, d X ) has exponential growth, thenthe same holds for ( Y, d Y ) .Proof . — We may assume again that X and Y are uniformly discrete. Because α is acoarse embedding, there exists C > d X ( x, y ) ≥ C , then d Y ( α ( x ) , α ( y )) ≥
1. Now let Λ ⊂ X be a C -metric lattice in X . It means that d ( x, x ′ ) ≥ C if x = x ′ in Λ, and also that there is a constant D > x ∈ X is at distance atmost D from Λ. From the fact that all balls of radius D have at most k D points, itis easy to check that if x ∈ Λ, then(2) β Λ ( x , r ) ≤ β X ( x , r ) ≤ k D β Λ ( x , r + D ) . CHARLES FRANCES
Let us call y = α ( x ). From Lemma 2.3, there exist a, b > α ( B X ( x , r )) ⊂ B Y ( y , ar + b ). By definition of C , α is one-to-one in restric-tion to Λ, so that β Λ ( x , r ) ≤ β Y ( y , ar + b ). From (2), we infer β X ( x , r ) ≤ k D β Y ( y , ar + aD + b ), which concludes the proof.If we consider ( M n +1 , g ) a Lorentz manifold, and G ⊂ Iso(
M, g ) a closed, compactlygenerated subgroup, then we inherits a coarse embedding D x : G → H n (see Section2.2.3). Since the growth of H n is exponential, Lemma 2.4 does not put any restrictionon G . But when some extra geometric informations are available, then some usefulconclusions can be drawn, as we see now (see also Section 3.4). . — It is plain that one canrestrict the derivative cocycle D x to closed subgroups of Iso( M, g ), still getting acoarse embedding. This can be especially interesting when such a subgroup preservesa reduction of the bundle ˆ M . We have indeed: Corollary 2.5 . —
Let ( M n +1 , g ) be a Lorentz manifold. Assume that there exists aclosed subgroup G ⊂ Iso(
M, g ) , leaving invariant a compact subset K ⊂ M , as wellas a reduction of ˆ M above K , to an H -subundle with H ⊂ O(1 , n ) a closed subgroup.Then there exists a coarse embedding α : G → H .Proof . — The proof is a straigthforward rephrasing of that of Lemma 2.2, looking ata bounded section σ : K → ˆ N , where ˆ N is the H -subbundle over K .As a toy application in the realm of Lorentz geometry, let us formulate the following Proposition 2.6 . —
Let ( M n , g ) be a compact, ( n + 1) -dimensional Lorentz man-ifold. Assume that there exists on M a nontrivial Killing field X which is every-where lightlike , namely g ( X, X ) = 0 . Then every closed, finitely generated, subgroup Λ ⊂ Iso(
M, g ) which commutes with X is virtually nilpotent, of growth degree at most n − .Proof . — It is classical that nontrivial lightlike Killing fields on Lorentz manifolds cannot have singularities. Thus, the field X defines a reduction of the frame bundle ˆ M to the group L ⊂ O(1 , n ), where L is the stabilizer of a lightlike vector in Minkowskispace R ,n . This group L is isomorphic to O( n − ⋉ R n − . Any closed finitelygenerated Λ ⊂ Iso(
M, g ) commuting with X preserves the reduction, hence coarselyembeds into L by corollary 2.5. Lemma 2.4 implies that Λ has polynomial growth,with growth degree ≤ n −
1. The proposition follows from Gromov’s theorem aboutgroups with polynomial growth. . — The notion of asymptoticdimension of a metric space appears in [
Gr1 , Section 4], and was developped in[
Gr2 ].One says that (
X, d X ) has asymptotic dimension at most n , written Asdim( X ) ≤ n ,if for every r >
0, there exists a covering X = S i ∈ I U i with the following properties – All the U i ’s are uniformly bounded. SOMETRY GROUP OF LORENTZ MANIFOLDS – The U i ’s can be splitted into ( n + 1) families U , . . . , U n , with the property thatwhenever U i and U j , i = j , belong to a same family U k , then d X ( U i , U j ) > r .The asymptotic dimension of ( X, d X ) is the minimal n for which one hasAsdim( X ) ≤ n .The following lemma follows almost directly from the definitions. Lemma 2.7 . —
Let ( X, d X ) and ( Y, d Y ) be two metric spaces, and α : X → Y acoarse embedding. Then Asdim( X ) ≤ Asdim( Y ) . In particular, two spaces which are coarse-equivalent will have the same asymptoticdimension.It is known that Asdim( R n ) = Asdim( H n ) = n for every n ∈ N (see for instance[ BS , chap. 10]). Hence, Lemma 2.7 says, for instance, that if ( M n , g ) is a compact,( n + 1)-dimensional manifold, and if Λ ⊂ Iso(
M, g ) is a discrete subgroup isomorphicto Z k , then k ≤ n . One has the sharper upper bound k ≤ n −
1, as the followingstatement shows:
Proposition 2.8 . —
Let n ≥ . Then there is no coarse embedding α : R n → H n . Observe that horospheres in H n yield coarse embeddings of R n − into H n . Proof . — The proposition can not be derived in a straigthforward way, because Lem-mas 2.4 and 2.7 are obviously useless in our situation. The ingredients of the proofare more elaborate (though classical for the experts), and involve “coarse topologicalarguments”. The details can be found in [ DK , section 9.6], and especially Theorem9 .
69 there. This theorem states that if a coarse embedding α : R n → H n would exist,then it should be almost surjective (namely the image is cobounded). Then it is easyto build a coarse inverse β : H n → R n to α . But the existence of such a β is forbiddenbecause of Lemma 2.4. O(1 , k ). — To illustrate how the basic tools ofcoarse geometry presented so far can already say interesting things about isometricactions on Lorentz manifolds, let us prove part of Corollary C, without appealing toTheorem A.
Proposition 2.9 . —
Let ( M n +1 , g ) be a compact Lorentz manifold. Then if k ≥ n + 1 , Iso(
M, g ) does not contain any discrete subgroup Λ isomorphic to a lattice of O(1 , k ) .Proof . — Assume that Λ ⊂ Iso(
M, g ) is discrete and isomorphic to a lattice in O(1 , k ).Observe first that Λ is closed and finitely generated, hence we can apply all what wedid so far. If Λ is uniform in O(1 , k ), then its asymptotic dimension is that of H k ,and by the coarse embedding D x : Λ → H n of Section 2.2.3 and Lemma 2.7, we get k ≤ n , and we are done. If Λ is not uniform, then the thick-thin decomposition (see[ Th , Section 4.5]) provides a discrete subgroup of Λ, virtually isomorphic to Z k − .But then Proposition 2.8 forces k − ≤ n −
1, which concludes the proof. CHARLES FRANCES
The statement is optimal. Indeed, we saw in the introduction that on any torus T n +1 ( n ≥ ⋉ T n +1 , whereΛ is some lattice in O(1 , n ). . — At thispoint, it is worth noticing that one can produce examples of compact Lorentz ( n +1)-manifolds ( n ≥
3) admitting an isometry group which is virtually a lattice inO(1 , n − q on R n , n ≥
3, which is rational. Then, Borel Harish-Candra theorem ensures thatΛ := O( q, Z ) is a lattice in O( q ). The quadratic form q defines a Lorentz metric g on R n , which is flat and translation invariant. We consider R × R n , with coor-dinates ( t, x , . . . , x n ), and we endow it with the Lorentz metric g a := dt + a ( t ) g ,where a : t a ( t ) is a positive, 1-periodic function on R . Now, let Γ be the sub-group generated by γ : ( t, x , . . . , x n ) ( t + 1 , − x , . . . , − x n ) and the translations τ i : ( t, x , . . . , x i , . . . , x n ) ( t, x , . . . , x i + 1 , . . . , x n ), 1 ≤ i ≤ n . This is a discretesubgroup, acting by isometries for g a .The quotient manifold M := ( R × R n ) / Γ is topologically a T n -bundle over the circle.It inherits a Lorentz metric g a , for which the action of Λ is isometric. Actually, fora generic choice of the 1-periodic function a , Iso( M, g a ) will coincide virtually withΛ (see [ Fr1 , Sec. 2.2, Lemma 2.1] for the precise genericity condition that has to beput on a ). All the invariants of coarseembeddings (growth, asymptotic dimension) presented so far are very basic. Moresofisticated tools, leading to obstructions for a group to admit a coarse embeddinginto some real hyperbolic space H d can be found in [ HS ]. For instance, it is shownin [ HS , Cor. 1.3] that a finitely generated, virtually solvable group, which is notvirtually nilpotent does not admit such a coarse embedding (see also Theorem 2.10below). It follows that those groups do not appear as closed subgroups of isometriesfor a compact Lorentz manifold. Observe that this result can also be infered fromTheorem A, because it is easy to check that a discrete subgroup of O(1 , d ) which isvirtually solvable has to be virtually abelian.Recently, several authors ([ DKLMT ], [
HMT ], [ LG ]) studied the behaviour ofnotions such as separation, and isoperimetric profiles, with respect to coarse embed-dings. This led to the following quite amazing result of R. Tessera: Theorem 2.10 . — [ Te , Th. 1.1] Let Λ be a finitely generated, amenable group. Ifthere exists a coarse embedding from Λ to H n , then Λ is virtually nilpotent of growthdegree at most n − . This result, combined with Theorem A implies easily Theorem B. Indeed, let( M n +1 , g ) be a compact, ( n +1)-dimensional, Lorentz manifold, and let Λ ⊂ Iso(
M, g )be a discrete, finitely generated, subgroup. We endow Λ with a word metric, andconsider its growth function, namely β : n
7→ | B (1 Λ , n ) | . If the growth of Λ is subex-ponential, namely if lim n →∞ n log( β ( n )) = 0, then Λ is amenable. Because Λ coarsely SOMETRY GROUP OF LORENTZ MANIFOLDS embeds into H n by Lemma 2.2, Tessera’s theorem ensures that Λ is virtually nilpotent,of growth degree at most n − ′ ⊂ Λ,and a proper homomorphism ρ : Λ ′ → PO(1 , n ), n ≥
2. The image ρ (Λ ′ ) is a discretesubgroup Λ ρ of PO(1 , n ), and because the kernel of ρ is finite, Λ is virtually isomorphicto Λ ρ . Now discrete subgroups of PO(1 , n ) split into two categories (see [ MT ] foran introduction to Kleinian groups). The elementary ones, for which the limit setis finite. Those groups are virtually abelian, hence don’t have exponential growth.This case is not compatible with our growth assumption on Λ. The non-elementarygroups of PO(1 , n ) have infinite limit set. It is known that such groups contain pairsof loxodromic elements α, β with pairwise disjoint fixed points (see for instance [ MT ,Lemma 2.3]). It is then clear that suitable powers α n and β n satisfy the Ping-ponglemma, hence generate a free group. The inverse image of this free group in Λ ′ is alsoa free subgroup, what concludes the proof of Theorem B.The remaining of the paper is devoted to the proof of Theorem A.
3. The limit set of an isometric action3.1. Dynamical definition of the limit set. —
Let (
M, g ) be a compact Lorentzmanifold. Following [ Z1 , Section 9.1], one can introduce a notion of (fiberwise) limitset for the action of the isometry group Iso( M, g ). For each x ∈ M the “limit set”Λ( x ) ⊂ P ( T x M ) comprises all nonzero lightlike directions [ u ] ∈ P ( T x M ), for whichthere exists a sequence ( f k ) ∈ Iso(
M, g ) tending to infinity in Iso(
M, g ), and a sequence( u k ) ∈ T M tending to u , such that Df k ( u k ) is bounded in T M . It follows from thedefinition that if Iso(
M, g ) is compact, then Λ( x ) = ∅ for every x ∈ M . Conversely,it is pretty easy to check that Λ( x ) = ∅ for every x ∈ M as soon as Iso( M, g ) isnoncompact. As we will see later, the existence of a “big” limit set at each x ∈ M has strong geometric consequences on the Lorentz manifold ( M, g ). To estimate thesize of Λ( x ), and give a precise meaning of big limit set, we introduce the map card Λ : M → N ∪ {∞} which to a point x ∈ M associates the number of points in Λ( x ). Itwill be also useful to consider E Λ ( x ), the vector subspace of T x M spanned by Λ( x ),and denote by d Λ ( x ) the dimension of E Λ ( x ). A totally geodesic, codimension one, lightlike foliation on M , is a codimensionone foliation, the leaves of which are totally geodesic, and such that the restriction ofthe metric g to the leaves is degenerate (one says lightlike ). The work [ Z1 ] makes alink between directions in the limit set, and totally geodesic codimension one lightlikefoliations on M . To be precise, A. Zeghib introduces, for every sequence ( f k ) goingto infinity in Iso( M, g ), the asymptotically stable distribution AS ( f k ) ⊂ T M of ( f k )as follows: the vectors of AS ( f k ) are those u ∈ T M for which there exists a sequence( u k ) of T M converging to u , and such that Df k ( u k ) is bounded. CHARLES FRANCES
Theorem 3.1 . — [ Z1 , Theorem 1.2] Let ( M, g ) be a compact Lorentz manifold, andlet ( f k ) be a sequence of Iso(
M, g ) going to infinity. Then, after considering a sub-sequence of ( f k ) , the asymptotically stable set AS ( f k ) is a codimension one lightlikedistribution, which is tangent to a Lipschitz totally geodesic codimension one lightlikefoliation. If [ u ] is a direction belonging to Λ( x ), then it follows from the definitions thatthere exists a sequence ( f k ) going to infinity in Iso( M, g ) such that u ∈ AS ( f k ).Actually, after considering a subsequence, AS ( f k )( x ) is a lightlike hyperplane of T x M coinciding with u ⊥ . Let us call F the totally geodesic codimension one lightlikefoliation integrating AS ( f k ), the existence of which is asserted by Theorem 3.1. Forevery y ∈ M , if v ∈ T y M is a nonzero lightlike vector such that v ⊥ is tangent to theleaf of F containing y , then [ v ] ∈ Λ( y ). It is known that totally geodesic codimensionone lightlike foliations have Lipschitz transverse regularity. We thus get: Corollary 3.2 . —
Each direction [ u ] ∈ Λ( x ) can be extended to a Lipschitz field oflightlike directions y ∆ [ u ] ( y ) on M , such that ∆ [ u ] ( y ) ∈ Λ( y ) for every y ∈ M . This yields the following semi-continuity property.
Corollary 3.3 . —
The map x d Λ ( x ) is lower semi-continuous. For any m ∈ N ∗ ,the sets K ≥ m := { x ∈ M | card Λ ( x ) ≥ m } are open. Let us consider a subset G ⊂ O(1 , n ), and a point o ∈ H n . We call O G the set: O G := { g − .o | g ∈ G } . Then we introduce the limit set of G , denoted Λ G , as Λ G := O G ∩ ∂ H n . The closure O G is taken in the topological ball H n . We notice that Λ G = ∅ if and only if G hascompact closure in O(1 , n ).If G ⊂ Iso(
M, g ) is a closed subgroup and if for x ∈ M , D x : G → O(1 , n ) is thecoarse embedding given by the derivative cocycle (see Section 2.2.2), then it makessense to consider the limit set Λ D x ( G ) , that we will rather write Λ D ( x ) to ease thenotations.It turns out that the sets Λ( x ) and Λ D ( x ) encode the same object. Let us explainwhy. First, denote by R ,n the space R n +1 endowed with the quadratic form q ,n =2 x x n +1 + x + . . . + x n . The group O(1 , n ) is the subgroup of GL( n + 1 , R ) preserving q ,n . We will use the projective model for real hyperbolic space H n , namely we seethe H n as the set of timelike lines (satisfying q ,n ( x ) <
0) in R ,n . Its boundary ∂ H n coincides with the set of lightlike lines. The action of O(1 , n ) on those sets isthe obvious one, coming from the linear action of O(1 , n ) on R ,n . Let us consider abounded section σ : M → ˆ M as in section 2.2.2, which defines the coarse embedding D x : Iso( M, g ) → O(1 , n ). Each frame σ ( x ) can be seen as a linear isometry σ ( x ) : R ,n → T x M . Let us denote by ι x : T x M → R ,n the inverse map. We claim: Lemma 3.4 . —
For every x ∈ M , we have Λ D ( x ) = ι x (Λ( x )) . SOMETRY GROUP OF LORENTZ MANIFOLDS Proof . — The very definition of D x yields, for every f ∈ Iso(
M, g ), every x ∈ M , andevery u ∈ T x M , the equivariance relation(3) D x f ( u ) = σ ( f ( x ))( D x f ( ι x ( u ))) . Now if [ u ] ∈ Λ( x ), there exists a sequence ( u k ) in T x M , and a sequence ( f k ) goingto infinity in Iso( M, g ) such that | D x f k ( u k ) | remains bounded. Relation (3) showsthat D x f k ( ι x ( u k )) remains bounded in R ,n . Let us perform a Cartan decompositionof D x f k in O(1 , n ), namely write D x f k = m k a k l k where ( m k ) and ( n k ) stay in amaximal compact group of O(1 , n ) and a k is a diagonal matrix of the from a k = λ k I n −
00 0 λ − k , where λ k → + ∞ . We may also consider a subsequence, so that m k converges to m ∞ and l k to l ∞ . Then it is plain that because D x f k ( ι x ( u k )) is bounded and ι x ( u ) islightlike, one must have ι x ( u ) ∈ R .l − ∞ ( e n +1 ). Also quite obvious is the fact that forevery timelike vector v , ( D x f k ) − ( v ) tends projectively to [ l − ∞ ( e n +1 )]. This shows[ ι x ( u )] ∈ Λ D ( x ). The inclusion Λ D ( x ) ⊂ ι x (Λ( x )) is shown in the same way. We con-sider G ⊂ Iso(
M, g ) a closed, compactly generated subgroup. We choose S a sym-metric compact set generating G , and we consider the associated word metric d S (seeSection 2.1.1). For g ∈ G , we will denote by ℓ ( g ) the distance d S (1 G , g ).Given a function ψ : G → R + , one says that ψ has exponential growth, if thereexists λ >
0, and a sequence ( g i ) in G , which tends to infinity, and such that ψ ( g i ) ≥ e λℓ ( g i ) .We fix from now on an auxiliary Riemannian metric h on M . The statementsbelow will involve estimates with respect to this metric, but by compactness of M ,their conclusions won’t depend on the choice of h . The norm of a vector u ∈ T M with respect to the metric h will be denoted by | u | . For x ∈ M , and ϕ ∈ Diff( M ), wedenote by | D x ϕ | := sup | u | =1 | D x ϕ ( u ) | . Proposition 3.5 . —
Let ( M, g ) be a compact Lorentz manifold. Let G ⊂ Iso(
M, g ) be a closed, compactly generated subgroup. If G has exponential growth, then for every x ∈ M , the function g ∈ G
7→ | D x g | has exponential growth.Proof . — We consider, for each x ∈ M , the coarse embedding D x : G → O(1 , n )given by the derivative cocycle (see Section 2.2.2). Because D x is defined relatively toa bounded section σ : M → ˆ M , Proposition 3.5 amounts to proving that g
7→ | D x g | has exponential growth, where we put any norm | . | on the space of matrices.Let us recall Iwasawa’s decomposition in O(1 , n ). Each D x g ∈ O(1 , n ) can bewritten in a unique way as D x g = k ( g ) a ( g ) n ( g ), where k ( g ) belongs to a maximalcompact subgroup K ⊂ O(1 , n ) CHARLES FRANCES a ( g ) = e λ ( g ) I n −
00 0 e − λ ( g ) , λ ( g ) ∈ R n ( g ) = v ( g ) −
Assuming that G has exponential growth, either g e | λ ( g ) | or g v ( g ) | has exponential growth.Proof . — Assume for a contradiction that both maps have subexponential growth.Working in the upper-half space model R ∗ + × R n − for H n , with coordinates ( t, x ),the action of a ( g ) is given by ( t, x ) ( e λ ( g ) t, e λ ( g ) x ), and that of n ( g ) by ( t, x ) ( t, x + v ( g )). We consider the point o = (1 , ∈ R ∗ + × R n − , which is precisely thepoint fixed by the compact group K .We already observed in Section 2.2.3 that the map: D x : G → H n defined by D x g := ( D x g ) − .o = n ( g ) − a ( g ) − .o is a coarse embedding.Our subexponential growth assumption tells that for every ǫ >
0, there exists N ǫ ∈ N such that for every k ≥ N ǫ , and every g satisfying ℓ ( g ) ≤ k , e | λ ( g ) | ≤ e kǫ and | v ( g ) | ≤ e kǫ . Geometrically, it means that for k ≥ N ǫ , the ball B (1 G , k ) is mapped by D x into a rectangle R k,ǫ = [ e − kǫ , e kǫ ] × [ − e kǫ , e kǫ ] n − ⊂ R ∗ + × R n − . Let us computethe hyperbolic volume of this rectangle in H n :vol hyp ( R k,ǫ ) = 2 n − e k ( n − ǫ Z e kǫ e − kǫ t n dt = 2 n − n − e k ( n − ǫ − . Now D x being a coarse embedding, there exists c > g, h ∈ G , d S ( g, h ) ≥ c implies d H n ( D x g, D x h ) ≥
1. Let us choose a c -metriclattice L in G . Because G has exponential growth, there exists µ > n k = ♯ ( L ∩ B (1 , k )) ≥ e µk for k big enough. On the other hand, D x ( L ∩ B (1 G , k )) ⊂ R k,ǫ contains n k points at mutual hyperbolic distance at least 1. Hence there exists aconstant C > k and ǫ ), such that n k ≤ C vol hyp ( R k,ǫ ). We endup with the inequality e µk ≤ C n − n − ( e k ( n − ǫ − k big enough. If wechosed ǫ small, for instance 0 < ǫ < µ n − , this yields a contradiction as k tends toinfinity. SOMETRY GROUP OF LORENTZ MANIFOLDS We see that if a ( g ) = e λ ( g ) Id n −
00 0 e − λ ( g ) and n ( g ) = v ( g ) | v ( g ) | Id n − − t v ( g )0 0 1 then: a ( g ) n ( g ) = e λ ( g ) e λ ( g ) v e λ ( g ) | v ( g ) | Id n − − t v e − λ ( g ) . If g e | λ ( g ) | has exponential growth, then so has | a ( g ) n ( g ) | (look at diagonalterms). If g
7→ | v ( g ) | has exponential growth, then so has | a ( g ) n ( g ) | (look at thelast column). Since D x g = k ( g ) a ( g ) n ( g ) with k ( g ) in a compact set, it follows fromLemma 3.6 that g
7→ | D x g | has exponential growth. This concludes the proof. Proposition 3.7 . —
Let ( M, g ) be a compact Lorentz manifold. Let G ⊂ Iso(
M, g ) be a closed, compactly generated subgroup. If at some x ∈ M , the map g
7→ | D x g | hasexponential growth, then the limit set Λ( x ) has at least two points.Proof . — The main ideas of the proof are borrowed from [ HKR ].We will use again the map D y : G → O(1 , n ) defined in Section 2.2.2, whichis a coarse embedding for every y ∈ M . We consider S a compact, symmetricgenerating set for the group G . Let us call Σ = S Z the set of bi-infinite words w = ( . . . w − w − w w w . . . ) in the alphabet S . The shift σ : Σ → Σ is defined as σ ( w ) = w ′ where w ′ i = w i +1 (we shift to the right). We call O the closure of the orbit G.x , and we define θ : Σ × O → Σ × O by θ ( w, y ) = ( σ ( w ) , w .y ). We observe that θ − ( w, y ) = ( σ − ( w ) , w − − .y ).The action of G on M defines naturally a map A : Σ × M → O (1 , n ), by the formula A ( w, y ) := D y w . Let us put A ( k ) ( w, y ) = A ( θ k − ( w, y )) A ( θ k − ( w, y )) . . . A ( θ ( y )) A ( y ) , which by the cocycle property of the derivative cocycle is nothing but D y w k − ....w w .In the same way we define A ( − k ) ( w, y ) = A − ( θ − k ( w, y )) . . . A − ( θ − ( w, y )) , which coincides D y w − − k . . . w − − w − − .We obtain in this way a cocycle over θ : Σ × M → Σ × M with values into O(1 , n ).Because Σ × O is compact, there exists ν an ergodic θ -invariant Borel probabilitymeasure on Σ × O . Observe that ( w, y ) log || A ( w, y ) || and ( w, y ) log || A − ( w, y ) || are integrable for ν because those are Borel maps which are bounded (since S iscompact and the derivative cocycle ( x, f ) D x f is defined relatively to a boundedframe field).Let us recall the conclusions of Oseledec theorem in this context (see for instance[ L , Theorem 4.2]). CHARLES FRANCES
Theorem 3.8 (Oseledec Theorem) . —
There exists a θ -invariant Borel subset B ⊂ Σ × O with ν ( B ) = 1 , a measurable decomposition R n +1 = W z ⊕ . . . ⊕ W rz , z ∈ B , and real Lyapunov exponents λ < λ < . . . < λ r satisfying the properties:1. The decomposition R n +1 = W z ⊕ . . . ⊕ W rz is invariant in the sense that A ( z ) W iz = W iθ ( z ) for all z ∈ B and ≤ i ≤ r .2. A vector v belongs to W iz if and only if (4) lim k →±∞ k log || A ( k ) ( z ) .v || = λ i . Recall that the matrices A ( k ) ( z ) preserve a Lorentz scalar product <, > ,n on R n +1 hence have determinant 1. It follows that the sum λ + . . . + λ r = 0 (see[ L , Prop. 1.1]). One infers easily from equation (4) that the Lyapunov spaces W iz are mutally orthogonal for <, > ,n , and spaces associated to nonzero exponents areisotropic (hence 1-dimensional). We thus see that only two cases may occur. Eitherthere is a single exponent and this exponent is 0, or there are exactly three exponents(if n ≥ λ + > , λ − = − λ + . In this last case, we say that the measure ν is partially hyperbolic . We then denote by W − z , W z and W + z the Lyapunov spacesassociated to λ − , λ + respectively.Let us proceed with the proof of the proposition, assuming that we have foundsuch a partially hyperbolic measure ν . Oseledec theorem above yields a point z =( w, y ) ∈ B , and linearly independent vectors u + ∈ W + z and u − ∈ W − z such that,lim k → + ∞ k log( || A ( k ) ( w, y ) .u − || ) = − λ + and lim k → + ∞ k log( || A ( − k ) ( w, y ) .u + || ) = − λ + . Let w = ( . . . w − w − w w w . . . ). We define g k = w k − . . . w and g ′ k = w − − k . . . w − − .Then: lim k → + ∞ k log || D y g k ( u − ) || = − λ + and lim k → + ∞ k log || D y g ′ k ( u + ) || = − λ + . In particular both sequences ( g k ) and ( g ′ k ) tend to infinity, and u − (resp. u + ) is alightlike asymptotically stable vector for ( g k ) (resp. for ( g ′ k )). It follows that thedirections [ u ± ] belong to the limit set Λ( y ), showing that this set has at least twopoints. In an open set U around y , we must have d Λ ≥ y ∈ O , it follows that for some point, hence any (byisometric invariance) point of O , d Λ ≥
2. This concludes the proof of Proposition 3.7in this case.It remains to show the existence of a θ -invariant, ergodic, partially hyperbolic mea-sure on Σ × O . This is here that our assumption of exponential growth for thederivatives of G comes into play. We fix an auxiliary Riemannian metric h on M , anddenote by U T M the unit tangent bundle of h . Exponential growth of derivatives at x yields λ >
0, a sequence ( g k ) in G such that ℓ ( g k ) → ∞ , as well as a sequence v k of U T x M , such that || D x g k ( v k ) || ≥ e λℓ ( g k ) . SOMETRY GROUP OF LORENTZ MANIFOLDS This growth condition can also be formulated using the sequence A ( k ) in the fol-lowing way. For each k , g k can be written as g k = s ( k ) ℓ ( g k ) − s ( k ) ℓ ( g k ) − . . . s ( k )1 s ( k )0 , which isa word of length ℓ ( g k ) in elements of S . Then, let us consider the bi-infinite, periodicword w ( k ) ∈ S Z which is defined by w ( k ) i = s ( k ) i for 0 ≤ i ≤ ℓ ( g k ) −
1, and completedby periodicity. The growth condition yields the existence of u k a sequence of vectorsin R n +1 , such that || u k || = 1 (we use here any norm) and || A ( ℓ ( g k )) ( w ( k ) , x ) .u k || ≥ e λℓ ( g k ) . We now lift the map θ to a map ˆ θ : Σ × O × R P n → Σ × O × R P n by the formulaˆ θ ( w, y, [ u ]) = ( θ ( w, y ) , A ( w, y ) . [ u ])Let us also introduce ψ : Σ × O × R P n → R defined by ψ ( w, y, [ u ])) =log( || A ( w, y ) .u || / || u || )Now look at the sequence of measures on Σ × O × R P n , defined by ˆ ν k = ℓ ( g k ) Σ ℓ ( g k ) m =1 (ˆ θ m ) ∗ δ ( w k , u k ).We compute R Σ × O × R P n ψd ˆ ν k = ℓ ( g k ) Σ ℓ ( g k ) m =1 ψ (ˆ θ m ( w ( k ) , x, [ u k ])). This expression isnothing but ℓ ( g k ) log || A ( ℓ ( g k )) ( w ( k ) , x ) .u k || , so that R Σ × O × R P n ψd ˆ ν k ≥ λ .Because the space Σ × O × R P n is compact, we can find a subsequence (ˆ ν i k )converging for the weak-star topology to a probability measure ˆ ν on Σ × O × R P n . It iseasily checked that ˆ ν is ˆ θ -invariant, and we still have R Σ × O × R P n ψd ˆ ν ≥ λ . Performingan ergodic decomposition, we get an ergodic, ˆ θ -invariant measure ˆ ν e on Σ × O × R P n satisfying(5) Z Σ × O × R P n ψd ˆ ν e ≥ λ. We push ˆ ν e forward to an ergodic, θ -invariant measure ν e on Σ × O . From (5) and[ L , Prop. 5.1], we conclude that the cocycle A ( k ) admits a Lyapunov exponent whichis ≥ λ (hence positive). It means precisely that ν e is partially hyperbolic, and theproof is complete.Proposition 3.7 uses only an hypothesis involving growth of derivatives. Whenthe group G itself has exponential growth, the conclusions of Proposition 3.7 can bestrengthen in the following way: Corollary 3.9 . —
Let ( M, g ) be a compact Lorentz manifold. Let G ⊂ Iso(
M, g ) bea closed, compactly generated subgroup. If G has exponential growth, then the limitset Λ( x ) is infinite for every x ∈ M . In particular d Λ ( x ) ≥ for every x ∈ M .Proof . — By Propositions 3.5 and 3.7, we now that card Λ ( x ) ≥ x ∈ M .We call c min ∈ { , , . . . } ∪ {∞} the minimal value achieved by x card Λ ( x ) on M .If the minimal value c min is + ∞ , we are done. If on the contrary c min is finite, weare going to get a contradiction. To see this, we define: K min := { x ∈ M | card Λ ( x ) = c min } . CHARLES FRANCES
By corollary 3.3, the set K ≥ c min +1 is open, hence K min is a compact subset of M .If c min = 2 then we have two lightlike Lipschitz directions on K , which are pre-served by G , or an index two subgroup of G . Because the subgroup of O(1 , n ) leavinginvariant two linearly independent lightlike directions is isomorphic to R × O( n − G -invariant (Lipschitz) reduction of the bundle ˆ M to the group R × O( n − K min . Lemma 5.11 then provides a coarse embedding α : G → R × O( n − G is assumed to have exponential growth,while R × O( n −
1) has linear growth. It means that we have + ∞ > c min ≥ , n ) leaving individually invariant a finite family of lightlikedirections spanning a subspace of dimension ≥
3, is a compact group isomorphic tosome group O( k ), 1 ≤ k ≤ n −
2. Again, looking at a finite index subgroup of G ,we have a G -invariant reduction of the bundle ˆ M to the group O( k ). Lemma 5.11then provides a coarse embedding α : G → O( k ). This is a new contradiction sinceno noncompact group can be coarsely embedded into a compact one.
4. Exponential growth and Killing fields
We have shown in Corollary 3.9 that the existence of a closed subgroup with expo-nential growth in the isometry group of a compact Lorentz manifold ( M n +1 , g ) forcesthe limit set to be infinite at each point. The aim of the present section is to derivethe first geometric consequences of this fact.Recall that a local Killing field on M is a vector field defined on some open subset U ⊂ M , such that the Lie derivative L X g = 0. In other words, the local flow of X acts isometrically for g . In the neighborhood of each point x ∈ M , the algebra oflocal Killing fields is a finite dimensional Lie algebra, that will be denoted kill loc ( x ).The isotropy algebra i x is the subalgebra of kill loc ( x ) comprising all local Killing fieldsvanishing at x .The main theorem of this section shows that the existence of a big limit set forIso( M, g ) produces many local Killing fields, at least on a nice open and dense subset M int , called the integrability locus, to be defined later on. Theorem 4.1 . —
Let ( M, g ) be a compact Lorentz manifold. Assume that the limitset Λ( x ) of Iso(
M, g ) is infinite for every x ∈ M . Then for every x in the integrabilitylocus M int , the isotropy Killing algebra i x is isomorphic to o (1 , k x ) , with k x ≥ . Observe that by Corollary 3.9, Theorem 4.1 will hold as soon as Iso(
M, g ) containsa closed, compactly generated subgroup of exponential growth.The Killing fields appearing in Theorem 4.1 will be obtained using integrabilityresults, which were first proved in [
Gr1 ], and that we present below.
In all what follows, we will denote by g the Lie algebra o (1 , n ) ⋉ R n +1 . We consider( M n +1 , g ) a ( n +1)-dimensional Lorentz manifold. Let π : ˆ M → M denote the bundleof orthonormal frames on M . This is a principal O(1 , n )-bundle over M , and it isclassical (see [ KN ][Chap. IV.2 ]) that the Levi-Civita connection associated to g can SOMETRY GROUP OF LORENTZ MANIFOLDS be interpreted as an Ehresmann connection α on ˆ M , namely a O(1 , n )-equivariant 1-form with values in the Lie algebra o (1 , n ). Let θ be the soldering form on ˆ M , namelythe R n +1 -valued 1-form on ˆ M , which to every ξ ∈ T ˆ x ˆ M associates the coordinates ofthe vector π ∗ ( ξ ) ∈ T x M in the frame ˆ x . The sum α + θ is a 1-form ω : T ˆ M → g called the canonical Cartan connection associated to ( M, g ).Observe that for every ˆ x ∈ ˆ M , ω ˆ x : T ˆ x ˆ M → g is an isomorphism of vector spaces,and the form ω is O(1 , n )-equivariant (where O(1 , n ) acts on g = o (1 , n ) ⋉ R n +1 viathe adjoint action).The notion of Riemannian curvature for g , as well as its higher order covariantderivatives have a counterpart in ˆ M . The curvature of the Cartan connection ω isa 2-form K on ˆ M , with values in g , defined as follows. If X and Y are two vectorfields on ˆ M , the curvature is given by the relation: K ( X, Y ) = dω ( X, Y ) + [ ω ( X ) , ω ( Y )] . Because at each point ˆ x of ˆ M , the Cartan connection ω establishes an isomorphismbetween T ˆ x ˆ M and g , it follows that any k -differential form on ˆ M , with values insome vector space W , can be seen as a map from ˆ M to Hom( ⊗ k g , W ). This remarkapplies for the curvature form K itself, yielding a curvature map κ : ˆ M → W , wherethe vector space W is a sub O(1 , n )-module of Hom( ∧ ( R n +1 ); g ) (the curvature isantisymmetric and vanishes when one of its arguments is tangent to the fibers of ˆ M ).We now differentiate the map κ , getting a map Dκ : T ˆ M → W . The connection ω allows to identify Dκ with a map D κ : ˆ M → W , where W = Hom( g , W ). the r th-derivative of the curvature D r κ : ˆ M → Hom( g , W r ) (with W r defined inductively by W r = Hom( g , W r − )). The generalized curvature map of our Lorentz manifold ( M, g )is the map κ g = ( κ, D κ, D κ, . . . , D dim( g ) κ ). The O(1 , n )-module Hom( g , W dim( g ) )will be rather denoted W κ g in the sequel. . — One defines the integrability locus of ˆ M , denoted ˆ M int ,as the set of points ˆ x ∈ ˆ M at which the rank of Dκ g is locally constant. Notice thatˆ M int is a O(1 , n )-invariant open subset of ˆ M . Because the rank of a smooth mapcan only increase locally, this open subset is dense. We define also M int ⊂ M , theintegrability locus of M , as the projection of ˆ M int on M . This is a dense open subsetof M . . — Local flows of isometries on M clearly inducelocal flows on the bundle of orthonormal frames, which moreover preserve ω . It followsthat any local Killing field X on U ⊂ M lifts to a vector field ˆ X on ˆ U := π − ( M ),satisfying L ˆ X ω = 0. Conversely, local vector fields of ˆ M such that L ˆ X ω = 0, thatwe will henceforth call ω -Killing fields , commute with the right O(1 , n )-action on ˆ M .Hence, they induce local vector fields X on M , which are Killing because their localflow maps orthonormal frames to orthonormal frames. It is easily checked that a ω -vector field which is everywhere tangent to the fibers of the bundle ˆ M → M must CHARLES FRANCES be trivial. As a consequence, there is a one-to-one correspondence between local ω -Killing fields on ˆ M and local Killing fields on M . We will use this correspondence allalong the paper. The same remark holds for local isometries.Observe finally that if ˆ X is a ω -Killing field on ˆ M (namely L ˆ X ω = 0), then thelocal flow of ˆ X preserves κ g , hence ˆ X belongs to Ker( D ˆ x κ g ) at each point. Theintegrability theorem below says that the converse is true on the set ˆ M int . Theorem 4.2 (Integrability theorem) . —
Let ( M, g ) be a Lorentz manifold. Let M int ⊂ M denote the integrability locus. For every ˆ x ∈ ˆ M int , and every ξ ∈ Ker( D ˆ x κ g ) , there exists a local ω -Killing field ˆ X around ˆ x such that ˆ X (ˆ x ) = ξ . An akin integrability result for Killing fields of finite order first appeared in theseminal paper [
Gr1 ]. The results were recast in the framework of real analytic Car-tan geometry in [ M2 ], and [ P ] provides an alterative approach for smooth Cartangeometries, leading to the statement of Theorem 4.2 (see also Annex A of [ Fr2 ], whichelaborates slightly on the statement proved in [ P ]). M int , and “analytic continuation” of Killing fields .— A first important consequence of Theorem 4.2 is that on the set M int , Killingfields have a particularly nice behavior. To see that, let us recall that for any x ∈ M ,there is a good notion of local Killing algebra at x . Indeed, there exists U a smallenough neighborhood of x , such that for every neighborhood V ⊂ U containing x ,any Killing field on V will be the restriction of a Killing field of U . We then call kill loc ( x ) the (abstract) Lie algebra kill ( U ) of all Killing fields defined on U . Theorem4.2 shows that if M is a connected component of M int , then the dimension of kill loc ( x )does not depend of x ∈ M , because this dimension is just the corank of κ g on ˆ M .As a consequence, the local Killing fields on M behave much like Killing fields of areal analytic metric. In particular, given a Killing field X defined on some open set U ⊂ M , and given a path γ starting at a point of U , one can perform the “analyticcontinuation” of X along γ . It follows that if U is a 1-connected open subet of M ,and if X is a Killing field defined on V ⊂ U , then there exists a Killing field definedon U , whose restriction to V is X . Those nice properties will often be used implicitelyin the sequel. . — Another corol-lary of Theorem 4.2, which will be of particular interest to prove Theorem 4.1, is thefollowing: Corollary 4.3 . —
For every point x ∈ M int , the isotropy algebra i x is isomorphicto the Lie algebra s x of the stabilizer of κ g (ˆ x ) in O(1 , n ) , for any ˆ x ∈ ˆ M in the fiberof x .Proof . — Every element X ∈ i x defines a local Killing field in a neighborhood of x , that can be lifted to a ω -Killing field ˆ X on a neighborhood of ˆ x . Observe thatˆ X is tangent to the fiber of ˆ x since X vanishes at x . The map ρ : X ω ˆ x ( ˆ X (ˆ x ))yields a Lie algebra morphism from i x to s x . The map is injective since two local ω -Killing fields coinciding at ˆ x must coincide on an open subset around ˆ x . The map SOMETRY GROUP OF LORENTZ MANIFOLDS is onto, because if Y ∈ s x , and if ξ = ω − x ( Y ), then ξ ∈ Ker( D ˆ x κ g ). Theorem 4.2 thenprovides a local Killing field around x , such that ˆ X (ˆ x )) = ξ . In particular X ( x ) = 0so that X ∈ i x . We can now proceedto the proof of Theorem 4.1. In light of Corollary 4.3, it is enough to show that forevery ˆ x ∈ ˆ M int , the stabilizer of κ g (ˆ x ) in O(1 , n ) contains a subgroup isomorphic toSO o (1 , k ), for k ≥ σ : M → ˆ M , thanks to which we built the derivativecocycle D : M × Iso(
M, g ) → O(1 , n ), with D ( x, f ) = D x f (see Section 2.2.2). We fix x ∈ M in the sequel. Because σ ( M ) is included in a compact subset of ˆ M , we havethat κ g ( σ ( M )) is also contained in a compact subset K of W κ g . Let f ∈ Iso(
M, g ).By the definition of D x f , one has the relation f ( σ ( x )) . ( D x f ) − = σ ( f ( x )) . This yields κ g ( σ ( f ( x ))) = D x f.κ g ( σ ( x )) , and we infer that D x f.κ g ( σ ( x )) belongs to the compact set K for every f ∈ Iso(
M, g ).This leads to the following general notion of stability. Let n ≥
2, and ρ : O(1 , n ) → GL( V ) be a finite dimensional representation. If G is a subset of O(1 , n ), and v ∈ V is a vector, we say that v is stable under G , if ρ ( G ) .v is a bounded subset of V . Theprevious discussion shows that κ g ( σ ( x )) ∈ W κ g is stable under the set D x (Iso( M, g )).In full generality, we wonder if a vector v ∈ V is stable under a set G ⊂ O(1 , n )having a big limit set Λ G in ∂ H n (see Section 3.3), then v is fixed by a big subgroupof O(1 , n ).To make things a little bit precise, we see ∂ H n as the set of lightlike directions in R ,n , and we introduce the linear hull of the limit set Λ G , denoted E Λ G , as the linearspan of Λ G in R ,n . The dimension of E Λ G will be denoted d Λ G . We can now state: Proposition 4.4 (Big limit set implies big stabilizer)
Let ρ : O(1 , n ) → GL( V ) be a finite dimensional representation. Let G ⊂ O(1 , n ) such that the limit set Λ G ⊂ ∂ H n satisfies the property d Λ G ≥ . Then for every vector v ∈ V which is stable under G , the stabilizer of v in O(1 , n ) contains a subgroupisomorphic to SO o (1 , d Λ G − . If we take this proposition for granted, then Theorem 4.1 follows easily. Indeed,the asumption of Theorem 4.1 implies that d Λ ( x ) ≥
3. Proposition 4.4, applied forthe representation ρ : O(1 , n ) → GL( W κ g ), and G = D x (Iso( M, g )) then says that thestabilizer of κ g ( σ ( x )) in O(1 , n ) contains a subgroup isomorphic to SO o (1 , d Λ ( x ) − The remaining of this section is devoted to theproof of Proposition 4.4. CHARLES FRANCES . — By hypothesis, n ≥
2, hence o (1 , n ) is simple. The representation ρ : O(1 , n ) → GL( V ) is thus a direct sumof irreducible representations. It is enough to prove proposition 4.4 for irreduciblerepresentations ρ . To avoid cumbersome notations, we will denote in the following g.v instead of ρ ( g ) .v (the image of the vector v ∈ V under the linear transformation ρ ( g )).Let us consider a subset G ⊂ O(1 , n ), and recall the notion of limit set λ G intro-duced in Section 3.3. If K ⊂ O(1 , n ) is a compact subset, and if k : G → K , we candefine a new set G ′ := { k ( g ) g | g ∈ G } . If for some point ν ∈ H n , a sequence ( g k ) ofO(1 , n ) satisfies g − k .ν → p , where p ∈ ∂ H n , then for any compact subset C ⊂ H n , wehave g − k .C → p (the limit has to be understood for the Hausdorf distance betweencompact subsets of H n ). Thus it is clear that G and G ′ have the same limit set,and if ρ : O(1 , n ) → GL( V ) is a finite dimensional representation, then stable vectorsfor G and G ′ coincide. It follows from Iwasawa decomposition O(1 , n ) = KAN (seethe proof of Proposition 3.5) that we may assume, to prove Proposition 4.4, that G ⊂ AN . We will do this in the sequel.We recall that in the upper-half space model H n = R ∗ + × R n − , elements of A act as homothetic transformations a s : ( t, x ) ( e s t, e s x ), s ∈ R . The group N isabelian, isomorphic to R n − and it acts as n ( v ) : ( t, x ) ( t, x + v ), v ∈ R n − . Thedynamics of a s on H n has two distinct fixed points p + and p − on ∂ H n , and for every p ∈ H n , lim s → + ∞ a s .p = p + (resp. lim s →−∞ a s .p = p − ). The group N fixes p + andacts simply transitively on ∂ H n \ { p + } .We can now formulate the following dynamical lemma. Lemma 4.5 . —
Let us consider an unbounded sequence g k = a s k n k of AN ⊂ O(1 , n ) , and let us pick a point o ∈ H n . After considering maybe a subsequence, weare in one of the following four cases.1. The sequence ( s k ) converges to −∞ . Then g − k .o → p + .2. The sequence ( s k ) converges in R and n k → ∞ in N . Then g − k .o → p + .3. The sequence ( s k ) tends to + ∞ and n k → ∞ in N . Then g − k .o tends to p + .4. The sequence ( s k ) tends to + ∞ and ( n k ) tends to n ∞ ∈ N . Then g − k .o tendsto p = n − ∞ .p − = p + .4.4.2. Proof of Proposition 4.4 for n = 2. — We first do the proof of Proposition 4.4for a representation ρ : O(1 , → GL( V ). The Lie algebras o (1 ,
2) and sl (2 , R ) areisomorphic, and there is a 2-fold covering π : SL(2 , R ) → SO o (1 , ρ ′ : SL(2 , R ) → GL( V ) such that ρ ′ = ρ ◦ π . If V is m -dimensional, thenirreducible finite representations of SL(2 , R ) occur from the natural action of SL(2 , R )on homogeneous polynomials of degree m − m = 2 l + 1 mustbe odd for ρ ′ to induce a representation of SO o (1 , H = (cid:18) − (cid:19) , E = (cid:18) (cid:19) . We assume in the following that l ≥ ρ : sl (2 , R ) → gl ( V ) is the induced representation, then there isa suitable basis e , . . . , e l +1 of V where SOMETRY GROUP OF LORENTZ MANIFOLDS ρ ( H ) = l l − − l + 2 − l , ρ ( E ) = l . After identifying sl (2 , R ) and o (1 ,
2) under the adjoint action, we see that if n t = e tE , then ρ ( n t ) = a t a t . . . a m t m − a t . . . a m t m − a m − ,m t , for positive coefficients a i,j where 1 ≤ i ≤ j ≤ m = 2 l + 1. We also see thatthe vectors v ∈ V which are stable under a s := e sH , s ≥
0, are vectors in V − =Span( e l +1 , . . . , e l +1 ).We are now ready to prove the following lemma, which is a reformulation of Propo-sition 4.4 for n = 2. Lemma 4.6 . —
Let ρ : O(1 , → GL( V ) be a finite dimensional representation.Let o ∈ H be a point, and let ( g k ) , ( g ′ k ) and ( g ′′ k ) three sequences in O(1 , . Assumethat there exists three pairwise distinct points p, p ′ , p ′′ in ∂ H such that g − k .o → p , ( g ′ k ) − .o → q and ( g ′′ k ) − .o → r . Then any vector v ∈ V which is stable for ( g k ) , ( g ′ k ) and ( g ′′ k ) is actually SO o (1 , -invariant.Proof . — We first assume that ρ is irreducible, and V is not 1-dimensional. Weobserve that the conclusions of the Lemma are unaffected if we conjugate the threesequences ( g k ), ( g ′ k ) and ( g ′′ k ) by an element h ∈ O(1 ,
2) and replace p, p ′ , p ′′ by h.p, h.p ′ , h.p ′′ . Therefore, because the action of O(1 ,
2) is transitive on triplets in ∂ H , we may assume that p = p − , p ′ = p + and p ′′ ∈ ∂ H is different from p + and p − .As explained in Section 4.4.1, after having performed such a conjugacy, we still mayleft-multiply our sequences by sequences with values in a maximal compact group K ⊂ O(1 ,
2) without affecting the conclusions. Hence we will also assume that ( g k ),( g ′ k ) and ( g ′′ k ) are sequences in AN ⊂ SO o (1 , g k = a s k n t k , g ′ k = a s ′ k n t ′ k g ′′ k = a s ′′ k n t ′′ k .Our first assumption is that g − k .o → p + . We have seen in Lemma 4.5 that it canhappen in three different ways.1. First case: The sequence ( s k ) is bounded in R (and | t k | → ∞ ). Then, v isstable under e s k H e t k E if and only if it is stable under e t k E . This only occurs if v ∈ R .e . CHARLES FRANCES
2. Second case: The sequence ( s k ) tends to −∞ . Then if v is stable under e s k H e t k E ,the coordinates of e t k E .v along Vect( e l +2 , . . . , e l +1 ) must tend to 0. In particu-lar, v l +1 = 0. Then v l + a l, l +1 t k v l +1 → k → ∞ , hence v l = 0. We pro-ceed in the same way to get v l − = . . . = v l +2 = 0, and v ∈ Vect( e , . . . , e l +1 ).3. Third case: The sequence ( s k ) tends to + ∞ . Then g − k .o → p + means that | t k | → + ∞ . If the vector v is stable under e s k H e t k E , then necessarily, thecoordinates of n t k .v along Vect( e , . . . , e l ) tend to 0. Looking at the coordinatealong e we get that v + a t k v + . . . + a , l +1 t lk v l +1 → . because the coefficient a ij are positive, this only occurs when v = . . . , v l +1 = 0,namely v = 0.We now use our hypothesis that ( g ′ k ) − .o → p − . By Lemma 4.5, this happensexactly when s ′ k → + ∞ and t ′ k →
0. The vector v is stable under a s ′ k n t ′ k only when v belongs to V − = Vect( e l +1 , . . . , e l +1 ). Together with the conclusions of the threepossible cases above, we end up with v ∈ R .e l +1 .We write v = λe l +1 , and finally use our third assumption: ( g ′′ k ) − .o → p ′′ , with p ′′ = p + , p ′′ = p − . This assumption is equivalent to s ′′ k → + ∞ and t ′′ k → t ∞ .Observe that t ∞ = 0 since p ′′ = p − . Again, v can be stable under a s ′′ k n t ′′ k only whenthe coordinates of n t ∞ .v along the vectors e , . . . , e l are zero. But the coordinate of n t ∞ .e l +1 along e is a ,l +1 t l ∞ , which is nonzero since a ,l +1 > t ∞ = 0. Thisthird stability condition thus forces λ = 0, namely v = 0.To conclude te proof, we now write the respresentation ρ as a direct sum of ir-reducible representations. By what we showed above, a stable vector v has onlynonzero components on irreducible factors of dimension 1. Dimension 1 representa-tions of SO o (1 ,
2) being trivial, we infer that v is SO o (1 , Corollary 4.7 . —
Let ρ : O(1 , → GL( V ) be a finite dimensional representation.Let V − ⊂ V be the stable subspace of { a s } s ≥ . Let n = n t , n = n t and n = n t be three elements of N , that we assume to be pairwise distinct. Then any vector v ∈ n .V − ∩ n .V − ∩ n .V − is fixed by SO o (1 , .Proof . — Let us consider the three sequences g k := a k n − , g ′ k := a k n − and g ′′ k = a k n − . Any vector v belonging to n .V − ∩ n .V − ∩ n .V − is stable under ( g k ) , ( g ′ k )and ( g ′′ k ). On the other hand g − k .o = n a − k .o hence g − k .o → n .p − . In the same way,( g ′ k ) − .o → n .p − and ( g ′′ k ) − .o → n .p − . Because n , n , n are pairwise distinct, thepoints n .p − , n .p − and n .p − are pairwise distinct too. Lemma 4.6 ensures that v isfixed by SO o (1 , . — Let n ≥
2, and let ρ : O(1 , n ) → GL( V ) be a finite dimensional representation. We consider G ⊂ O(1 , n ). The sim-plifications mentioned in Section 4.4.1 are still in force. In particular we still assumethat G ⊂ AN . Let us recall that we identify ∂ H n with the set of lightlike directions in SOMETRY GROUP OF LORENTZ MANIFOLDS R ,n . The linear hull of a set S ⊂ ∂ H n is then the linear span of S in R ,n . If E is thislinear hull, then E has Lorentz signature as soon as S contains at least two points.We then denote by SO o ( E ) the subgroup of SO o (1 , n ) which leaves E invariant andacts trivially on E ⊥ . Lemma 4.8 . —
Let p , q and r be three pairwise distinct points in Λ G , and let E ⊂ R ,n be their linear hull. Then any vector v ∈ V which is stable for G is fixed by SO o ( E ) .Proof . — As already seen, we may conjugate G into O(1 , n ) to prove the lemma.Hence, we may assume that ∆ = ∆ and p, q, r different from p + . Our assump-tions imply the existence of ( g k ) , ( g ′ k ) and ( g ′′ k ) such that g − k .o → p , ( g ′ k ) − .o → q ,( g ′′ k ) − .o → r . We write g k = a s k n k , g ′ k = a s ′ k n ′ k and g ′′ k = a s ′′ k n ′′ k . We must bein case 3 of Lemma 4.5, namely the three sequences s k , s ′ k , s ′′ k tend to + ∞ and n k , n ′ k , n ′′ k tend to ν, ν ′ , ν ′′ respectively, with p = ν.p − , q = ν ′ .p − and r = ν ′′ .p − .Actually, because we assume that p, q, r belong to S ∆ , we get that ν, ν ′ and ν ′′ be-long to N ∩ S ∆ . Stability of v under ( g k ) , ( g ′ k ) and ( g ′′ k ) implies that v belongs to ν − .V − ∩ ( ν ′ ) − .V − ∩ ( ν ′′ ) − .V − . By Corollary 4.7, v must be fixed by SO o ( E ).We are now in position to prove Proposition 4.4 by induction on the integer d G .Lemma 4.8 settles the case d G = 3. We assume that we proved Proposition 4.4 forall subsets G ′ ⊂ O(1 , n ) satisfying 3 ≤ d G ′ ≤ d G −
1. We pick p , . . . , p d G +1 pairwisedistinct points in Λ G . For each index 1 ≤ i ≤ d G + 1, there exists by assumption asequence ( g ( i ) k ) in G , such that given o ∈ H n , we have ( g ( i ) k ) − .o → p i . The linear hullof { p , . . . , p d G } is a subspace E ⊂ E Λ G , and the linear hull of { p d G − , p d G , p d G +1 } is a subspace F ⊂ E Λ G . We can apply our induction hypothesis to the subset G ′ comprising all elements g ( i ) k for 1 ≤ i ≤ d G and k ∈ N , thus obtaining that the vector v , which is obviously stable by G ′ is fixed by SO o ( E ). We also apply the inductionhypothesis to the set G ′′ comprising the elements g ( d G − k , g ( d G ) k , g ( d G +1) k , k ∈ N , andwe get that v is fixed by SO o ( F ). Since it is readily checked that SO o ( F ) and SO o ( E )generate SO o ( E Λ G ), we have finally proved that v is fixed by SO o ( E Λ G ), which isindeed isomorphic to SO o (1 , d G −
5. Invariant locally homogeneous Lorentz submanifolds
We say that a Lorentz manifold ( M n +1 , g ) is locally homogeneous , when M consistsin a single kill loc -orbit (see Section 5.1.1 below for the definition). It is plain that forsuch manifolds, M int = M . Also, if x and y are two points of M , the isotropyalgebras i x and i y are conjugated in o (1 , n ), because there exists a local isometry f : U → V such that f ( x ) = y . We say that a locally homogeneous Lorentz manifold( M n +1 , g ) has semisimple isotropy , if for some (hence any) x ∈ M , the isotropy algebra i x contains a subalgebra isomorphic to o (1 , d ), for d ≥
2. It actually follows fromTheorem 4.1, that if ( M n +1 , g ) is a compact locally homogenous Lorentz manifold,and if Iso( M, g ) contains a compactly generated, closed subgroup with exponentialgrowth, then ( M n +1 , g ) must have semisimple isotropy. The aim of this section is CHARLES FRANCES to explain that when proving Theorem A, we may actually make this hypothesis oflocal homogeneity without losing any generality. This is the content of the followingtheorem.
Theorem 5.1 . —
Let ( M, g ) be a compact Lorentz manifold. Assume that Iso(
M, g ) contains a closed, compactly generated subgroup G having exponential growth. Thenthere exists a finite index subgroup Iso ′ ( M, g ) ⊂ Iso(
M, g ) and a compact Lorentzsubmanifold Σ ⊂ M , which is locally homogeneous with semisimple isotropy, preservedby Iso ′ ( M, g ) , and such that the restriction morphism Iso ′ ( M, g ) → Iso(Σ , g | Σ ) is one-to-one and proper. Is loc -orbits. — We have seen inSection 4.2.1 the existence of a dense open subset M int ⊂ M , the integrability locusof M , where the behavior of the Killing fields is aspecially nice (see also Section 4.2.3).We are now going to see that there exists a “regular” subset M reg ⊂ M int , which isstill open and dense, where the behavior of the kill loc -orbit is also nice. This propertywas first observed by M. Gromov in [ Gr1 ]. kill loc and Is loc -orbits . — Let us recall that if x ∈ M , the kill loc -orbit of x isthe set of points y ∈ M that can be reached from x by flowing along (finitely many)successive local Killing fields. We will be also interested by the Is loc -orbit of x , namelythe set of points y ∈ M for which there exists a local isometry f : U ⊂ M → V ⊂ M such that y = f ( x ).We will also, for ˆ x ∈ ˆ M , define the kill loc -orbit of ˆ x as the set of points ˆ y ∈ ˆ M thatcan be reached by flowing along (finitely many) successive local ω -Killing fields. It ispretty clear from the discution at the begining of Section 4.2.2 that kill loc -orbits in M are exactly the projections of kill loc -orbits on ˆ M . Is loc -orbits in the regular locus . — Theorem 4.2, together withthe nice structure for orbits of algebraic group actions, ensures that if we restrict ourattention to a smaller (but still dense) open subset of M int , that will be called theregular locus of M , kill loc -orbits define a simple foliation. By a simple foliation, wemean a foliation, such that each point is contained in a transversal meeting each leafat most once. Those nice properties of kill loc -orbits were first proved in Gromov’spaper [ Gr1 ]. Theorem 5.2 (Structure of Is loc -orbits, see [ Gr1 ] ) . — Let ( M, g ) be a Lorentzmanifold. There exists a dense open set M reg called the regular locus of M , satisfying M reg ⊂ M int ⊂ M , and having the following properties:1. The set M reg is Iso(
M, g ) -invariant and saturated in kill loc -orbits.2. There exists a smooth manifold Y , as well as a smooth map κ g : M reg → Y withlocally constant rank, such that kill loc -orbits of M reg coincide with the connectedcomponents of the fibers of κ g .3. The kill loc -orbits and Is loc -orbits of M reg are closed in M reg , and kill loc -orbitsdefine a simple foliation on any connected component of M reg . SOMETRY GROUP OF LORENTZ MANIFOLDS Proof . — For the reader’s convenience, we recall the proof of the theorem. The mainingredient is a classical result of M. Rosenlicht, about orbits of algebraic group actions(see [ R1 ]). Theorem 5.3 . —
Let H be an R -algebraic group, acting on a real algebraic variety X . Then there exists a stratification into H -invariant Zariski closed sets F ( F ( . . . ( F m = X , and for each ≤ i ≤ m , a variety Y i and a regular map ψ i : F i \ F i − → Y i , such that the fibers of ψ i coincide with the H -orbits on F i \ F i − . In the statement we put F − = ∅ . The field R ( X ) H of H -invariant rationnalfunctions is generated by a finite number of function f , . . . , f r . The theorem followsfrom the fact that outside a Zariski closed set F (the set F m − in the statementabove), f , . . . , f r separate the orbits, so that we can put Y = Y m − = R r , and ψ = ψ m − = ( f , . . . , f r ). One then applies the same result to F m − and so on.For every 0 ≤ i ≤ m , we call ˆΩ i the interior of the set ( κ g ) − ( F i \ F i − ). One checksthat S mi =0 ˆΩ i is open and dense in ˆ M , and thus so is ˆΩ := ˆ M int ∩ (cid:16)S mi =0 ˆΩ i (cid:17) . Let uscall Y = F mi =0 Y i , and define ˆ κ g : ˆΩ → Y , by ˆ κ g = ψ i ◦ κ g on ˆΩ i ∩ M int . Since ˆ κ g isO(1 , n )-invariant, it induces a smooth map κ g : Ω → Y , where Ω is the projection ofˆΩ on M . We consider the open set M reg ⊂ Ω ⊂ M int where the rank of κ g is locallyconstant. We thus obtain a dense open set called the regular locus of M .The theorem is now a direct consequence of Theorems 4.2 and 5.3. Invariance of κ g by local isometries show that M reg is Is loc -invariant and saturated by kill loc -orbits.From Theorem 4.2, we infer that if ˆ x ∈ ˆ M int , the kill loc -orbit of ˆ x coincides with theconnected component of ( κ g ) − ( w ) ∩ ˆ M int containing ˆ x . It follows that if x ∈ M reg ,the kill loc -orbit of x coincides with the connected component of the fiber ( κ g ) − ( κ g ( x ))containing x .Because κ g has locally constant rank on M reg , the fibers of κ g are submanifolds of M reg , and are closed in M reg . The same property holds for kill loc -orbits (resp. theIs loc -orbits) which are connected components (resp. unions of connected components)of those submanifolds. kill loc -orbits. — We haveseen in Corollary 3.9 that the presence of a subgroup of exponential growth inIso(
M, g ) forces the limit set Λ( x ) to be infinite for every x ∈ M . We are nowgoing to see that such a property forces the existence of compact kill loc -orbits. Proposition 5.4 . —
Let ( M n +1 , g ) be a compact Lorentz manifold. Assume thatthe limit set Λ( x ) is infinite at each x ∈ M . Then there exists a compact, Lorentz kill loc -orbit Σ contained in M reg . The proof will show that there are actually infinitely such Lorentz compact kill loc -orbits. CHARLES FRANCES
Proof . — Because of our asumption card Λ ( x ) = ∞ , Theorem 4.1 shows that for every x ∈ M int , the isotropy algebra i x contains a subalgebra isomorphic to o (1 , k ), for some k ≥
2. We thus infer:
Lemma 5.5 . —
Under the asumption that card Λ ( x ) = ∞ for all x ∈ M , then every kiloc -orbit Σ contained in M int has Lorentz signature.Proof . — Let x ∈ M int , and let Σ be the kill loc -orbit of x . Since i x contains a copy ofthe Lie algebra o (1 , k ), for k ≥
2, there is a local Killing field X around x , vanishingat x , and such that the flow { D x φ tX } ⊂ O( T x M ) is a hyperbolic 1-parameter group.Linearizing X around x thanks to the exponential map, we see there are two distinctlightlike 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 tX . In particular, for s = 0 close to0, ˙ γ u ( s ) and ˙ γ v ( s ) are colinear to X , hence tangent to the kill loc -orbits O ( γ u ( s )) and O ( γ v ( s )) respectively. By continuity, this property must still hold for s = 0. We inferthat T x (Σ) contains the two distinct lightlike directions u and v , hence has Lorentzsignature. This holds on all of Σ by local homogeneity of the kill loc -orbit.We now consider M a connected component of M reg where the rank of the map κ g is the maximal value r max that the rank of κ g does achieve on ˆ M . We considerthe pullback ˆ M ⊂ ˆ M .We pick x ∈ M , and we choose ˆ x ∈ ˆ M in the fiber of x . If U ⊂ ˆ M is a smallopen set around ˆ x , κ g ( U ) is a r max -dimensional submanifold of W κ g . Let us now callˆΛ the closed subset of ˆ M where the rank of κ g is ≤ r max −
1. By Sard’s theorem,the r max -dimensional Hausdorff measure of κ g (ˆΛ) is zero. We infer the existenceof w ∈ κ g ( U ) \ κ g (ˆΛ). Moving ˆ x inside U , we assume that w = κ g (ˆ x ), and wedenote by O ( w ) the O(1 , n )-orbit of w in W . By O(1 , n )-equivariance of κ g , theinverse image ( κ g ) − ( O ( w )) avoids ˆΛ, hence the rank of κ g is constant equal to r max on ( κ g ) − ( O ( w )). Let us observe that because the rank can not drop locally, anypoint of ˆ M where the rank of κ g is r max must belong to ˆ M int , hence the inclusion( κ g ) − ( O ( w )) ⊂ ˆ M int . From the discussion following Theorem 5.2, the connectedcomponents of ( κ g ) − ( w ) are kill loc -orbits. Because ˆ M reg is saturated in kill loc -orbits,and is O(1 , n )-invariant, we infer that ( κ g ) − ( O ( w )) ⊂ ˆ M reg . By Theorem 5.2, theprojection of ( κ g ) − ( O ( w )) on M is a submanifold, whose connected components are kill loc -orbits. Hence proposition 5.4 will be proved if we show that ( κ g ) − ( O ( w )) isclosed in ˆ M (because O(1 , n )-invariant closed sets of ˆ M project on compact subsetsof M ). This is a direct consequence of: Lemma 5.6 . —
Under the hypothesis card Λ ( x ) = ∞ , the orbit O ( w ) is a closedsubset of W κ g .Proof . — As already observed the isotropy algebra i x contains a subalgebra isomor-phic to o (1 , k ), for some k ≥
2. Recall that i x is identified with the Lie algebraof the stabilizer of w in O(1 , n ) (see Corollary 4.3). Since it contains a copy of SOMETRY GROUP OF LORENTZ MANIFOLDS o (1 , k ), i x contains a conjugate of the Cartan algebra a ⊂ o (1 , n ). Hence the Sta-bilizer of w in O(1 , n ) contains a conjugate of the Cartan group A (the exponenti-ation of a ), that we may assume to be A itself, after moving ˆ x in its fiber if neces-sary. Let now g k .w be a sequence in O ( w ) that converges to some point w ′ ∈ W κ g .We write O(1 , n ) = KAN (Iwasawa decomposition), and decompose g k accordingly: g k = m k a k u k , with m k ∈ K , a k ∈ A , and u k ∈ N . We may assume that m k → m ∞ .Because w is fixed by a k , we observe that g k .w = m k a k u k a − k .w = m k u ′ k .w , for somesequence u ′ k ∈ N (the group N is normalized by A ). It follows that u ′ k .w converges to m − ∞ .w ′ . But the orbit N.w is closed in W κ g , because in a linear representation, theorbits of unipotent groups are closed (this is a general property for algebraic actionsof unipotent groups, see [ R2 , Theorem 2]). We finally get m − ∞ .w ′ ∈ O ( w ), hence w ′ ∈ O ( w ). kill loc -orbits. — Wenow use the nice structure of kill loc -orbits on M reg , to prove the following embeddingproperty: Theorem 5.7 . —
Let ( M n +1 , g ) be a compact Lorentz manifold of dimension n + 1 ,and let M reg be its regular locus. Assume that Σ ⊂ M reg is a compact kill loc -orbit, ofLorentz signature. Then there exists a finite index subgroup Iso ′ ( M, g ) ⊂ Iso(
M, g ) ,such that Σ is left invariant by Iso ′ ( M, g ) , and such that the restriction morphism Iso ′ ( M, g ) → Iso(Σ , g | Σ ) is one-to-one and proper. The proof will be made in several steps. The first one is to exhibit distinguishedneighborhoods for compact Lorentz kill loc -orbits. kill loc -orbits . — When there exists a com-pact kill loc -orbit in M reg having Lorentz signature, the nice structure described inTheorem 5.2 can be strengthen in the following way. Lemma 5.8 (Existence of standard foliated neighborhoods)
Let Σ ⊂ M reg be a compact kill loc -orbit. Then Σ admits a connected open neigh-borhood U ⊂ M reg satisfying the following properties:1. The closure U is saturated in kill loc -orbits, all of which are compact Lorentzsubmanifolds.2. There exists a total transversal B to the kill loc -orbits of U , which is a submanifoldwith boundary diffeomorphic to a closed d -ball.3. Every Is loc -orbit of M intersects B at most once. In particular, every kill loc -orbits of U intersects B exactly once.Proof . — Let Σ ⊂ M ⊂ M reg be a compact kill loc -orbit having Lorentz signature.We fix x ∈ Σ a reference point. We introduce, for x ∈ Σ the following notations: T x Σ ⊥ := { u ∈ T x M | u ⊥ T x Σ } ,T x Σ ⊥ := { u ∈ T x Σ ⊥ | g ( u, u ) = 1 } , CHARLES FRANCES and for ǫ > B ǫ := { exp( x , su ) | s ∈ [0 , ǫ ) , u ∈ T x Σ ⊥ } ,B ǫ := { exp( x , su ) | s ∈ [0 , ǫ ] , u ∈ T x Σ ⊥ } . We recall that on M , the smooth map κ g : M → Y has constant rank, hencedefines a foliation F on M . Each leaf of this foliation is a kill loc -orbit (see Section5.1.2). Moreover the Is loc -orbits, intersected with M , are contained in the fibers of κ g | M .For ǫ > B ǫ is a transversal to F . Observe that (taking ǫ > g restricts to a Riemannian metric on B ǫ . Indeed, Σ isLorentzian by assumption, hence the spaces T x Σ ⊥ are spacelike. A direct applicationof the inverse mapping theorem shows that if ǫ is chosen small enough, κ g realizesa smooth diffeomorphism from B ǫ onto a submanifold Y ′ ⊂ Y . It follows that thefibers of κ g intersect B ǫ at most once. We call V the saturation of B ǫ by leaves of F . We then have a natural smooth surjective submersion π : V → B ǫ .For 0 < ǫ < ǫ small enough, we define V ǫ := { exp( x, su ) | x ∈ Σ , u ∈ T x Σ ⊥ , s ∈ [0 , ǫ ] } . If ǫ is small enough, V ǫ is a compact neighborhood of Σ contained in V .We finally call U the saturation of B ǫ by kill loc -orbits. We claim that U satisfies theproperties of Lemma 5.8. Actually the only point which is still unclear and that wemust prove is that U is compact in V , and saturated by kill loc -orbits. The compactnessof all kill loc -orbits in U will follow because kill loc -orbits are closed in V . Let us call U ′ the closure of U in V , which is nothing but the saturation of B ǫ by kill loc -orbits.Let us pick x ∈ U ′ ∩ V ǫ , and let X be a Killing field defined in a neighborhoodof x . Because x ∈ V ǫ , there exists z ∈ Σ, u ∈ T z Σ ⊥ and a ∈ [0 , ǫ ] such that x = exp( z, au ). Let us call, for every s ∈ [0 , γ ( s ) = exp( z, asu ). This is a geodesicsegment, homeomorphic to the closed unit interval (this is so if ǫ is small enough).We can consider a 1-connected open neighborhood W ⊂ V containing γ , and extend X to a Killing field on W (see Section 4.2.3). For t ∈ ( − δ, δ ) with δ > ϕ tX ( γ )is well defined. It is a geodesic segment joining z ( t ) := ϕ tX ( z ) to x ( t ) := ϕ tX ( x ),orthogonal to Σ at z ( t )), and of positive Lorentz length a . Hence one can write x ( t ) = exp( z ( t ) , au ( t )), where u ( t ) ∈ T x ( t ) Σ ⊥ . This means x ( t ) ∈ U ′ ∩ V ǫ for all t ∈ ( − δ, δ ). What we have proved is that in any kill loc -orbit of U ′ , the set of pointsbelonging to V ǫ is open. Since it is obviously closed, and because B ǫ ⊂ V ǫ , weinfer that all kill loc -orbits of U ′ are included in V ǫ .We thus have the inclusion U ′ ⊂ V ǫ . The compactness of U ′ follows because U ′ = π − ( B ǫ ) is closed in V , and V ǫ is compact in V . We conclude that U = U ′ ,and the lemma is proved. . — We consider U a standard foliated neighborhood ofΣ, as given in Lemma 5.8, and π : U → B the projection, whose fibers are exactly the kill loc -orbits in U . For f ∈ Iso(
M, g ), we denote by Λ f the set of points x ∈ U suchthat the kill loc -orbit O ( x ) satisfies f ( O ( x )) = O ( x ). Observe that because of property(3) of Lemma 5.8, if f ( U ) ∩ U = ∅ , then Λ f = ∅ . SOMETRY GROUP OF LORENTZ MANIFOLDS Lemma 5.9 . —
The set Λ f is open and closed in U .Proof . — We first prove that Λ f is closed. For this, let us consider ( x k ) a sequenceof Λ f converging to x ∞ ∈ U . Then f ( x k ) converges to f ( x ∞ ), and f ( x ∞ ) ∈ U . Now π ( f ( x k )) = π ( x k ) for all k , so π ( f ( x ∞ )) = π ( x ∞ ). It follows that x ∞ and f ( x ∞ )belong to the same kill loc -orbit, hence x ∞ ∈ Λ f .To check that Λ f is open, let us pick x ∈ Λ f . By assumption, f ( x ) ∈ U . Because U is open, there exists a small open set V ⊂ U containing x such that f ( V ) ⊂ U .Then, by the third property of Lemma 5.8, V ⊂ Λ f . Lemma 5.10 . —
The stabilizer of U in Iso(
M, g ) has finite index in Iso(
M, g ) .Proof . — Let S denote the stabilizer of U in Iso( M, g ), and let ( f i ) i ∈ I be a familyof elements in Iso( M, g ), such that f i S = f j S whenever i = j . We oberve that f i ( U ) ∩ f j ( U ) = ∅ . Indeed, if f i ( U ) ∩ f j ( U ) = ∅ , then by the remark before Lemma5.9, we would have Λ f − j f i = ∅ . By connectedness of U and Lemma 5.9, this wouldimply Λ f − j f i = U , or in other words f − j f i ∈ S : Contradiction. Because all sets f i ( U ) have a same positive Lorentz volume, there can be only finitely many ( f i ) suchthat f i ( U ) are pairwise disjoint, and we are done.We call in the following Iso ′ ( M, g ) the stabilizer of U in Iso( M, g ). The kill loc -orbitΣ is left invariant by Iso ′ ( M, g ), because of point (3) in Lemma 5.8.
Lemma 5.11 . —
The restriction morphism ρ : Iso ′ ( M, g ) → Iso(Σ , g | Σ ) is one-to-one and proper.Proof . — Properness comes from the fact that Iso( M, g ) acts properly on the bundleof orthonormal frames on M . Thus, because Σ has Lorentz signature, if the 1-jet of ρ ( f k ) remains bounded along a sequence ( x k ) of Σ, then the 1-jet of ( f k ) along ( x k )remains bounded. In particular if ( ρ ( f k )) has compact closure in Iso(Σ , g ), then ( f k )has compact closure in Iso ′ ( M, g ).The fact that ρ is one-to-one comes from the properties of the neighborhood U .Assume indeed that some f ∈ Iso ′ ( M, g ) acts trivially on Σ. We pick x ∈ Σ andlook at the transformation D x f . The tangent space T x M splits as an orthogonal sum T x M = T x Σ ⊕ T x Σ ⊥ . The linear transformation D x f acts trivially on T x Σ. If theaction of D x f on T x Σ ⊥ is nontrivial, then because the exponential map conjugatesthe action of D x f around 0 x and that of f around x , we would get, for a small disc D ⊂ T x Σ ⊥ , two points y and y ′ of exp( x, D ) in the same Iso ′ ( M, g ) orbit, hence inthe same kill loc -orbit because of Lemma 5.8. This is absurd since exp( x, D ) must betransverse to kill loc -orbits if D is small enough. As a conclusion, the map D x f istrivial, hence f is the identity map on M (Lorentz isometries having a trivial 1-jet atone point are trivial). This concludes the proof of Theorem 5.7. CHARLES FRANCES
Theorem 5.1 readily follows from what wehave done so far. By Corollary 3.9 and Proposition 5.4, the presence of G ⊂ Iso(
M, g )with exponential growth yields a compact kill loc -orbit Σ ⊂ M reg having Lorentzsignature. We can then apply Theorem 5.7, which yields a finite index subgroupIso ′ ( M, g ) leaving Σ invariant, and such that the inclusion Iso ′ ( M, g ) → Iso(Σ , g | Σ )is one-to-one and proper. In particular, the group Iso(Σ , g | Σ ) contains a compactlygenerated subgroup of exponential growth. Theorem 4.1 then ensures that the locallyhomogeneous manifold (Σ , g | Σ ) has semisimple isotropy.
6. Geometry of locally homogeneous Lorentz manifolds with semisimpleisotropy
Under the assumptions of Theorem A, we showed in Theorem 5.1 the existence of aone-to-one and proper homomorphism ρ : Iso ′ ( M, g ) → Iso(Σ , h ), where Iso(
M, g ) ′ ⊂ Iso(
M, g ) is a finite index subgroup, and (Σ , h ) is a compact, locally homogeneousLorentz manifold, with semisimple isotropy. This section is thus devoted to the generaldescription of this class of Lorentz manifolds, which, beside being a crucial step inthe proof of Theorem A, is a topic of independent interest.Precisely, the situation we are investigating is that of a compact, connected, locallyhomogeneous, Lorentz manifold (
M, g ). The isotropy algebra at x , denoted i x , is thealgebra of vector fields in g vanishing at x . By local homogeneity, all the algebras i x are pairwise isomorphic, so that we will speak about the isotropy algebra i of ( M, g ).Our standing assumption is that (
M, g ) has semisimple isotropy , which means that i is isomorphic to o (1 , k ) ⊕ o ( m ), k ≥
2. The following proposition will be one of thecrucial steps needed to prove Theorem A. The reader may take it for granted on afirst reading, and go directly to Section 7.
Proposition 6.1 . —
Let ( M, g ) be a compact Lorentz manifold. Assume that M islocally homogeneous, and that its isotropy algebra i is isomorphic to o (1 , k ) ⊕ o ( m ) ,with k ≥ . Assume that Iso(
M, g ) contains a closed, compactly generated subgroupwith exponential growth. Then:1. There exists a simply connected complete homogeneous Riemannian manifold ( N, g N ) , and a smooth function w : N → R ∗ + , such that the universal cover ( ˜ M , ˜ g ) is isometric to the warped product N × w X , where ( X, g X ) is isometricto either the -dimensional anti-de Sitter space g ADS , (in which case k = 2 )or Minkowski space R ,k .2. The isometry group Iso( ˜
M , ˜ g ) is included in Iso( N ) × Homot( X ) . In particular,the manifold ( M, g ) is the quotient of N × X by a discrete subgroup Γ ⊂ Iso( N ) × Homot( X ) . In the proposition Homot( X ) stands for the group of homothetic transformationsof X , namely those ϕ : X → X such that ϕ ∗ g X = λg X , for some nonzero scalar λ . When X = g ADS , , this group coincides with the isometry group. The proof ofProposition 6.1 will be done in two steps. The first one is a geometric description SOMETRY GROUP OF LORENTZ MANIFOLDS of the universal cover ( ˜ M , ˜ g ) (see Proposition 6.2). It will be the aim of Section 6.1,which is pretty close to [ Z2 , Sec. 2 and 3 ].The second step is more difficult, and establishes completeness results, under theassumption that the limit set of Iso( M, g ) is infinite at each point. Those problemswill be tackled in Sections 6.3 and 6.4 (see Theorem 6.7). ˜ M . — Our first aim is to prove
Proposition 6.2 . —
Let ( M, g ) be a compact, locally homogeneous, Lorentz man-ifold with semisimple isotropy. Then the universal cover ( ˜ M , ˜ g ) is isometric to awarped product N × w X , where ( N, g N ) is a simply connected, homogeneous, Rieman-nian manifold, and ( X, g X ) is Lorentzian of constant curvature and dimension ≥ .Moreover Iso( ˜
M , ˜ g ) ⊂ Iso( N ) × Homot( X ) , so that the manifold ( M, g ) is obtained asa quotient of N × w X by a discrete subgroup Γ ⊂ Iso( N ) × Homot( X ) .6.1.1. Bifoliation on the universal cover ˜ M . — We begin with an important remarkabiut Killing fields on ( ˜ M , ˜ g ). The manifold ˜ M is locally homogeneous, hence theintegrability locus ˜ M int coincides with ˜ M . The process of extending analyticallyKilling fields along pathes in ˜ M int (see Section 4.2.3), and the simple connectednessof ˜ M shows that any local Killing field in ˜ M extends to a global one. Thus in thefollowing, all Killing fields on ˜ M will be globally defined (this does of course not meanthat those fields are complete).For any x ∈ ˜ M , the isotropy representation i x → o ( T x ˜ M ) defined by X
7→ ∇ X ( x ) isfaithful. We thus identify i x with a subalgebra of o (1 , n ). Our assumption says that i x contains a subalgebra isomorphic to o (1 , k ), k ≥ k maximal for this property).Thus i x splits, in a unique way, as a sum i x = s x ⊕ m x , where s x is isomorphic to o (1 , k ) and m x is isomorphic to a subalgebra of o ( n − k ). This provides a i x -invariantsplitting T x ˜ M = F x ⊕ F ⊥ x , where F x has Lorentz signature and dimension k + 1,and F ⊥ x is n − k dimensional, orthogonal to F x (and hence of Riemannian signature).The Lie algebra s x (resp. m x ) acts on F x (resp. on F ⊥ x ) by the standard ( k + 1)-dimensional representation of o (1 , k ) (resp. through the standard ( n − k )-dimensionalrepresentation of o ( n − k )) and trivially on F ⊥ x (resp. on F x ).We thus inherits two (mutually orthogonal) distributions F and F ⊥ on ˜ M . Ob-serve that any local isometry sending x to y will map i x to i y . This implies that thedistributions F and F ⊥ are invariant by any local isometry of ˜ M . In particular,those distributions are smooth. Lemma 6.3 . —
The two distributions F and F ⊥ are integrable. Moreover, theleaves of F ⊥ are totally geodesic, and those of F are totally umbilic, with constantsectional curvature.Proof . — Let x ∈ ˜ M . Denote by Z x the subset of ˜ M , where all elements of s x vanish.It is a classical fact that Z x is a totally geodesic submanifold of ˜ M (it is easily checkedusing the fact that exponential map linearizes local flows of isometries). If y ∈ Z x ,then s y = s x , because we clearly have s x ⊂ s y , and those two algebras have same CHARLES FRANCES dimension (namely the dimension of o (1 , k )). It follows that Z x is actually a leaf of F ⊥ , and it proves the assertions about F ⊥ .Let us now check that F is integrable as well. Let us consider Y, Z two local vectorfields tangent to F , and let us call [ Y, Z ] ⊥ (resp. II ( Y, Z )) the component of [
Y, Z ](resp. of ∇ Y Z ) on F ⊥ . One readily checks that those maps are tensorial, namelyfor any pair of functions f and g , then [ f Y, gZ ] ⊥ = f g [ Y, Z ] ⊥ , and II ( f Y, gZ ) = f gII ( Y, Z ). Let us consider x ∈ ˜ M , and a bilinear map b x : F x × F x → R . Wecan write b x ( , ) = g x ( A , ) for some endomorphism A : F x → F x . Let us denoteby I x the isotropy group at x . If b x is I x -invariant, then A must commute with I x ,and because the action of I x on F x is irreducible, it means that A is an homothetictransformation. We have shown that any bilinear form on F x which is I x -invariantis a scalar multiple of g x (restricted to F x ). In particular [ , ] ⊥ x must be zero, whatshows that F is an involutive distribution, hence is integrable. We also get that II x ( , ) = g x ( , ) ν x , for some vector ν x ∈ F ⊥ x , what means precisely that the leaves of F are totally umbilic.Let us conclude the proof of Lemma 6.3 by showing that the leaves of F haveconstant sectional curvature. Let x ∈ ˜ M , and F ( x ) the leaf of F containing x . Forany Z ∈ i x , the local flow D x ϕ tZ acts on the manifold of non degenerate 2-planes of T x M . If P is such a 2-plane, then K ( P ) = K ( D x ϕ tZ ( P )), where K ( P ) stands for thesectional curvature of P (relatively to the curvature tensor of g ). Because i x containsa subalgebra isomorphic to o (1 , k ), one easily gets that K ( P ) is the same for everynondegenerate 2 plane P ⊂ T x M . Now let K ( P ) be the sectional curvature of P ,computed with respect to the metric induced by g on F ( x ). Because F ( x ) is totallyumbilic, we have that(6) K ( P ) = K ( P ) + g x ( ν x , ν x )To check this, let us consider two local vector fields X and Y , tangent to F ( x ),such that g ( X, Y ) = 0 and g ( X, X ) = ǫ = ± g ( Y, Y ) = 1. Let ∇ be theLevi-Civita connection of the restriction g | F ( x ) . Using the property that F ( x ) istotally umbilic, we compute ∇ X ∇ Y X = ∇ X ∇ Y X + g ( X, ∇ Y X ) ν = ∇ X ∇ Y X. But
Y.g ( X, X ) = 0 = 2 g ( ∇ Y X, X ). We get ∇ X ∇ Y X = ∇ X ∇ Y X .Writting g ( X, X ) = ǫ , we also have ∇ Y ∇ X X = ∇ Y ( ∇ X X + ǫν ) = ∇ Y ∇ X X + ǫ ∇ Y ν. This yields ∇ Y ∇ X X = ∇ Y ∇ X X + g ( Y, ∇ X X ) + ǫ ∇ Y ν = ∇ Y ∇ X X + ǫ ∇ Y ν. We thus obtain R ( X, Y, X, Y ) = R ( X, Y, X, Y ) − ǫg ( ∇ Y ν, Y ) . But
Y.g ( ν, Y ) = 0 = g ( ∇ Y ν, Y ) + g ( ν, ∇ Y Y ) . It follows that g ( ∇ Y ν, Y ) = − g ( ν, ∇ Y Y + g ( Y, Y ) ν ) = − g ( ν, ∇ Y Y ) − g ( ν, ν ) = − g ( ν, ν ) . Finally, R ( X, Y, X, Y ) = R ( X, Y, X, Y ) + g ( ν, ν ), which is precisely (6). SOMETRY GROUP OF LORENTZ MANIFOLDS Because K ( P ) is the same for every nondegenerate 2-plane P ⊂ T x M , equation (6)ensures that the same property holds for K ( P ). This remark, together with Schur’slemma, implies that the leaves of F have constant sectional curvature.Actually, under the assumption that ( M, g ) is locally homogeneous, the sectionalcurvature κ ( x ) of the leaf F ( x ) does not depend on x , because we already noticedthat local isometries of ˜ M preserve the distributions F and F ⊥ . ˜ M . — The arguments until Lemma 6.4 belowalready appear in [ Z2 , Section 4.3]. We already noticed that F and F ⊥ are invariantby local isometries. In particular, those two foliations are π ( M )-invariant, and thusinduce two mutually orthogonal foliations F and F ⊥ on M . One can then choose aRiemannian metric on T F , and still put the restriction of g on T F ⊥ in order to builda Riemannian metric h on M , for which F and F ⊥ are still orthogonal. We observethat F ⊥ remains totally geodesic for h . Indeed, the property for F ⊥ to be totallygeodesic is equivalent to F being transversally Riemannian, namely the holonomylocal diffeomorphisms, between small open sets of leaves of F ⊥ are isometries (see[ JW , Prop 1.4]). It was the case for the metric g and it is still the case for h sincethose two metrics coincide in restriction to leaves of F ⊥ . One can then use [ BH1 ,Theorem A] to infer that the two foliations F and F ⊥ define a product structureon ˜ M . More precisely, if we pick z ∈ ˜ M , and call X and N the leaves of F and F ⊥ containing z , then there is a diffeomorphism ψ : N × X such that each factor N × { x } (resp. { n } × X ) is sent to a leaf of F ⊥ (resp. of of F ), with moreover ψ ( N × { x } ) = F and ψ ( { n } × X ) = F ⊥ .Let us now discuss the form of the metric ˜ g on N × X . The metric ˜ g restricts toa Riemannian metric g N on the leaf N , and to a Lorentzian metric g X of constantcurvature κ on the leaf X . Observe that ( N, g N ) is complete, because leaves of F ⊥ are complete for the Riemannian metric h constructed above, and h coincides with g on the leaves of F ⊥ . As already mentioned, the maps ( n, x ) ( n, x ′ ) are isometriesbecause leaves N × { x } are totally geodesic. Moreover, because leaves { p } × X areumbilic, the maps ( n, x ) ( n ′ , x ) are conformal (see [ BH2 , lemma 5.1]). It followsthat there exists a Riemannian metric g N on N , a Lorentz metric g X on X withconstant curvature κ , and a function w : N × X → R ∗ + such that ˜ g = g N ⊕ wg X . Lemma 6.4 . —
The function w : ( n, x ) w ( n, x ) does not depend on x .Proof . — Let us fix n ∈ N , and let us pick x ∈ X . We will denote w n : X → R the function defined by w n ( x ) = w ( n, x ). Let z = ( n, x ), and Z be a vector field in s z . Recall that Z is globally defined. Because the local flow of s z must preserve F and F ⊥ , there exist Z a Killing field on ( N, g N ) and Z a vector field on X , suchthat Z ( p, x ) = ( Z ( p ) , Z ( x )) for every p ∈ N and x ∈ X . We already observed thatbecause Z belongs to s z , it vanishes on N × { x } . It follows that Z = 0. Thus,Because Z is a Killing field on ( ˜ M , ˜ g ), the vector field Z is a Killing field for allthe metrics w p g X , p ∈ N . In particular, it is Killing for both metrics g X and w n g X , CHARLES FRANCES hence w n is left invariant by Z . Now, because the Lie algebra s z acts on T x X by thestandard ( k + 1)-dimensional representation of o (1 , k ) the local punctured lightcone of X at x is a local pseudo-orbit of s z . It follows that D x w n ( u ) = 0 for every u in thelightcone of T x X . This lightcone spans T x X as a vector space, hence D x w n = 0.Because x was chosen arbitrarily in X , the lemma follows. N × w X . — We have established the warped-product structure of( ˜ M , ˜ g ) announced in Proposition 6.2. It remains to show that ( N, g N ) is homogeneous,and that ( M, g ) is obtained as a quotient of N × w X by a discrete subgroup Γ ⊂ Iso( N ) × Homot( X ). This will be obtained thanks to points (2) and (3) of Lemma6.5 below.We will call in the following f Iso(
M, g ) the subgroup of Iso( ˜
M , ˜ g ) obtained as allpossible lifts of isometries in Iso( M, g ). In particular, if Γ ≃ π ( M ) denotes the groupof deck transformations of the covering ˜ M → M , then Γ ⊂ f Iso(
M, g ).Let us consider the group Iso(
N, g N ), acting on C ∞ ( N ) in the following way: Forevery f ∈ Iso(
N, g N ), ( f.v )( n ) := v ( f ( n )) , for every v ∈ C ∞ ( N ) and n ∈ N . We denote by G the stabilizer of the line R w underthis representation. We observe that G is a closed subgroup of Iso( N, g N ), hence aLie subgroup. We call g its Lie algebra (that may be trivial at this stage). The group G admits a continuous homomorphism λ : G → R ∗ + , satisfying w ( g.n ) = λ ( g ) w ( n ) forevery g ∈ G and n ∈ N .We call Homot( X ) the group of homothetic transformations of X , namely diffeo-morphisms ϕ ∈ Diff( X ) for which there exists some real number λ ∈ R ∗ + such that ϕ ∗ g X = λg X . We can now describe more precisely the isometries of the manifold N × w X . The assumptions are still those of Proposition 6.2. Lemma 6.5 . —
1. A transformation ϕ = ( f, h ) ∈ Diff( N ) × Diff( X ) acts isomet-rically on N × w X if and only if f ∈ G and h ∗ g X = λ ( f ) − g X .2. The group G × Homot( X ) contains f Iso(
M, g ) , and in particular Γ ≃ π ( M ) .3. The group G acts transitively on N . In particular ( N, g N ) is a homogeneousRiemannian manifold.Proof . — Let us consider ϕ = ( f, h ) ∈ Diff( N ) × Diff( X ), and let ξ = ( u, v ) ∈ T n N × T x X . We compute that | ξ | = | u | g N + w ( n ) | v | g X while | Dϕ ( ξ ) | = | Df ( u ) | g N + w ( f ( n )) | Dh ( v ) | g X . We get that ϕ will be an isometry if and only if f ∈ Iso(
N, g N ) and | Dh ( v ) | g X = w ( n ) w ( f ( n )) | v | g X for every v ∈ T X . In particular w ( n ) w ( f ( n )) does not depend on n ∈ N , what proves f ∈ G , and | Dh ( v ) | g X = λ ( f ) − | v | g X as claimed in the lemma.Point (2) is a direct consequence of point (1).Let us prove point (3).Let us consider X = ( Y, Z ) a Killing field on N × w X . Recall that X is definedglobally. The same computations as in point (1), for Killing fields, imply that Y mustbe a (global) Killing field of ( N, g N ), satisfying moreover that n D n w ( Y ( n )) w ( n ) is aconstant function. We already observed in Section 6.1.2 that ( N, g N ) is complete. Asa consequence, every Killing field on N must be complete as well. This is a classical SOMETRY GROUP OF LORENTZ MANIFOLDS property, which follows from the obvious fact that Killing fields have constant normalong their integral curves. Thus incomplete integral curves would yield curves offinite Riemannian length, leaving every compact subset of N . This is not possible ona complete manifold.Constancy of the map n D n w ( Y ( n )) w ( n ) implies that the 1-parameter group ϕ tY integrating Y belongs to the group G . In other words, Y ∈ g . By local homogeneityof ˜ M , the evaluation map kill ( ˜ M ) → T ˜ M is onto at each point, what implies that theevaluation map g T N is also onto at each point. Transitivity of the action of G on N follows. Remark 6.6 . —
Observe that the first point of Lemma 6.5 holds for general warpedproducts N × w X , not necessarily the universal cover of a compact locally homogeneousLorentz manifold with semisimple isotropy. A weakness in the description made in Proposition6.2 is that the factor (
X, g X ) might not be a complete , simply connected manifold ofconstant curvature. This is a serious limitation in the full understanding of ( ˜ M , ˜ g ),thus of the manifold ( M, g ). Those completeness issues are quite subtle. When thefactor N in Proposition 6.2 is reduced to a point, then ( X, g X ) is complete by adeep theorem of Y. Carri`ere and B. Klingler (see Theorem 6.8 below). For arbitraryfactors ( N, g N ) (even homogeneous ones), it is not clear how to prove that ( X, g X )is complete, eventhough it is likely to be true (this problem is evocated in [ Z2 , Sec.4.3]. As the following sections show, some serious difficulties were overlooked there).When proving the completeness of ( X, g X ), we will actually need an extra assumptionabout the group Iso( M, g ). To explain this, let us consider the situation of a compactLorentz manifold (
M, g ), obtained as a quotient of a warped product N × w X bya discrete subgroup Γ ⊂ Iso( N ) × Homot( X ). The manifold M is endowed withtwo foliations F and F ⊥ , whose leaves are respectively the projections of the sets { n } × X ans N × { x } . The subgroup of Iso( M, g ) preserving the bifoliation ( F , F ⊥ )is denoted by Iso × ( M, g ). Let us remark that whenever (
M, g ) is locally homogeneouswith semisimple isotropy, and N × w X is the natural warped product structure on( ˜ M , ˜ g ) exhibited in Section 6.1.2, then Iso × ( M, g ) = Iso(
M, g ).We can now state the completeness theorem we will need:
Theorem 6.7 . —
Let ( M, g ) be a compact Lorentz manifold, such that the universalcover ( ˜ M , ˜ g ) is isometric to a warped product N × w X where ( N, g N ) is Riemannianand ( X, g X ) is Lorentzian of dimension ≥ and constant sectional curvature. Assumethat M is obtained as a quotient of N × X by a discrete subgroup Γ ⊂ Iso( N ) × Homot( X ) . Assume moreover that Iso × ( M, g ) has an infinite limit set at every pointof M .Then the factor ( X, g X ) is complete and isometric to either the -dimensional anti-de Sitter space g ADS , or Minkowski space R ,k , k ≥ . Observe that there is no local homogeneity assumption in the previous statement.In the case of a locally homogeneous Lorentz manifold with semisimple isotropy,Theorem 6.7 implies directly Proposition 6.1. Indeed, the growth assumption and CHARLES FRANCES
Corollary 3.9 imply that Iso(
M, g ) has an infinite limit set at each point, and wenoticed the equality Iso × ( M, g ) = Iso(
M, g ) for those manifolds.We will prove the completeness of (
X, g X ) in two quite different ways, accordingto the nature of the group Γ (see Sections 6.3 and 6.4 below). . — In the following, we will call X κ the complete, simply connected, Lorentz manifold of constant curvature κ . Tounderstand what follows, we must recall a few fundamental facts about those spaces,and their geometry.First of all, the model for X is Minkowski space R ,k , namely the space R k +1 endowed with the flat Lorentz metric defined by the quadratic form q ,k ( x ) = − x + x + . . . + x k +1 .Next, to describe X κ when κ >
0, we consider the space R ,k +1 , and the quadric X κ defined by the equation q ,k +1 = √ κ . Inducing q ,k +1 on X κ , one gets a Lorentzmetric of constant sectional curvature κ . This yields the model space X κ . When κ = +1, the usual name for X κ is de Sitter space , denoted DS ,k .When κ <
0, we consider R ,k , namely the space R k +2 endowed with the quadraticform q ,k . The quadric X κ is then defined by the equation q ,k = − √− κ . The restrictionof q ,k to X κ is a complete Lorentz metric of constant sectional curvature κ . However, X κ is not simply connected. To get the model space X κ , one has to consider theuniversal cover ˜ X κ endowed with the lifted metric. We will call π : X κ → X κ thecovering map (this is an infinite cyclic covering). For κ = −
1, the usual name of X κ is anti-de Sitter space , denoted g ADS ,k .One thing we will have to know about the geometry of X κ , for any κ , is the notionof lightlike hyperplane . For Minkowski space, this is the usual notion of an affinehyperplane, on which the Minkowski metric is degenerate.When κ > κ < lightlike hyperplane of X κ as a connectedcomponent of the intersection u ⊥ ∩ X κ , where u ⊂ R ,k +1 (resp. u ⊂ R ,k ) is anyisotropic vector.When κ > X κ = X κ and we have thus defined the notion of a lightlike hyperplaneof X κ .When κ <
0, we define a lightlike hyperplane of X κ as a connected component ofthe lift π − ( H ) ⊂ X κ of any lightlike hyperplane H ⊂ X κ . Let us just mention onepeculiarity of the case κ <
0, that will be used later on (this is detailed in [ Kl , p.368], where the terminology “semi-coisotropic hyperplane” is used). Let us consider H a lightlike hyperplane of X κ (here κ < H ′ on X κ is a lightlike hyperplane of X κ . But the preimage π − ( H ′ ) has infinitely manyconnected component, naturally indexed by Z , and H is just one of them. The secondpoint is that X κ \ π − ( H ′ ) also has infinitely many components. It turns out thatonly two of thosecomponents contain H in their closure, and we will denote them by U + H and U − H . Details about this can be found in [ Kl , p. 369]. ( X, g X ) when Γ is a subgroup of Iso( N ) × Iso( X ) . — Mutiplying if necessary the metric g by a constant, we don’t loose any SOMETRY GROUP OF LORENTZ MANIFOLDS generality if we take the curvature κ of X equal to 0 , +1 or −
1, what we will do fromnow on.The Lorentz manifold (
X, g X ) has constant curvature κ , thus there exists an isomet-ric immersion δ : X → X κ , as well as a holonomy morphism ρ : Iso( X, g X ) → Iso( X κ ),such that δ ◦ f = ρ ( f ) ◦ δ , for every f ∈ Iso(
X, g X ). Our aim is to show that underthe hypotheses of Theorem 6.7, the map δ is an isometry between ( X, g X ) and X κ .The situation is reminiscent of the following celebrated theorem of Y. Carri`ere (in theflat case κ = 0), completed by B. Klingler (for any constant curvature κ ). Theorem 6.8 . — [ Ca ] , [ Kl ] Let ( X, g X ) be a simply connected Lorentz manifold ofconstant sectional curvature κ . Assume that there exists Γ X ⊂ Iso(
X, g X ) a dis-crete subgroup acting properly discontinuously on X with compact quotient, then thedevelopping map δ is an isometry between ( X, g X ) and X κ . In our situation, the projection Γ X of Γ on Iso( X, g X ) does act cocompactly on( X, g X ), since Γ acts cocompactly on N × X . However the group Γ X is not necessarilydiscrete. Reading carefully the proof of B. Klingler in [ Kl ], one realizes that part ofthe arguments actually do not use the discreteness of Γ X , but only the cocompactnessassumption. In particular, the begining of the proof carries over to the case whereΓ X is only assumed to be cocompact, until [ Kl , Proposition 3]. Actually, this propo-sition uses the discreteness of Γ X only at the very end, in order to use cohomologicaldimension arguments. If we just assume cocompactness of Γ X , Klingler’s proof yieldsthe slightly weaker version: Proposition 6.9 . — [ Kl , Proposition 3] Under the asumption that there exists Γ X ⊂ Iso(
X, g X ) which acts cocompactly, then the developping map δ is injective. If δ isnot onto, the boundary of Ω := δ ( X ) in X κ is a disjoint union H ∪ P , where H is alightlike hyperplane, and P is either empty, or a lightlike hyperplane. To conclude the proof of the completeness of (
X, g X ), we have to show that theconclusions of Proposition 6.9 contradict the hypotheses of Theorem 6.7, so that δ will be a diffeomorphism. In [ Kl ], arguments of cohomological dimension, and resultsabout flat affine manifolds are used, that don’t clearly carry over to our situation.We thus adapt them to conclude. κ = ±
1. — We refer to Section 6.2.1 for the notion oflightlike hyperplane of X κ , and the notations therein. Lemma 6.10 . — [ Kl , Sec 4, p. 370] Let H be a lightlike hyperplane in X κ . Let U + H and U − H the associated components. Let G H the stabilizer of U + H and U − H in Iso( X κ ) . Then there exists a complete vector field Y on U + H (resp. on U − H ), which ispreserved by G H , and which have constant nonzero divergence λ ∈ R ∗ with respect tothe restriction of g X κ to U + H (resp. U − H ). Corollary 6.11 . —
Let H be a lightlike hyperplane in X κ , and assume that ρ (Γ X ) ⊂ G H . Then the image δ ( X ) can not contain, nor be contained, in a connected compo-nent U ± H . CHARLES FRANCES
Proof . — Let us call Ω = δ ( X ). We thus have that N × X is identified with N × Ω.We consider the vector field Y given by Lemma 6.10 and we construct the vector field˜ Y on N × U + H by the formula ˜ Y = (0 , Y ) ∈ T N × T X κ . If Ω ⊂ U + H , then ˜ Y induces avector field on ( M, g ) with constant nonzero constant divergence. This is impossible.If U + H ⊂ Ω. Then the quotient Γ \ ( N × U + H ) is an open subset of ( M, g ) (hence hasfinite Lorentzian volume) and Y induces on it a complete vector field with nonzeroconstant divergence. This is again impossible.Corollary 6.11 allows to settle easily the case κ = +1 (actually as in [ Kl ]). Twolightlike hyperplanes must intersect in de Sitter space, so that P = ∅ in Proposition6.9. Then δ ( X ) = U H and Corollary 6.11 yields a contradiction.In the case κ = − P = ∅ in Proposition 6.9, then δ ( X ) contains U + H or U − H , and we get a contradiction by Corollary 6.11. If P = ∅ , we also get acontradiction as follows.First, assume that P is parallel to H . It means that if H ′ and P ′ denote theprojections of H and P on X κ , then H ′ and P ′ are connected components of theintersections u ⊥ ∩ X κ and v ⊥ ∩ X κ with u and v isotropic and orthogonal for the form q ,k .Then either P meets U + H (resp. U − H ) and then δ ( X ) ⊂ U + H (resp. δ ( X ) ⊂ U − H ). Or δ ( X ) does not meet U + H and U − H , in which case it contains U + H or U − H . Whatever thecase, we get a contradiction by Corollary 6.11.If P is not parallel to H , then we call G H,P the subgroup of ^ O(2 , k ) leaving thepair (
H, P ) invariant. The holonomy group ρ (Γ X ) is a subgroup of G H,P . Thehyperplanes H and P project onto two hyperplanes H ′ and P ′ in X κ . The openset Ω = δ ( X ), which is bounded by H and P , projects onto a connected componentΩ ′ of X κ \ { H ′ ∪ P ′ } . By definition of lightlike hyperplanes, there are two lightlikedirections u and v in R ,k such that H ′ = X κ ∩ u ⊥ and P ′ = X κ ∩ v ⊥ . Since P and H are not parallel, u is not orthogonal to v , and calling E the span of u and v , weget a decomposition R ,k = E ⊕ E ⊥ . If we split any x ∈ R ,k as x = x E + x E ⊥ , thefunction λ : x
7→ | x E | is continuous and unbounded on Ω ′ . But then, recalling theprojection π : X κ → X κ , we can define ˜ λ : Ω → R by the formula ˜ λ ( x ) = λ ( π ( x )).This is a continuous function, unbounded on Ω, and moreover G H,P -invariant. Thiscontradicts the fact that ρ (Γ X ) ⊂ G H,P acts cocompactly on Ω.
0. — It remains to show that when the developpingmap is not onto, Proposition 6.9 leads to a contradiction when (
X, g X ) is flat. Thearguments used in Carri`ere’s work are based on former results of W. Goldman and M.Hirsh about flat affine manifolds (see [ GH ]), that we can not use straigthforwardly inour product situation. This is here that we will use for the first time our asumptionthat Iso × ( M, g ) has an infinite limit set at every point (actually here, an infinite limitset at some point would be enough).To this aim, it seems to be useful to isolate the following incompleteness propertyfor Lorentzian (actually affine) manifold.Let (
M, g ) such a manifold. Let u ∈ T M . We denote by γ u the parametrizedgeodesic t exp( tu ). This geodesic has a maximal open interval of definition SOMETRY GROUP OF LORENTZ MANIFOLDS ( T − ( u ) , T + ( u )), with T − ( u ) < < T + ( u ) (and maybe T − ( u ) = −∞ or T + ( u ) =+ ∞ ). We say that the direction u is uniformly incomplete in the future (resp. in thepast) whenever there exist a constant C >
C < V of u in T M , such that for every v ∈ V , T + ( v ) ≤ C (resp. T − ( v ) ≥ C ).We make the following remark: Lemma 6.12 . —
Let ( M, g ) be a compact Lorentz manifold, let x ∈ M , and let Λ( x ) ⊂ P ( T x M ) be the limit set of Iso(
M, g ) at x . If [ u ] ∈ Λ( x ) , then u is neitheruniformly incomplete in the future, nor in the past.Proof . — If [ u ] ∈ Λ( x ), then u ∈ T x M is a lightlike vector which is asymptoticallystable for some sequence ( f i ) in Iso( M, g ). Namely there exists ( x i ) a sequence in M converging to x , and v i ∈ T x i M converging to u such that D x i f i ( v i ) remains bounded.After considering a subsequence, we may assume that y i = f i ( x i ) converges to y .There are also sequences of orthogonal frames ( e ( i )1 , . . . , e ( i ) n ) at x i , and ( ǫ ( i )1 , . . . , ǫ ( i ) n ) at y i , converging in the bundle of frames, so that the differential D x i f i has the followingmatrix in those frames: D x i f i = λ i I k −
00 0 λ i with lim i →∞ λ i = + ∞ . In particular the stability property of u shows that e ( i ) n tends to αu , for some α ∈ R ∗ . Let us call v i = D x i f i ( e ( i ) n ). This is a sequence ofvectors in T y i M which tends to 0. As a consequence, T + ( ± v i ) → + ∞ . Now, because α − e ( i ) n = α − D y i f − i ( v i ), we get T + ( ± α − e ( i ) n ) to + ∞ . Since ± α − e ( i ) n tends to ± u ,this shows that u is neither uniformly incomplete in the future, nor in the past.Now, for open subsets of Minkowski space, one has the obvious lemma: Lemma 6.13 . —
Let Ω ⊂ R ,k be an open subset, such that ∂ Ω contains a lightlikehyperplane H = u ⊥ . Then all directions of T Ω different of u , are uniformly incompleteeither in the future, or in the past. Now our standing assumption is that Iso × ( M, g ) has an infinite limit set at ev-ery point x . Lifting everything to ( ˜ M , ˜ g ), we get a point ˜ x = ( n, z ) ∈ ˜ M and, byLemma 6.12, infinitely many lightlike directions in T ˜ x ˜ M which are neither uniformlyincomplete in the future, nor in the past. On the other hand, assuming for a contra-diction that δ : X → R ,k is not a diffeomorphism, Proposition 6.9 says that ( X, g X )is isometric to an open subset Ω ⊂ R ,k satisfying the hypotheses of Lemma 6.13.In particular, this lemma says that the only directions in T ˜ x ˜ M which are neitheruniformly incomplete in the future, nor in the past, are of the form v + u , where v ∈ T n N is any direction and u ∈ T z X is a specific lightlike direction. Among them,only one, namely 0 + u , is lightlike, and we get a contradiction. CHARLES FRANCES ( X, g X ) is g ADS , or Minkowski space . — Having established thecompleteness of the factor ( X, g X ) under the assumption Γ ⊂ Iso( N ) × Iso( X ), itremains to check that not all spaces X κ are possible for ( X, g X ), as announced inTheorem 6.7. Proposition 6.14 . —
Let ( M, g ) be a compact Lorentzian manifold. Assume that M is a quotient of N × X κ by a discrete subgroup of Iso( N ) × Iso( X κ ) , with thedimension of X κ ≥ . If Iso × ( M, g ) has an infinite limit set at every point of M ,then X κ is either the -dimensional anti-de Sitter space g ADS , , or Minkowski space R ,k .Proof . — Let x ∈ M . If the limit set of Iso × ( M, g ) is infinite at x , then ( M, g )admits infinitely many codimension one, totally geodesic, lightlike foliations whichare transverse at x (see Theorem 3.1). The proposition is then the content of points(1) and (2) of [ Z1 , Theorem 15.1]. The proof can be found in [ Z1 , Sec. 15.1 and15.2] ( X, g X ) in the general case where Γ ⊂ Iso( N ) × Homot( X ) . — We assume here that Γ is not included in Iso( N ) × Iso( X ),since this situation was already handled in Section 6.3. The first observation is thata Lorentz manifold of constant sectional curvature κ = 0 does not admit homothetictransformations which are not isometric. It follows that the factor ( X, g X ) is flat, andthere exists an isometric immersion δ : X → R ,k . It is easily checked that any diffeo-morphism between connected open subsets of R ,k which is homothetic with respectto the Minkowski metric is the restriction of a global homothetic transformation of R ,k . It thus follows that we have a morphism ρ : Homot( X ) → Homot( R ,k ) , such that the following equivariance relation holds: δ ( h.x ) = ρ ( h ) .δ ( x ) , for every h ∈ Homot( X ), and x ∈ X . Our aim is, as in the previous section, to showthat δ : X → R ,k is an isometry.Before doing this, let us make a few reductions. Theorem 5.1 yields a compactLorentz submanifold Σ ⊂ M , which is locally homogeneous with semisimple isotropy,and which is preserved by a finite index subgroup of Iso( M, g ). This last propertyensures that the limit set of Iso(Σ) is infinite at each point of Σ. Moreover, thesubmanifold Σ is actually obtained as a kill loc -orbit (see Section 5.3). In particular, if ˜Σdenotes the lift of Σ to ˜ M , then ˜Σ is a union of leaves { n }× X . Thus ˜Σ = N ′ × X , where N ′ is a submanifold (maybe not connected) of N . As a consequence, the universalcover of Σ is also of the form N × w X , for some simply connected Riemannianmanifold N . It follows that replacing if necessary M by Σ, we may assume in thefollowing that M is locally homogeneous with semisimple isotropy.To make our second reduction, recall the group G that we introduced in Section6.1.3. The group G acts transitively and isometrically on the Riemannian manifold( N, g N ) (Lemma 6.5). The same is true for its identity component G o . We observe SOMETRY GROUP OF LORENTZ MANIFOLDS that G o has finite index in G . Indeed, if K is the stabilizer, in G , of a point n ∈ N ,then the isotropy representation identifies K as a compact subgroup of O( n − k ).In particular K has finitely many connected components, and because G/K = N isconnected, the group G has finitely many connected components too.Let us denote in the following f Iso × ( M, g ) the subgroup of Iso( ˜
M , ˜ g ) comprising alllifts to ˜ M of isometries in Iso × ( M, g ). Since G has finitely many connected compo-nents, there exist finite index subgroups Γ ′ ⊂ Γ, and f Iso × ( M, g ) ′ ⊂ f Iso × ( M, g ) whichare both contained in G o × Homot( X ) (see Lemma 6.5). Let g G be any left-invariantRiemannian metric on G o . Let us denote by π : G o → N = G o /K o the naturalprojection and let ˜ w : G o → R ∗ + be the function defined by ˜ w ( g ) = w ( π ( g )). Then f Iso × ( M, g ) ′ acts isometrically on the warped-product G o × ˜ w X (see Lemma 6.5 point(1)). Let us call M ′ the quotient of G o × X by Γ ′ . This manifold is compact becauseit fibers over M with compact fibers. The group f Iso × ( M, g ) ′ surjects on a finite indexsubgroup Iso × ( M, g ) ′ ⊂ Iso × ( M, g ). The limit set of Iso × ( M, g ) ′ is thus infinite atevery point, and because the fibers of the fibration M ′ → M are Riemannian, thelimit set of f Iso × ( M, g ) ′ / Γ ′ on M ′ is infinite. Since f Iso × ( M, g ) ′ / Γ ′ is a subgroup ofIso × ( M ′ , g ′ ), we conclude that the limit set of Iso × ( M ′ , g ′ ) is also infinite at eachpoint.All those remarks show that we don’t loose any generality if we assume that ˜ M = N × w X is actually the space G o × ˜ w X , and moreover f Iso × ( M, g ) ⊂ G o × Homot( X ).We will make those asumptions from now on.Recall (see Section 6.1.3) that the group G o admits a continuous homomorphism λ : G o → R ∗ + , satisfying ˜ w ( g g ) = λ ( g ) ˜ w ( g ) for every g , g in G o . This homomor-phism is nontrivial. Indeed triviality of λ would mean that ˜ w is constant. Stickingto the notations of Lemma 6.5, ˜ w constant implies that the group G coincides withIso( N, g N ), and λ ( f ) = 1 for every f ∈ Iso(
N, g N ). It follows from Lemma 6.5and Remark 6.6 that Iso × ( ˜ M , ˜ g ) coincides with the product Iso( N, g N ) × Iso(
X, g X ),and we are then in the realm of Section 6.3. A nontrivial λ yields a decomposition G o = AH , where H = Ker λ , and A is a 1-parameter group { a s } satisfying for every g ∈ G o ˜ w ( a s g ) = e αt ˜ w ( g ), with α >
0. Let us denote by ˜ w the value taken by˜ w on the subgroup H . Let g ∈ G o that we write g = a t h , for some h ∈ H . Weget ˜ w ( g ) = e αt ˜ w . We also compute ˜ w ( ga s ) = ˜ w ( a t + s a − s ha s ) = e α ( t + s ) ˜ w , the lastequality holding because H is normalized by A . We end up with the relation(7) ˜ w ( ga s ) = e αs ˜ w ( g ) . The 1-parameter group of transformations ˜ ψ s : ( g, x ) ∈ G o × X ( ga s , x ) com-mutes with the action of G o × Homot( X ). In particular, it commutes with the actionof f Iso(
M, g ) (see point (2) of Lemma 6.5), hence induces a flow ψ s on M which com-mutes with Iso( M, g ). Let us consider z ∈ M a recurrent point (in the future) for theflow ψ s . Such a point exists since M is compact. Let ( g , x ) ∈ G o × X projecting on z . Saying that z is recurrent in the future means that there exist a sequence s i → ∞ ,and a sequence ( α i , β i ) in Γ ⊂ G o × Homot( X ) such that g i = α i .g .a s i tends to g ,and x i = β i .x tends to x . By equation (7), we see that ˜ w ( α i g a s i ) = e αs i λ ( α i ) ˜ w ( g ). CHARLES FRANCES
Since this quantity must converge to ˜ w ( g ) we get that λ ( α i ) ∼ + ∞ e − αs i . By point (1)of Lemma 6.5, β i is an homothetic transformation of X , with dilatation λ i = λ ( α i ) − .In particular, λ i ∼ + ∞ e αs i .Then the holonomy ρ ( β i ) satisfies ρ ( β i ) .δ ( x ) → δ ( x ), and ρ ( β i ) = λ i A i + T i , for( A i ) a sequence in O(1 , k ), and ( T i ) a sequence in R k +1 . Lemma 6.15 . —
The sequence ( A i ) is bounded.Proof . — We consider the foliation ˜ F on G o × X , whose leaves are the sets { g } × X .It induces a foliation F on M , the leaves of which are Lorentzian. For each x ∈ M , wedenote by Λ × ( x ) the limit set of the group Iso × ( M, g ). We observe that if x ∈ M , and[ u ] ∈ Λ × ( x ), then u is tangent to F . Now the flow ψ s maps each leaf of F conformallyto another leaf. In particular, Dψ s preserves the set of lightlike vectors tangent to F .We now use our asumption that Λ × ( x ) is infinite for every x ∈ M . In particular thereare three distinct directions [ u ] , [ u ] , [ u ] belonging to Λ( z ). Associated to thosedirections are three sequences ( f (1) i ) , ( f (2) i ) , ( f (3) i ) going to infinity in Iso( M, g ), suchthat u ⊥ , u ⊥ and u ⊥ coincide with the asymptotically stable distributions (see Section3.2) AS ( f (1) i )( z ), AS ( f (2) i )( z ) and AS ( f (3) i )( z ). Theorem 3.1 yields three Lipschitzfields of lightlike directions x ξ j ( x ), j = 1 , ,
3, such that ξ j ( x ) ⊥ = AS ( f ( j ) i )( x ) forevery x ∈ M , j = 1 , ,
3. Now, we already observed that ψ s commutes with Iso( M, g ).In particular, Dψ s maps AS ( f ( j ) i ) to itself, for j = 1 , ,
3. Because ψ s maps lightlikedirections tangent to F to lightlike directions tangent to F , we moreover infer that Dψ s ( ξ j ) = ξ j . We lift the fields of directions ξ j to Lipschitz fields of directions ˜ ξ j on G o × X . They are tangent to the leaves { g } × X , are Γ-invariant, and also invariantby ˜ ψ s . Projecting ξ j ( g , x ) on the factor { } × T x X ⊂ T ( g ,x ) M , and taking theimages by D ( g ,x ) δ , we get three distinct lightlike directions ξ , ξ and ξ at δ ( x ),such that A i .ξ j → ξ j , for j = 1 , ,
3. Because A i is a sequence in O(1 , k ), this forces A i to stay in a compact subset. Lemma 6.16 . —
Let U and V be open subsets of X on which δ is injective. Assumethat U ∩ V = ∅ , and that δ ( U ) ( δ ( V ) . Then U ( V . We pick U an open subset containing x such that δ restricted to U is injective.We know that for every i large enough β i ( U ) ∩ U = ∅ , and δ is injective on β i ( U )as well, because of the equivariance relation δ ◦ β i = ρ ( β i ) ◦ δ . Now ρ ( β n i )( U ) is anincreasing sequence of open subsets exhausting R ,k , for a suitable subsequence n i . Itfollows from Lemma 6.16 that β n i ( U ) is an increasing sequence of open subsets in X .The union S k ∈ N β n i ( U ) is an open subset Ω ⊂ X , which is mapped diffeomorphicallyby δ onto R ,k . We then must have Ω = X , because otherwise, looking at point on ∂ Ω, we would check that δ could not be injective on Ω. This finishes the proof of thecompleteness of ( X, g X ), and that of Theorem 6.7. SOMETRY GROUP OF LORENTZ MANIFOLDS
7. Proof of Theorem A and conclusion
This section is devoted to the proof of Theorem A. The proof will be done in twosteps. First, we will deal with two particular cases, namely the manifolds (
M, g )which are quotients of N × g ADS , , and those which are locally homogeneous withsemisimple isotropy (see Sections 7.1 and 7.2 below). Thanks to all the work done sofar, and a last important extension result (see Theorem 7.7) we will be able to derivethe general statement from those two particular cases. N × g ADS , . — We recall the notation Iso × introduced in Section 6.2. We are going to prove: Proposition 7.1 . —
Let ( M, g ) be a compact Lorentz manifold. Assume that thereexists N a Riemannian manifold, such that M is a quotient of a warped product N × w g ADS , by a discrete subgroup Γ ⊂ Iso( N ) × Iso( g ADS , ) . Assume that the limitset of Iso × ( M, g ) is infinite at each point. Then Iso(
M, g ) is virtually an extension of PSL(2 , R ) by a compact Lie group. Observe that the hypothesis involves Iso × ( M, g ), but the conclusion is about thefull isometry group Iso(
M, g ).The proof is discussed in [ Z1 , Section 15.2]. The model for 3-dimensional anti-de Sitter space g ADS , is the Lie group ^ PSL(2 , R ) endowed with the left Lorentzianmetric obtained from the Killing form on sl (2 , R ). This metric g AdS turns out to bebi-invariant, and we get an isometric action of the product ^ PSL(2 , R ) × ^ PSL(2 , R )(by left and right translations). The action is not faithful because ^ PSL(2 , R ) hasa nontrivial center Z , so that the isometry group of g ADS , is up to finite index( ^ PSL(2 , R ) × ^ PSL(2 , R )) /Z . As already mentioned, conformal transformations of g ADS , are isometric. It follows from Proposition 6.1 that ( M, g ) is the quotientof N × g ADS , by a discrete subgroup Γ ⊂ Iso( N ) × ^ PSL(2 , R ) × ^ PSL(2 , R ). Theprojection of this group on each factor is denoted by Γ N , Γ L (left) and Γ R (right).The splitting N × g ADS , induces two transverse foliations F ⊥ and F on M ,which are preserved by Iso × ( M, g ). In particular, each direction [ u ] belonging tothe limit set of Iso × ( M, g ) must be tangent to F . This remark, together with thehypothesis that the limit set of Iso × ( M, g ) is infinite yields, by Theorem 3.1, infinitelydistinct codimension 1 lightlike geodesic foliations on M , whose lightlike direction istangent to F . As explained in [ Z1 , Section 15.2], this forces Γ L or Γ R to be trivial.Let say that Γ R is trivial. Then the action of ^ PSL(2 , R ) by right-multiplicationon N × w ^ PSL(2 , R ) induces a nontrivial isometric action of ^ PSL(2 , R ) on ( M, g ).Under these circomstances, the action is not faithfull, but one gets that Iso o ( M, g )is finitely covered by PSL(2 , R ) ( m ) × K , for K a connected compact Lie group (see[ Gr1 ], [
AS1 ], [ Z4 ]). Here PSL(2 , R ) ( m ) denotes the m -fold cover of PSL(2 , R ). Inparticular, Iso o ( M, g ) is a compact extension of PSL(2 , R ). CHARLES FRANCES
The action of PSL(2 , R ) ( m ) on ( M, g ) is locally free (see [
Gr1 , Th 5.4.A]), andits orbits have Lorentz signature. As observed in [ ZP , Corollary 6.2], this impliesthat Iso( M, g ) has finitely many connected components. Proposition 7.1 follows. Weobserve that we are precisely in the first case of Theorem A. ( M, g ) is locally homogeneous with semisimpleisotropy. — Proposition 7.2 . —
Let ( M, g ) be a compact, locally homogeneous, Lorentz mani-fold. Assume that the isotropy algebra i is isomorphic to o (1 , k ) ⊕ o ( m ) , with k ≥ .Assume that the limit set of Iso(
M, g ) is infinite at each point. Then the conclusionsof Theorem A hold. By Proposition 6.1, the universal cover ( ˜
M , ˜ g ) is isometric to a warped product N × w g ADS , , or N × w R ,k , where N is a homogeneous Riemannian manifold. More-over, it follows from point (2) of Proposition 6.1 that Iso( ˜ M , ˜ g ) ⊂ Iso( N ) × Iso( g ADS , )(resp. Iso( ˜ M , ˜ g ) ⊂ Iso( N ) × Homot( R ,k )), and Iso × ( M, g ) = Iso(
M, g ).When ( ˜
M , ˜ g ) is isometric to N × w g ADS , , we can directly apply Proposition 7.1.The group Iso( M, g ) is then virtually a compact extension of PSL(2 , R ), and we arein the first case of Theorem A.We now prove Proposition 7.2 when ( ˜ M , ˜ g ) is isometric to N × w R ,k . ρ : Iso( M, g ) → PO(1 , d ). — The argumentsuntil Section 7.2.3 are essentially discussed in [ Z1 , Section 15.3]. The manifold M is a quotient of N × R ,k by a discrete subgroup Γ ⊂ Iso( N ) × Homot( R ,k ). Wecall Γ N and Γ X the projections of Γ on each factor. As explained above, since thelimit set is infinite at each point, M admits infinitely many codimension one, totallygeodesic lightlike foliations. The lift of each of those foliations to ˜ M is preserved byΓ. Now, codimension one, totally geodesic, lightlike foliations of N × w R ,k are easyto describe (see [ Z1 , Section 15.3]). They are parametrized by lightlike directions[ u ] in R ,k . The leaves of F u determined by such a direction, are products N × H u ,where H u is an affine hyperplane of R ,k , parallel to u ⊥ . Let us call E the span of alllightlike directions u which give rise to a foliation F u , coming from the asymptoticallystable foliation of a sequence ( f k ) in Iso( M, g ) (see Theorem 3.1). Because the limitset Λ( x ) has more than 3 elements at each x ∈ M , the space E ⊂ R ,k is Lorentzof dimension d + 1, with d ≥
2. This yields an orthogonal splitting R ,k = E ⊕ F ,with F Riemannian. Let us call f Iso(
M, g ) the group comprising all lifts to ˜ M ofelements of Iso( M, g ). Then f Iso(
M, g ) preserves the splitting ˜ M = N × E × F ,and f Iso(
M, g ) ⊂ Iso( N ) × Homot( E ) × Homot( F ). The projection on the secondfactor yields a morphism π E : f Iso(
M, g ) → Homot( E ) ≃ ( R + ∗ × O(1 , d )) ⋉ R d +1 .Postcomposing with the natural morphism ( R + ∗ × O(1 , d )) ⋉ R d +1 → PO(1 , d ), weget a homomorphism ˜ ρ : f Iso(
M, g ) → PO(1 , d ). The projection of the group Γ oneach factors will be denoted Γ N , Γ E and Γ F . Because each lightlike direction in Λ( x )corresponds to a Γ E -invariant direction in E , we see that elements of Γ E have a linear SOMETRY GROUP OF LORENTZ MANIFOLDS part acting by similarities: x λx, for λ ∈ R ∗ .In other words, Γ ⊂ Ker ˜ ρ , and we finally get a well-defined morphism ρ : Iso( M, g ) = f Iso(
M, g ) / Γ → PO(1 , d ) . Lemma 7.3 . —
The homomorphism ρ : Iso( M, g ) → PO(1 , d ) is a proper map.Proof . — We want to show that the map ρ is proper, namely the preimage of everybounded sequence is a bounded sequence. The splitting ˜ M = N × E × F is preservedby Γ, hence induces an orthogonal splitting T M = T M N ⊕ T M E ⊕ T M F . For each x , the space T x M E is Lorentz, while T x M N ⊕ T M F is Riemannian. Let ( f k ) bea sequence of Iso( M, g ). The condition ρ ( f k ) bounded is easily seen to imply that Df k | T M E is bounded. But since T M E has Lorentz signature, it shows that the 1-jetof f k is bounded. Because Iso( M, g ) acts properly on the orthonormal bundle ˆ M (seeSection 2.2.1), this implies ( f k ) bounded in Iso( M, g ). Iso o ( M, g ) is compact . — Sticking tothe previous notations, we consider the homomorphism ρ : Iso( M, g ) → PO(1 , d ),and we have shown that it is a proper map. It means that ρ (Iso( M, g )) = H is aclosed subgroup of PO(1 , d ), and H o = ρ (Iso( M, g ) o ) is a compact, normal subgroupof H . In particular, the set of fixed point of the action of H o on H d is nonempty.This set of fixed points Fix( H o ) is a totally geodesic submanifold of H d , isometricto some H d ′ . Because H normalizes H o , it acts on Fix( H o ), yielding a morphism ρ ′ : H → Iso( H d ′ ) ≃ PO(1 , d ), which is proper. The image of ρ ′ is discrete since H o does not act on Fix( H o ). We are then in the second case of Theorem A. Iso o ( M, g ). — It remains to provethe compactness of Iso o ( M, g ). Let us introduce a bit of notations. We call Z thecentralizer of Γ E in Homot( E ). This is an algebraic group, with Lie algebra z . We willdenote by Z L the projection of Z ⊂ Homot( E ) on PO(1 , d ), and Z O the projectionof Z ∩ O(1 , d ) on PO(1 , d ). Lemma 7.4 . —
1. There is an action of Z on M , which is an isometric actionin restriction to Z ∩ Iso( E ) .2. One has the inclusions Z O ⊂ Iso(
M, g ) ,ρ (Iso ( M, g )) ⊂ Z L , and ρ (Iso( M, g )) ⊂ Nor
O(1 ,d ) ( Z L ) . In the statement, Nor
O(1 ,d ) ( Z L ) denotes the normalizer of Z L in O(1 , d ). Proof . — For every h ∈ Z , one defines ˜ h ∈ Iso( N ) × Homot( E ) × Homot( F ) bythe formula ˜ h ( n, x, y ) = ( n, h ( x ) , y ) for every ( n, x, y ) ∈ N × E × F . Obviously, ˜ h centralizes Γ, hence induces a diffeomorphism on M . If moreover h ∈ Z ∩ Iso( E ), CHARLES FRANCES then ˜ h acts isometrically on ( ˜ M , ˜ g ), and the induced action on M is isometric. Thisproves point 1).As for point 2), Z O ⊂ Iso(
M, g ) is a direct consequence of point 1). Every flow f t inIso o ( M, g ) lifts to ˜ f t ∈ Iso( ˜
M , ˜ g ) centralizing Γ. The component ˜ f tE on Homot( E, g )belongs to Z , and the definition of ρ yields ρ ( f t ) ∈ Z L .Finally, every element ˜ f of ^ Iso(
M, g ) normalizes Γ. It follows that the component˜ f E on Homot( E ) normalizes Z , hence the last inclusion ρ (Iso( M, g )) ⊂ Nor
O(1 ,d ) ( Z L ) . We observed in the proof that for every ˜ f ∈ ^ Iso(
M, g ), the component ˜ f E normalizes Z , hence induces an automorphism of z . Since Z centralizes Γ, this automorphism istrivial when ˜ f ∈ Γ. We thus inherits a well-defined representation ζ : Iso( M, g ) → z .The first point of Lemma 7.4 shows that each ξ ∈ z yields a vector field X ξ on M .The very definition of the representation ζ lead to the tautological but useful relation:(8) f ∗ X ξ = X ζ ( f ) ξ We are now in position to prove:
Lemma 7.5 . —
If the group Γ E ⊂ Homot( E ) does not contain only translations,then Iso o ( M, g ) is compact.Proof . — We already observed that elements γ ∈ Γ E are of the form x λ γ x + T γ ,for some λ ∈ R ∗ . We assume, for a contradiction, that some element γ satisfies λ γ = 1. Conjugating everything in Iso( ˜ M ), we may assume T γ = 0. Then, it is clearthat Z ⊂ R × O(1 , d ) ⊂ Homot( E ). Let us recall the group Z O , and its Lie algebra z O . The inclusion Z ⊂ R × O(1 , d ) forces z O to be ζ (Iso( M, g ))-invariant. By the verydefinitions of the representations ζ and ρ , we infer that if ξ ∈ z O , ζ ( f ) .ξ = Ad( ρ ( f )) .ξ. Here Ad is the adjoint representation of PO(1 , d ) on its Lie algebra.The group Z is real algebraic, hence can be decomposed as a semi-direct product( S.T ) ⋉ U , where S is semisimple, T is a torus, and U a unipotent subgroup.We observe that S and U are included in O(1 , d ). We first claim that S is compact.If not, it would contain a subgroup S isomorphic to SO(1 , S would act isometrically on M . But looking carefully at the action of S on E ,we see that its orbits have dimension 0 or 2. The same property should hold for theisometric action of S on M , but we already mentioned (see [ Gr1 , Thm. 5.4.A])that isometric actions of SO(1 ,
2) on compact Lorentz manifolds must be locally free,yielding a contradiction.We now check that U is trivial. First, observe that U is the unipotent radicalof the algebraic group Z ∩ O(1 , d ). Because we already observed that ρ (Iso( M, g ))normalized z O , it must normalize he Lie algebra u .We begin our discussion with the case where U is nontrivial. Observe that then U ⊂ O(1 , d ). The normalizer of u in PO(1 , d ) is a group of the form ( R ∗ + × K ) ⋉ U max ,where K is compact, U max is a maximal unipotent of PO(1 , d ) (isomorphic to R d − ), SOMETRY GROUP OF LORENTZ MANIFOLDS and R ∗ + acts on U max ≃ R d − by homothetic transformations u αu , α >
0. Thegroup ρ (Iso( M, g )) normalizes u , hence ρ (Iso( M, g )) ⊂ ( R ∗ + × K ) ⋉ U max .If actually ρ (Iso( M, g )) ⊂ K ⋉ U max , then every compactly generated subgroupof Iso( M, g ) must have polynomial growth (because ρ is proper), contradicting ourhypothesis. If ρ (Iso( M, g )) K ⋉ U max , we get f ∈ Iso(
M, g ), and ξ ∈ u , such thatAd( ρ ( f k )) ξ → k → + ∞ . On the lightcone through the origin of E , the orbitsof the flow { e tξ } are spacelike (except at the origin). It means that the vector field X ξ (see the discussion after Lemma 7.4) is spacelike on an open subset Ω of M . But D y f k ( X ξ ( y )) → k → + ∞ by the relation (8). When y ∈ Ω, this contradicts thefact that f k are isometries.The previous discussion shows that U is trivial and S is compact. We now lookat the torus T . It may be written as a product T s × T e , where elements of T s are R -split and those of T e are diagonalisable over C , with eigenvalues of modulus 1. Theprojection Z L of Z on PO(1 , d ) is then a product T ′ s × K , where K is compact and T ′ s is trivial, or a 1-dimensional R -split torus in PO(1 , d ). The normalizer of Z L inPO(1 , d ) must normalize T ′ s . Now if T ′ s is nontrivial, its normalizer in PO(1 , d ) is agroup of the form T ′ s × K ′ , where K ′ is compact. The inclusion ρ (Iso( M, g )) ⊂ Nor( Z L )proved in Lemma 7.4 would imply ρ (Iso( M, g )) ⊂ T ′ s × K ′ . Again, this forces everyclosed compactly generated subgroup of Iso( M, g ) to have polynomial (actually linear)growth: Contradiction. We conclude that T ′ s is trivial, hence Z L is compact. But byLemma 7.4, ρ (Iso o ( M, g )) ⊂ Z L . Because ρ is proper, we conclude that Iso o ( M, g ) iscompact.The compactness of Iso o ( M, g ) will follow from Lemma 7.5 and the following
Lemma 7.6 . —
If the group Γ E ⊂ Homot( E ) contains only translations, then Iso o ( M, g ) is compact.Proof . — If Γ E comprises only translations in Homot( E ). Then, all translations ofHomot( E ) commute with Γ E , hence are contained in Z . By Lemma 7.4 this inducesan isometric action of R d +1 on M , which is locally free. Obviously, elements of thisaction stay in Ker ρ , which is a compact Lie group. Let us consider k the Lie algebraof Ker ρ . it splits as a sum a ⊕ m , where a is abelian and m is the Lie algebra of acompact semisimple group. The previous remark shows that a contains R d +1 . It thusintegrates into a torus T ⊂ Iso o ( M, g ), which is normalized by Iso(
M, g ). It is thuscentralized by Iso o ( M, g ). There are timelike translations in Homot( E ), thus thereexists a Killing field Y in a which is everywhere timelike on M , and commutes withIso o ( M, g ). The vector field Y yields a reduction of the bundle ˆ M to a subbundleˆ M ′ with compact structure group. Hence Iso o ( M, g ) preserves the compact subsetˆ M ′ ⊂ ˆ M . Since its action on ˆ M is proper, Iso o ( M, g ) is compact.
We are now ready to proveTheorem A. By hypothesis, (
M, g ) is a ( n + 1)-dimensional compact Lorentz mani-fold, n ≥
2. The group Iso(
M, g ) is assumed to have a closed, compactly generatedsubgroup with exponential growth. By Theorem 5.1, there exists a compact, locally CHARLES FRANCES homogeneous Lorentz submanifold Σ ⊂ M , a finite index subgroup Iso ′ ( M, g ) leav-ing Σ invariant, and a proper homomorphism ρ : Iso ′ ( M, g ) → Iso(Σ , g ). Moreover,still by Theorem 5.1, (Σ , g ) has semisimple isotropy, and Iso(Σ , g ) contains a closed,compactly generated subgroup of exponential growth.We apply Proposition 7.2 to (Σ , g ). Two cases may then occur. In the first case(which corresponds to the second case in Theorem A for the group Iso(Σ , g )), Iso(Σ , g )is virtually a compact extension of a discrete subgroup Λ ⊂ O(1 , d ), 2 ≤ d ≤ n . Theproper homomorphism ρ : Iso ′ ( M, g ) → Iso(Σ , g ) thus shows that Iso(
M, g ) is alsovirtually a compact extension of some subgroup of Λ. We are thus in the second caseof Theorem A for the group Iso(
M, g ).In the second case, there exists an epimorphism ρ ′ : Iso(Σ , g ) → PSL(2 , R ), withcompact kernel, yielding a proper homomorphism ρ ′ ◦ ρ : Iso ′ ( M, g ) → PSL(2 , R ).We are not done, because we don’t know if this homomorphism is onto, and this isthe last difficulty we have to overcome. At this stage, we just get that Iso( M, g ) isvirtually a compact extension of a closed subgroup H of PSL(2 , R ). Moreover, thisclosed subgroup must have exponential growth. Considering a finite index subgroupif necessary, there are only four possibilities.(i) The group H is PSL(2 , R ).(ii) The group H is a nonelementary discrete subgroup of PSL(2 , R ).(iii) The group H is conjugated in PSL(2 , R ) toAff( R ) = (cid:26)(cid:18) λ t λ − (cid:19) | λ ∈ R ∗ , t ∈ R (cid:27) . (iv) There exists λ ∈ R ∗ + \ { } , such that he group H is conjugated in PSL(2 , R ) to Z ⋉ λ R = (cid:26)(cid:18) λ m t λ − m (cid:19) | m ∈ Z , t ∈ R (cid:27) . Observe that those four cases are mutually exclusive.Cases ( i ) and ( ii ) in the list above lead to respectively the first, and the secondcase of Theorem A. Theorem A will thus be proved if we show that cases ( iii ) and ( iv )actually do not occur. This is basically known for case ( iii ). Indeed, it was shown in[ AS2 ] and [ Z4 , Th. 1.1] that if the group Aff( R ) acts isometrically (and faithfully)on a compact Lorentz manifold, then it yields an isometric action of a finite cover ofPSL(2 , R ). Hence if H = Aff( R ) in the list above, it actually implies H = PSL(2 , R ).Our last task is to extend this result to the smaller group Z ⋉ λ R , and this is thecontent of our last statement: Theorem 7.7 (Compare [ AS2 ] , [ Z4 ] ) . — Let ( M, g ) be a compact Lorentz mani-fold. Assume that Iso(
M, g ) contains a closed subgroup G which is a compact extensionof Z ⋉ λ R , λ ∈ R ∗ + \ { } . Then Iso(
M, g ) contains a finite cover of PSL(2 , R ) .Proof . — We consider the following subgroup of PSL(2 , R ): H = (cid:26)(cid:18) λ m t λ − m (cid:19) | m ∈ Z , t ∈ R (cid:27) . SOMETRY GROUP OF LORENTZ MANIFOLDS We will call a := (cid:18) λ λ − (cid:19) , { u t } t ∈ R := (cid:18) t (cid:19) , and F := (cid:18) (cid:19) .Observe that aF a − = λF . Our assumption is the existence of an epimorphismbetween Lie groups ρ : G → H .Let us consider ϕ ∈ G such that ρ ( ϕ ) = a . Let us call U ⊂ G the inverse image ρ − ( { u t } t ∈ R ). The group U is a closed Lie subgroup of G , and its Lie algebra is asum u = R ⊕ k , where k integrates into a compact Lie subgroup of G . The R -factorin this decomposition is mapped onto R F by ρ ∗ . The algebra u is normalized by ϕ ,and with respect to the splitting u = R ⊕ k , the action reads like:ad( ϕ ) = (cid:18) λ B C (cid:19) . We infer the existence of Y ∈ u satisfying Ad( ϕ )( Y ) = λY , and such that ρ ∗ ( Y ) = F .In what follows, we will see Y as a Killing vector field on M , and denote by { Y t } t ∈ R the 1-parameter group it generates. We observe that { Y t } t ∈ R is closed andnoncompact in G . It is clearly noncompact because it is mapped onto { u t } t ∈ R by ρ .Would it not be closed, its closure in G would be a torus, which would be mappedonto { u t } t ∈ R by ρ : Contradiction. From all this discussion, one infers easily that thegroup < ϕ, { Y t } > , generated by ϕ and the 1-parameter group { Y t } , is closed in G ,hence in Iso( M, g ). This group is clearly a compact extension of H . Thus, in thefollowing we will assume G = < ϕ, { Y t } > . Moreover, we will also assume, withoutloss of generality, that 0 < λ < AS2 ] and [ Z4 ]. The relation ϕ ∗ ( Y ) = λY implies that Y is a lightlike vector field, and it is a classical fact that nonzero lightlikeLorentzian Killing fields are nowhere vanishing. Moreover, one sees that log λ is anegative Lyapunov exponent for ϕ . One infers that there is a measurable Oseledecsplitting for ϕ of the form: T M = E + ⊕ E o ⊕ E − . The bundles E + , E o and E − are respectively associated to Lyapunov exponents − log λ , 0 and log λ . The bundles E + and E − are 1-dimensional and lightlike. More-over E − = R Y , and there exists a unique measurable vector field Z such that g ( Z, Y ) = 1 and E + = R Z . One also checks E o = ( E − ⊕ E + ) ⊥ . It is shown in[ AS2 , Lemma 5.1] that the Oseledec splitting extends to an everywhere defined, con-tinuous splitting. Precisely, Z extends to a continuous vector field, and ϕ ∗ Z = λ − Z everywhere. It follows from Zeghib’s theorem 3.1 that Z is actually Lipschitz. ByRademacher’s theorem there exists Ω ⊂ M a subset of full measure, on which Z isdifferentiable. Hence T := [ Y, Z ] makes sense on Ω, and defines there a measurablevector field. Moreover, from the relation D x ϕZ ( x ) = λ − Z ( ϕ ( x )), available for every x ∈ M , we see that Ω is ϕ -invariant and ϕ ∗ T = T . For every s ∈ R ∗ , and every x ∈ M , we define F sx = span { Z ( x ) , ( Y s ) ∗ Z ( x ) , Y ( x ) } . Lemma 7.8 . — [ Z4 , Fact 3.4] For every x ∈ M , the space F sx is -dimensional,Lorentzian, and does not depend on s ∈ R ∗ . CHARLES FRANCES
Proof . — Let us fix s ∈ R ∗ . We first observe that for every x ∈ M , F sx is 3-dimensional. If this is not the case, there is a closed subset F where ( Y s ) ∗ Z belongs tospan { Z, Y } , hence is colinear to Z . The relation < Z, Y > = 1 = < ( Y s ) ∗ Z, ( Y s ) ∗ Y > then shows that for every y ∈ F , we have ( Y s ) ∗ Z ( y ) = Z ( y ). Since ( Y s ) ∗ Y = Y , weget that in the splitting T M = E + ⊕ E o ⊕ E − above F , the differentials DY ms havethe form DY ms = K m
00 0 1 , where K m is a compact subset of O( n − { Y ms } m ∈ Z has compactclosure in Iso( M, g ), but we already checked at the begining of the proof, that thisis not the case. We conclude that F sx is 3-dimensional for every x ∈ M , and thisspace is Lorentzian because it contains two linearly independent lightlike directions Z ( x ) and Y ( x ). Actually, the previous arguments show that ( Y s ) ∗ Z ( x ) , ( Y s ′ ) ∗ Z ( x )and Y ( x ) are linearly independent for every s = s ′ in R ∗ , and x ∈ M .The vector field T = [ Z, Y ] is well-defined, and measurable on the subset of fullmeasure Ω. We next show that for x ∈ M , the space F sx does not depend on s ∈ R ∗ .To check this, one considers, for a given s ∈ R ∗ , V = ( Y s ) ∗ Z − Z + sT . The vectorfield V is defined on Ω. For x ∈ Ω the definition of the Lie bracket yields(9) ( Y t ) ∗ Z ( x ) = Z ( x ) − t [ Y, Z ]( x ) + tǫ ( t ) , with lim t → ǫ ( t ) = 0. One gets for m ∈ Z , and x ∈ Ω, ( ϕ m ) ∗ V ( x ) = λ − m (( Y λ m s ) ∗ Z ( x ) − Z ( x )) + sT ( x ) . By (11), we obtain lim m → + ∞ ( ϕ m ) ∗ V ( x ) = 0 , or in other words(10) lim m → + ∞ D x ϕ m V ( ϕ − m x ) = 0 . Let us denote by µ the measure defined by our Lorentzian metric g , renormalized toensure µ ( M ) = 1. For every k ∈ N ∗ , Lusin’s theorem yields a compact set K k ∈ M ofmeasure at least 1 − k , such that T is continuous on Ω k = Ω ∩ K k . Poincar´e recurrenceimplies that there exists E k ⊂ Ω k a conull set in Ω k such that if x ∈ E k , one can finda sequence ( m i ) going to infinity in N , with ϕ − m i ( x ) belonging to E k for every i , andlim i → + ∞ ϕ − m i ( x ) = x . Because V is continuous on Ω k , the sequence V ( ϕ − m i x ) tendsto V ( x ) = ( Y s ) ∗ Z ( x ) − Z ( x ) + sT ( x ). By (10) and Osseledec splitting properties,we get that V ( x ) is colinear to Y ( x ). It follows that F sx = Span { Z ( x ) , T ( x ) , Y ( x ) } for every x ∈ E k , and every s = 0. In particular, if s, s ′ are in R ∗ , then F sx = F s ′ x for every x ∈ S k ∈ N ∗ E k . Since S k ∈ N ∗ E k has full measure, hence is dense in M ,and because the distributions F s and F s ′ are Lipschitz (each one is spanned by 3Lipschitz vector fields), we conclude that F s = F s ′ everywhere on M .In the sequel, we will write F instead of F s , since there is no dependence in s .This is a 3-dimensional distribution, which is Lipschitz and Lorentzian. Lemma 7.8shows that it is invariant by ϕ and { Y t } t ∈ R , hence G -invariant.We denote by F ⊥ the distribution orthogonal to F . This distribution is Rieman-nian. An important remark is that F ⊥ is tangent to a Riemannian, totally geodesic,transversally Lipschitz foliation. To see this, we observe that the distribution Y ⊥ SOMETRY GROUP OF LORENTZ MANIFOLDS is the asymptotically stable distribution of { ϕ m } m ∈ N (see Section 3.2 for the defini-tion). Zeghib’s theorem 3.1 ensures that Y ⊥ is everywhere tangent to a codimensionone, totally geodesic lightlike foliation F (which is transversally Lipschitz). For thesame reasons, given s ∈ R ∗ , the distributions Z ⊥ and (( Y s ) ∗ Z ) ⊥ are tangent to codi-mension one, totally geodesic lightlike foliation F and F . Thus F ⊥ is tangent to F ∩ F ∩ F , which is Riemannian and totally geodesic. At this stage, we know thatthe leaves of F ⊥ are smooth, but the foliation is only transversally Lipschitz.We are going to show that F is integrable as well. Since this distribution is onlyLipschitz, we will have to use a Frobenius-type theorem for Lipschitz distributions.Such a result was proved in [ Ra ]. Before quoting it, we recall the following definition(see [ Ra , Def. 4.7]). Given a Lipschitz disctribution D , let us consider a Lipschitz localframe field ( X , . . . , X k ) of D . Each Lipschitz field X i is differentiable on some set Ω i .One says that D is involutive almost everywhere when for each ( i, j ) ∈ { , . . . , k } ,[ X i , X j ] belongs to D almost everywhere on Ω i ∩ Ω j . Theorem 7.9 . — [ Ra , Th. 4.11] Any Lipschitz distribution which is involutive al-most everywhere, is everywhere tangent to a transversally Lipschitz foliation, with C , leaves. To prove that F is integrable we are thus going to show: Lemma 7.10 . —
The distribution F is involutive almost everywhere. Moreover, itis of class C .Proof . — To show that the distribution F is involutive almost everywhere, we con-sider U ⊂ M an open subset. If U is small enough, the distribution F | U is spannedby Y , Z , and a third Lipschitz vector field X satisfying g ( X, X ) = 1 and g ( X, Y ) = g ( X, Z ) = 0. Observe that this property almost characterizes X , in the sense thatonly X and − X satisfy those relations. We denote by U ′ the subset of U where X, Y and Z are differentiable. This is a subset of full measure in U , and is contained in Ω.If x ∈ U ′ , then [ Y, Z ] = T and we saw that Z ( x ) , T ( x ) , Y ( x ) span F x . Hence[ Y, Z ]( x ) ∈ F x if x ∈ U ′ .Let us now show that [ X, Y ] ∈ F almost everywhere on U ′ . The vector field [ X, Y ]is measurable on U ′ , hence Lusin’s theorem yields for every k >> K k ⊂ U such that µ ( K k ) ≥ µ ( U ) − k , and [ X, Y ] is continuous on U ′ ∩ K k . Poincar´erecurrence theorem yields E + k ⊂ K k a subset of full measure in K k such that forevery y ∈ E + k , there exists a sequence ( m i ) satisfying ϕ m i ( y ) ∈ K k for all i , andlim i → + ∞ ϕ m i ( y ) = y . Let us now consider x ∈ E + k ∩ U ′ . Let ( m i ) be a sequence asabove, witnessing that x ∈ E + k . The vector fields ( ϕ m i ) ∗ X and ( ϕ m i ) ∗ Y are defined ina small neighborhood of ϕ m i ( x ) contained in U . Here they satisfy ( ϕ m i ) ∗ X = ǫ m i X ,where ǫ m i = ±
1, and ( ϕ m i ) ∗ Y = λ m i Y , what proves that ϕ m i ( x ) ∈ U ′ . Moreover,for every i ∈ N :(11) D x ϕ m i ([ X, Y ]( x )) = [( ϕ m i ) ∗ X, ( ϕ m i ) ∗ Y ]( ϕ m i ( x )) , Equation (11) reads: D x ϕ m i ([ X, Y ]( x )) = ǫ m i λ m i [ X, Y ]( ϕ m i ( x )) . CHARLES FRANCES
Since x ∈ E + k ∩ U ′ , we have that [ X, Y ]( ϕ m i ( x )) tends to [ X, Y ]( x ) as i → + ∞ . Weconclude that D x ϕ m i ([ X, Y ]( x )) tends to 0, so that [ X, Y ]( x ) is colinear to Y ( x ). Wefinally get that [ X, Y ]( x ) ∈ F x for every x ∈ S k ∈ N ( E k ∩ U ′ ), hence [ X, Y ] ∈ F almosteverywhere on U .We proceed in the same way to prove that [ X, Z ] ∈ F almost everywhere on U , what yields involutivity almost everywhere of the distribution F . We conclude,applying Theorem 7.9, that F is tangent to a foliation, whose leaves are of class C , .In particular, F is (tautologically) C in the direction of its leaves. Recall that F ⊥ is integrable as well, with totally geodesic, hence C , leaves. It follows that F is C in the direction of the leaves of F ⊥ . Since T M = F ⊕ F ⊥ , we conclude that F is C . Of course, the same is true for F ⊥ . Lemma 7.11 . —
The leaves of F are totally umbilic, and have constant sectionalcurvature.Proof . — For every x ∈ M , we call s F x the Lie algebra of local Killing fields X around x , satisfying X ( x ) = 0, and such that the 1-parameter group { D x X t } t ∈ R preserves F ⊥ x and acts trivially on it. Observe that if X ∈ s F x , then D x X t preserves the splitting T x M = F x ⊕ F ⊥ x . One expects that generally, s F x = { } , but we claim that this isnot the case. To check this, let us fix a bounded orthonormal frame field ( X , . . . , X n )on M , such that X , X , X (resp. X , . . . , X n ) span F (resp. F ⊥ ). This yields abounded section σ : M → ˆ M , defining a coarse embedding D x : Iso( M, g ) → O(1 , d )(see Section 2.2.2). Actually, because G preserves the splitting F ⊕ F ⊥ , the restrictionof D x to G takes values in a subgroup O(1 , × O( n − ⊂ O(1 , n ). Projecting tothe first factor, one gets for every x ∈ M a coarse embedding D ′ x : G → O(1 , G x := D x ( G ), and denoteby Λ G ( x ) ⊂ ∂ H the limit set of G x . We already observed that for every s ∈ R ∗ , Y ( x ), Z ( x ) and ( Y s ) ∗ Z ( x ) are asymptotically stable directions associated to the sequences( ϕ m ) m ∈ N , ( ϕ − m ) m ∈ N and ( Y s ϕ − m Y − s ) m ∈ N . The interpretation of the limit set asasymptotically stable lightlike directions (see Lemma 3.4) shows that Λ G ( x ) is infinitefor every x ∈ M and d Λ G ( x ) = 3 for every x ∈ M . Let us now choose x in theintegrability locus M int , and consider the generalized curvature map κ g (see Sections4.1 and 4.2.1). The vector κ g ( σ ( x )) is stable under G x . Proposition 4.4 then ensuresthat the stabilizer of κ g ( σ ( x )) inside O(1 , × O( n −
1) contains the factor SO o (1 , s F x contains a subalgebra isomorphic to o (1 , s F x = o (1 , s F x is at most 3-dimensional.Since we showed that the distribution F is of class C , it makes sense to consider,for every x ∈ M , the second fundamental form II x of the leaf F ( x ). Every Z ∈ s F x defines a 1-parameter group { D x Z t } t ∈ R ⊂ O( T x M ), which preserves the splitting F x ⊕ F ⊥ x and preserves II x . When x ∈ M int , the irreducibility of the action of s F x on F x forces, as in Lemma 6.3, the equality II x ( , ) = g x ( , ) ν x , for some vector ν x ∈ F ⊥ x . Because M int is dense in M , such an equality must hold everywhere. Thisshows that the leaves of F are totally umbilic. SOMETRY GROUP OF LORENTZ MANIFOLDS Lemma 7.12 . —
The distributions F and F ⊥ are C ∞ . The leaves of F haveconstant sectional curvature. The universal cover ( ˜ M , ˜ g ) is isometric to a warpedproduct N × w g ADS , or N × w R , , where N is a -connected complete Riemannianmanifold.Proof . — The key point is to show the smoothness of F . For that, we are going toshow that for every x ∈ M , the leaf F ( x ) of F containing x is a C ∞ (injectively)immersed submanifold of M . It will show that F is C ∞ of its leaves. But the leavesof F ⊥ are totally geodesic, hence C ∞ . We will conclude that F is also C ∞ in thedirection of the leaves of F ⊥ , yielding smoothness of F on M .Let us consider F a leaf of F , and let x ∈ F . Let us first remark that Z t -orbits arelightlike geodesics (the parametrization might not be affine). Indeed, Z ⊥ is at everypoint the asymptotically stable distribution of { ϕ − m } m ∈ N , and Theorem 3.1 ensuresthat Z ⊥ is tangent to a totally geodesic, lightlike foliation. In particular, the Z t -orbitof x , t ∈ ( − ǫ, ǫ ) is a piece of lightlike geodesic contained in F . Let y = Z − ǫ/ .x ,and assume ǫ <<
1. Let us choose U ⊂ T y M a small neighborhood of 0 y suchthat exp y is injective on U , and exp y ( U ) contains x . A second important remark isthat because F is totally umbilic, any lightlike geodesic of M which is somewheretangent to F must be contained in F . Thus, if C y ⊂ T y M denotes the lightcone of g y , then exp y ( U ∩ F y ∩ C y ) is included in F , and there exists x ′ ∈ U ∩ F y ∩ C y with exp y ( x ′ ) = x . Now choose V ⊂ U a small open subset containing x ′ , and callΣ := exp y ( V ∩ F y ∩ C y ). This is a piece of C ∞ lightlike surface in F , containing x ,and that we call Σ. Observe that Z ( x ) ∈ T x Σ. Because Y ( x ) is transverse to Z ( x )and lightlike, then Y ( x ) must be transverse to T x Σ. As a consequence, for δ > ψ : ( − δ, δ ) × ( V ∩ F y ∩ C y ) → M defined by ψ ( t, z ) := Y t . exp y ( z ) isa C ∞ immersion, whose image is an open neighborhood of x in F . Smoothness of F follows.We now consider a leaf F of F . Considering a small piece of it, it is an embedded,smooth, submanifold F ′ of M . The restriction of g to F ′ is called g , and its sectionalcurvature denoted by K . We already observed that for x ∈ M int , the Lie algebra s F x is isomorphic to o (1 , K ( x ) is constant (on the Grassmannian of non-degenerate 2-planes) forevery x ∈ M int . Again, density of M int in M show that this is true for every x ∈ M .Schur’s lemma then say that all leaves F have constant curvature.We then follow the same arguments as at the begining of Section 6.1.2 (beforeLemma 6.4), and get that the universal cover ˜ M is a product N × X , where X is a3-dimensional Lorentz manifold of constant sectional curvature, and N is a completeRiemannian manifold. The sets { n } × X (resp. N × { x } ) project on the leaves of F (resp. the leaves of F ⊥ ). The metric ˜ g has the form g N ⊕ wg X for some function w : N × X → R ∗ + . To check that we have a warped product structure, namely that w ( n, x ) does not depend on x , we recall that given t = 0, for all x ∈ M , the directions Z ( x ) , ( Y t ) ∗ Z ( x ) and Y ( x ) span F . Moreover, the distributions Z ⊥ , (( Y t ) ∗ Z ) ⊥ and Y ⊥ are the asymptotically stable distibutions of ( ϕ − m ) m ∈ N , ( Y t ϕ m Y − ) m ∈ N and( varphi m ) m ∈ N hence are tangent to three totally geodesic, lightlike, codimension onefoliations F , F , F , such that F ⊥ = F ∩ F ∩ F . Thus leaves of F ⊥ are included CHARLES FRANCES in leaves of F i for i = 1 , ,
3. We can then apply [ Z2 , Prop. 2.4] and conclude that( ˜ M , ˜ g ) is a warped product N × w X . The manifold ( M, g ) is obtained as a quotientof N × w X by a discrete subgroup Γ ⊂ Iso( N ) × Homot( X ) (see point (1) of Lemma6.5 and Remark 6.6). Theorem 6.7 ensures that X is actually isometric to g ADS , or R , .We are now ready to conclude the proof of Theorem 7.7. If in the previous Lemma,the factor X is isometric to R , , Proposition 7.2 ensures that Iso( M, g ) is virtually acompact extension of a discrete subgroup Λ ⊂ PO(1 , H is the group Z ⋉ λ R .It follows that X is isometric to g ADS , . Then Proposition 7.1 says that Iso( M, g )is virtually an extension of PSL(2 , R ) by a compact Lie group, which is precisely whatwe wanted to show. Remark 7.13 . —
The assumption that G is closed in Theorem 7.7 is crucial. In-deed, there are flat Lorentz tori T , with an isometric action of A = (cid:18) (cid:19) . Theisometry group of such a T clearly contains a (non closed) subgroup isomorphic to Z ⋉ λ R , but no subgroup locally isomorphic to PSL(2 , R ) . We finish thissection with the proof of Corollary C. Our assumption is that ( M n +1 , g ) is a ( n +1)-dimensional, compact, Lorentz manifold, and that Iso( M, g ) contains a discretesubgroup Λ isomorphic to a lattice in a noncompact simple Lie group G .By Theorem A, there exists a finite index subgroup Λ ′ ⊂ Λ, and a morphism ρ : Λ ′ → PO(1 , d ) with discrete image and finite kernel. Since Λ ′ is a lattice too, wewill assume Λ ′ = Λ in what follows.If G is not locally isomorphic to PO(1 , m ) or PU(1 , m ), then Λ has Kazhdan’sproperty (T). The morphism ρ : Λ → PO(1 , d ) provides an action of Λ on H d , and itis known that such an action should have a fixed point, [ BHV , prop 2.6.5], namely ρ (Λ) should be relatively compact. But ρ (Λ) is infinite discrete since ρ is proper:Contradiction.Assume now that G is isomorphic to SU(1 , m ), for m ≥
2. We first rule out thecase when Λ is not uniform. In this case, the thick-thin decomposition ensures that Λcontains a subgroup virtually isomorphic to a lattice in Heisenberg group Heis(2 m − ρ of this subgroup would yield a discrete, nilpotent subgroup of PO(1 , d ).Such subgroups are virtually abelian, hence can not be virtually isomorphic to latticesin Heis(2 m − , m ), we may assume that it is torsion-free(again by replacing Λ by a finite index subgroup). We can then conclude by anargument involving harmonic maps. I thank Pierre Py for pointing this out to me.First, observe that the Zariski closure of ρ (Λ) in PO(1 , d ) is reductive, because ifnot ρ (Λ), which is a discrete group, would be virtually abelian, contradicting that it SOMETRY GROUP OF LORENTZ MANIFOLDS is virtually isomorphic to Λ. Then it follows from [ CT , Corollary 3.7] that ρ (Λ) isvirtually contained in a surface group. The reader may also look at [ CDP , Section3] regarding this point. This forces ρ (Λ) to have asymptotic dimension ≤
2. This isa contradiction since the asymptotic dimension of Λ is 2 m , m ≥
2, and ρ : Λ → ρ (Λ)is proper, hence a coarse embedding (see Lemma 2.7).We have thus proved the first part of the corollary, namely G is locally isomorphicto PO(1 , m ). By Proposition 2.9, we must have m ≤ n .It remains to understand what happens when equality m = n holds. We then goback to Theorem 5.1. There exists a compact Lorentz submanifold Σ ⊂ M which ispreserved by a finite index subgroup Λ ′ ⊂ Λ. Moreover Σ is locally homogeneous,and its isotropy algebra contains a subalgebra o (1 , k ) (with n ≥ k ≥ ′ = Λ in the following. The frame bundleof Σ admits a reduction to O(1 , k ) × L , for some compact group L . Now Corollary2.5 says that Λ coarsely embeds into O(1 , k ) × L , hence into O(1 , k ). By the samearguments as in the proof of proposition 2.9, we have k ≥ n , hence k = n . It followsthat Σ = M , hence M is locally homogeneous, with isotropy algebra o (1 , n ). As aconsequence, M has constant sectional curvature. Since Λ has exponential growth,Proposition 6.1 applies and says that M is either 3-dimensional and of curvature − ≥ M . It is a quotient PSL(2 , R ) ( m ) / Γ, for some uniformlattice Γ.In the flat case, M is the quotient of R ,n by a discrete subgroup Γ of O(1 , n ) ⋉R n +1 .It was shown in Section 7.2 that R n +1 splits as a sum E ⊕ F , with E a lorentziansubspace of dimension d + 1, with Γ acting by ± Id on E . Lemma 7.3 says thatthere is a proper homomorphism from Λ to PO(1 , d ). Again, the same argumentsas in the proof of Proposition 2.9 show that d < n is impossible, hence d = n . Asa consequence, the linear part of Γ is contained in ± Id . This shows that M is aLorentzian flat torus, or a two-fold cover of such a torus. Corollary C is proved. Aknowledgment:
I warmly thank Pierre Py, Romain Tessera and AbdelghaniZeghib for enlightning conversations.
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